NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS

WITH THE TAU METHOD

. by

Peter Onumanyi, B.Sc.(Ibadan),M.Sc.(Imp. Coll.)

October, 1981

A thesis submitted for the degree of doctor of Philosophy

of the University of London.

Mathematics Department

Imperial College

London, S.W. 7.

To

Bgs§ and Jumok§,

with

Love and affection .

UNIVERSITY OF LONDON See over for

Abstract of thesis notes on

completion

a 4.1, f* n \ PETER ONUMANYI Author (full names) .

Title of thesis NUMERICAL^SOLUTION OF BOUNDARY^VALUE

PROBLEMS WITH THE TAU METHOD

Degree..??:?

This thesis concerns the application and extension of Ortiz's recursive formulation of the Tau method of Lanczos to the numerical, solution-of variety of problems in ordinary and some Partial differential equations.

Following a brief sketch of Ortiz's algebraic,algorithmic and analytic theory of the Lanczos's tau method, we apply the method to the numerical solution of initial value problems for nonlinear ordinary differential equations. Numerical solutions using a single and segmented tau approximations are considered and their relative merits discussed.

We then consider the numerical solution of linear and nonlinear boundary value problems, again by means of global and step-by-step tau approximations. The latter requires the use of an implicit matching approach. We also discuss in connection with this problem the use of the technique of improved accuracy at matching points.

As an application of our results on boundary value problems, we discuss its application in the formulation of the Method of lines( MOL ) for elliptic partial differential equations.

Problems with a singular behaviour in the interval in which the solution is required are also considered and consequently a new canonical sequence (Q-log) is introduced, which proves useful in the numerical treatment of logarithmic singularities.

Theoretical results on the convergence of the tau method are not easy to apply in day to day computation. For this reason a practical approach is followed which enables us to provide the user with an estimate of the order of accuracy of the numbers turned out by the computer. This approach is then related to correction techniques for differential equations and to the problem of adaptive c o n t r o l of the segmented process.

A new approach to the generalization of the Lanczos' tau method is based on collocation principle of the perturbed differential equation at equally spaced interior points of the interval. Unlike the Lanczos' method of selected points, this approach allows the explicit determination of the tau parameters simultaneously with the coefficients of the expansion.Moreover,global error estimation is more convenient with this present approach.

ACKNOWLEDGEMENTS

I wish to thank my Supervisor Dr. E« L. Ortiz for his

invaluable help throughout- m-y~ studies, and. with deep respect

I acknowledge his patience, interest and.scholarly guidance

throughout;^-the work reported in this thesis.

My sincere thanks to Mr. Colin Mason, a valuable friend

and colleague whose help in a variety of ways I am

greatly indebted. I am also grateful to my colleagues,

Mr. John Adeyeye and Mr. S . Namasivayam, who read this

thesis and made very useful comments.

During many difficult times, the company and moral support

of my close friends.saw me through to the end of my

studies and among the long list I wish to mention Dr.Muazu

Yisa, Dr. Felix Odaibo, D r . Olu Obafemi and Dr. Olugbenga

Qwajaiye. I owe them more than words can say.

My sincere thanks to Mrs. Christie lisa who helped in the

typing of this thesis.

Finally, I am grateful to the University of Ilorin,Ilorin,

Nigeria, for providing the grant for my studies at

Imperial College, London.

NOTATIONS

0(1)M

£

V

means

N

[a,b]

A

B

(g i, u , u ' , . . .

fu

( v : i )

) J

-1

0,1,2,3,...,M

belongs to

for all

modulus of

0,1,2,3,•«•

approximately equal to

closed interval of a,b

a matrix A

a row vector

transpose of C

functionals representing

the general form of any

initial or boundary

conditions

infinitesimal change

inverse matrix of A

identical

denotes or equivalent to

i \

CONTENTS

CHAPTER 1.

1 . 1

1.2

1.3

1 4

1.5

INTRODUCTION

A few words on the development of the

Lanczos tau method 1

The recursive formulation of the Lanczos

tau method .4

Ortizr

algebraic theory of the tau

method ,6

Algorithmic procedure for the

construction of the tau approximation 10

Existence and error estimation of the

recursive formulation of the tau method 11

CHAPTER 2.

2 .1

2.2

2.3

2 4

INITIAL VALUE PROBLEMS FOR NONLINEAR

ORDINARY DIFFERENTIAL EQUATIONS 15

Method for linear problems 15

Nonlinear problems 20

Method for nonlinear problems 20

Step-by-step tau method 22

CHAPTER 3.

3.1

3.2

3.3

APPLICATION OF THE GLOBAL TAU METHOD TO

NONLINEAR BOUNDARY VALUE PROBLEMS 28

Introduction 28

Initial Approximations 29

Numerical Examples 31

CHAPTER 4.

4.1

4.2

SEGMENTED TAU METHOD FOR BOUNDARY VALUE

PROBLEMS 37

Implicit matching of tau approximations 38

Choice of perturbation and improved

accuracy 43

4-.3 Error estimates and differential correction

of piecewise tau approximations 43

4..4- A minimization problem for adaptive control 4-4-

4-. 5 The Integrated system with Implicit

matching 4-7

4..6 Numerical Examples 50

CHAPTER 5 METHOD OF LINES BASED TAU METHOD FOR

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS 63

5.1 A Direct Formulation 64.

5.2 An iterative approach 69

5.3 Numerical Examples 70

CHAPTER 6 SINGULARITY TREATMENT IN THE RECURSIVE

FORMULATION OF THE TAU METHOD 76'

6.1 Implicit matching 77

6.2 A Logarithm differentiation sequence of

polynomials 81

6.3 Transformation of variables 88

6.4. Numerical Examples 92

CHAPTER 7 A POSTERIORI ERROR ESTIMATION AND A

DIFFERENTIAL CORRECTION IN THE TAU METHOD 95

7.1 Error estimate and a differential

correction 95

7.2 Numerical Examples 99

CHAPTER 8 A C0LL0CATI0N-TAU.METHOD 106

8.1 Introduction 106

8.2 The tau-method: a collocation approach 107

8.3 Error estimation 109

8.4 Canonical polynomials 111.

8.5 Nonlinear Problem 111

8.6 Numerical Results 112

REFERENCES 119'

APPENDIX A"

APPENDIX B

-1-

CHAPTER ONE

INTRODUCTION

1.1 A few words on the development of the Lanczos1

Tau Method

The approximating properties of Chebeyshev polynomials

attracted a lot of interest and rapidly developed in the 1930's.

Some interesting remarks on these polynomials can be frequently

found in major works on Mathematical Physics published in that

decade, such as Courant and Hilbert *£931- and-Sommerfield" 1'935~ .

Perhaps the most interesting paper of the early part of that

decade is Van der Rol-"3*935 where..the properties, graphs and

applications of Chebeyshev polynomials are discussed in the

context of electronics engineering. However, the origin of

these studies can be traced to early works of Lebesgue,

Bernstein, Szasz and Muntz (see Ortiz 1974-).

Towards the middle of the 1930's, C. Lanczos, a co-worker of

A.^Einstein, studied some applications of interpolation and

expansions 'in Chebeyshev polynomials to problems of

Mathematical Physics. He pointed out that the Fourier series

expansion in the solution of practical problems is severely

limited by the fact that the integrals giving the coefficients

of the expansion are in general not adapted to actual evaluation ?

and proposed alternative techniques. One of them, the so called

Tau Method is the concern of this thesis. The method is

related to the principle of economization of a differentiable

function y(x) implicitly defined by a linear differential

equation with polynomial coefficients. It enlarged

considerably the scope of previous methods, as it moved the

domain of applicability of interpolation and economization from

the spirere"6f~ analytic • funcv^€^^-tQv±hat...af'fini.tely . -

d-i-f-f e rent ia bl e fun c t-i on s:

-&ef ined- a— in t erval. .

Th e metho d-

-was- used" 'ex t en-si ve-ly —in- the- construction of ... .•

ma-th em'a t ic al -r

tables~ "ft ±12:er^~l94-6'Lanczos 1952 and machine tables

-f or-au to-mfirt-i-c • c-o mpu"ter-s- CLeushaw__19.62. _

1 ' 1

An attempt to generalize the idea of the tau method to

problems with nonpolynomial coefficients was discussed in

the same paper of Lanczos .1938 and called the technique the

" method of selected points ". The coefficients of the

approximation are obtained by requiring that they satisfy

the given differential equations at some selected points.

This idea is known in the current literature as orthogonal

collocation,whether the selected points for collocation

are chosen to be the zeroes of Chebeyshev,Legendre or

Hermite polynomials. This idea was independently develop-

ed by Frazer,Jones and Skan 1937 in an aeronautics report

published in 1937 by the British Air Ministry.

In 1955, Y. Luke started a series of very advanced

papers (see Luke 1969 ) dealing mainly with the construc-

tion of rational approximations with the tau method.

In 1956 , Lanczos published his celebrated book ,

"APPLIED ANALYSIS", which among many other topics concerns

the tau method. More systematic use of canonical polynomials

and numerical examples were given in that book. The

difficulties presented by the construction of such

polynomials limited the examples of application only to

first order equations with very simple coefficients.

In the following year, Minnick. 1957 -published? tables ~f:or the

• V .

conversion of power series into Chebeyshev series and

Clenshawl957' developed a method of direct replacement of

the Chebeyshev series expansion into the linear differential

equation with polynomial coefficients.

An application of the Clenshawf

s method to the one

dimensional heat equation was reported by Elliott 1961 using

1>he Method of lines as a reduction -technique from a partial

differential * equation "to." a system of ordinary differential

equations.

Fox 1962 discussed Clenshaw and Lanczos methods and he

proposed the use of prior integration with these techniques.

Lanczos however,had earlier suggested the use of the

derivatives of Chebeyshev polynomials as perturbation terms.

The application of Clenshaw's method to nonlinear

ordinary differential equations was reported in Clenshaw and

Norton 1963 and Norton 1964 .

Wright 1962 and 1964 has^ reported some interesting results

in this area.

Mason 1965 solved partial differential equations of

elliptic type and eigenvalue problem on an L - shaped region

using the method of selected points principle. The results

were later developed further in Mason 1969 . The book

of Fox and Parker 1968 discussed extensively various aspects

of the applications of Chebeyshev polynomials in numerical

analysis.

-4-

1.2 THE RECURSIVE FORMULATION OF THE LANCZOS TAU METHOD

In this section- and the rest of this chapter the notations

are chosen to be consistent with those of the references

quoted.

In Ortiz 1964,aJrecursive form, of.construction of the

solution by the tau method in terms of canonical polynomials

was reported for the first time. Ortiz 1968 discussed the

procedure in the framework of graph theory. A theoretical

analysis of the tau method was reported in Ortiz 1969,

providing a general algebraic theory for the method and

the methods of the type. He showed that linear differential

operators with polynomial coefficients are part of a class

is in one - to" - one correspondence with the classes of

.equivalence of sequences of canonical polynomials, for which

the algebraic kernel of the operator is the module of

this equivalence. On the basis of these results Ortiz

showed that the elements of the canonical sequences can be

generated by means of a simple recursive' relation which is

self starting and explicit* He also formulated an algorithmic

procedure for the construction of the tau approximate* • t

solution and gave examples of its application. Further

algebraic. problems',related to. this recursive formulation of

the tau method were discussed by Llorente and Ortiz 1968, and

a technique for solving eigenvalue problems based on the

same algebraic theory was discussed by Chaves and Ortiz 1968.

mapping polynomials into polynomials which

-5-

A Software for the automation of Ortiz1

s formulation of

Lanczosf

tau method was developed in 1969 - 1971 and reported

in the Conference on numerical analysis ,Dublin, 1972

Segmentation techniques for the recursive formulation of thB

tau method were discussed for linear initial value problems,

taking advantage of the fact that an approximate solution

expressed in terms of canonical polynomials can be used as

a master element and mapped over different subintervals

without having toZ - recompute them entirely.

The application to nonlinear initial value problems was

reported in Onumanyi 1978 based on the segmentation

technique (step-by-step)of the -recursive formulation of the

tau method. The application of the recursive formulation of

the tau method to nonlinear boundary value problems and

functional equations was . first reported in Ortiz 1978.

1.3 ORTIZ1

ALGEBRAIC THEORY OF THE TAU METHOD

Let D be a differential operator of order »v •'••belonging to the

class :

D : = 2 Pn-(x) d^ , where p.(x) e ^

i=0 j i i dx

and let ¥ ={ i|»-±-(x) } , ieN = { 0,1,2,...} be a polynomial

ba-sis- , where ^ ( x ) Let J be a compact set of the

real line.

Definition 1.1 :

a

P = Dip- (x) = Zn

a(

n

h (x) n n

r=0 r r

is called the generating polynomial of order n associated

with D ;

h = max (a - n ) nsN

n

is called the height of the operator D .

Let W = { an: = n+h , neN } and Z = N - W .

t Let be the set of finite linear con&inations

Z a P (x) a finite set of indices . neft

n 11

Definition 1.2:

Let S be the set of indices j such that no polynomial (x)

belongs to IP . We call = span {ip-.(x)} the subspace J

of residuals associated with D . Clearly S C Z .

Remark 1.1:

Card(Z) < v+h, since the degree of the coefficients P

as polynomial functions of n is v

* and h . . As a

consequence of this it follows that Card(S) = s is finite

and bounded by v +h .

Definition 1.3: Let = { *n-(-x) >

b e a b a s i s f o r t h e S p a c e

of polynomials. Qn (x) defined by the functional equation

D

* h .f x ) =

. ^n

( x ) + r

n .( x )

• r

n .( x )

4 r ± S c a a o a i c a l

; Po l

yn o B i a l

of order n associated with"D in T. • The index n^ takes into

account the possible multiplicity of $n( x ) and runs in the

set N-S .

Theorem 1.1:

Multiple canonical polynomials Qn (x) differ by an element

of the algebraic kernel U^ of D.

Proof:

Let us assume that Q ^ U ) ^ (x) - Q U ) ^ n , then Q e k 'J ^ ®

Q (x) is a polynomial and DQ which contradicts m m o

the definition 1.2.

Definition 1.4.:

We will call L = ;

{<J£,

n(x)}= {{ \

mU ) } / E }

a Lanczos sequence associated with D; the quotient set

is defined over the set of Qn (x) by the equivalence i

relation E : a(x) - 3(x) iff(a-3)eUD.

Theorem 1.2:

The mapping is bij ective .

Proof:

( i ) Let D^, D ^ e ^ b e both associated with the same

sequence^., for neN-S . Since D^ - D^ = then it

follows that

DiJJn< (x)e v -n^eN-S

which implies D^E as Card(S) is finite.

( ii X Assume a n d n e N - S , are both associated with

then there exists at least onje "index n=m such that

: Q_ fe Q

& (x)e£_ m-,

m

l m

2

'1

but this contradicts theorem 1.1

The next important result is a direct consequence of theore

1.2 and the universal factorization theorem,

see Birkoff and Maclane 1967.

Theorem 1.3

Let v neN-Z ; the elements <^"n(x) of the Lanczos

sequence D L are recursively related by

Z j x ) = _ 1 _ O J x ) - I ( a( n

"h )

i . 0 0 ) J ,

a(n-h)

L n

"h

rsA r r

cl n n

where A = { reN-S : r<n }. r

Proof;

Let meN be such that

a

P (x) = Zm

(x) m

n r

v

r r=u

where a =

n£N—S* and M be the class of equivalence m n

modulo Up of the polynomials of the form

Mn(x) = 4 _

} ( *

n(x) a ^ 4

r( x ) )., Q : - ^ , r

SA

m a

n a

where m=n-h . Clearly DMn(x) = +

r

n(

x

) »

and the result fol-lows.

Corollary 1.1 :

Let Qn(

x

) any realization o f J ^ C x ) , then

Qn(x) = 1 ( * .(*) - 2 a ^ Q (x) ) .. n

^ n - h ) n n

reA r r

an n

-10-

1.4. ALGORITHMIC PROCEDURE FOR THE CONSTRUCTION OF THE

TAU APPROXIMATION

The main task of this algorithm is to identify the

minimal set S, starting from the immediately accessible set Z.

Once S has been identified, after at mostv steps,the

recursive tau approximate solution is constructed by means

of equation 2.1..14 of chapter two,imposing on it the

conditions of the given problem,initial,boundary, or mixed

boundary types. The residual of DU-n(

x

) in

l Rs is set

equal to zero.

The steps of such an algorithm are: Let b.(x) be in J ^

( i ) Set n = order of approximation;generate Q=(Q^(x)},

set G =?, {i : ck £ i+h, ieN } = {g^, ..., g^}

( ii) Let B := Dq , i=l(l)q . s

i

E i t h e r ^ B = 0 store Q (x)-b.(x) in E, or

B £ 0 either p a eZ^(a), or

a £Z =^(b) " °i

(a) reduce Z by one element and include Q- In the list Q

or

(b) Eliminate between Q and Q , replace in B and return

to ( ii) until the list G is exhausted . Then the array 2

has been reduced to S; and Q contains all the

canonical polynomials with indicies jieN-S, j<n, and E i \

contains the elements b.(x) of the algebraic kernel of D.

This is essentially the procedure followed in the design

of Software for the recursive formulation of the tau method.

.( for more details see Ortiz 1969 and.^Ortiz et al 1972 and

for a more compact approach recently proposed by Ortiz and

Samara 1978 and Onumanyi,Ortiz and Samara 1981).

1.5 EXISTENCE AND ERROR ESTIMATION OF THE RECURSIVE

.v FORMULATION OF THE TAU METHOD.

The first attempts of error estimations of the tau method

were given by Lanczos 1938,1956. A more detailed study was

reported in Freilich and Ortiz 1975 who

developed error estimations for function and derivatives

which are directly applicable to systems of ordinary

differential equations. Freilich and Ortiz based their

work on a systematic use of sharp estimates derived for

the Greenes function of the error equation of the Tau

problem. It makes it possible to find upper and lower

errar bounds.

If applied to the harmonic oscillator:

D^(x)s 1

dx d

0 , X(Q) = xe [0,l] ,

and the following notation is used

where .e (x) = ^(x) - and ^n(

x

) is the vector

tau approximate solution, we find that

(n)

-12-

0.08518II(n) < 11 IL < 5 0 2 2 0 ( 1 + O f l /n ) )

2 2 n nl 2 2 n ( n + l ) i

and

0.49468n(n) < II epn^ IL < 1.76971(1- Q ( l /n) )

2 2 n n! 2 2 n ( n + l ) i

where

n ( n ) = ( 1 + 0 ( l / n ) ) $ (n )

and

<f>(n) = k co s ( jr ) ->• 1 ( 1 + 0 (1/n ) ) . 2k' k

2

i n -1

I f we take into account that H(n)~ ( 1 + Q ( l /n ) ) / (n+l ) ,

it follows that the previous result can be reformulated as

(n) = k( 1 + 0 ( l / n ) ) 2 " 2 n / ( n + l ) I

with

0.49468 < k < 1 .76971

This result is d o s e to the one given by the uniform

vectorial best approximation by algebraic polynomials, for

which it takes the value

k = 1 . 0

-13-

Exiatence and convergence result can also be obtained

following an entirely different path,which has been discus-

sed by Crisci and Ortiz 1981 in a recent paper.

Let us consider the first order linear differential

equation

Dy(x):= A(x)y'(x) + B(x)y(x) = f(x)

y(xQ) = y

Q , xe[a,b] ,x

de[a,b]

1.5.1

1.5.2

where A(x),B(x), and f(x) are assumed to be polynomials or

sufficiently close approximations of the given coefficient

functions (see Necas • 1967.:>and Pruess 1973).

Theorem 1.4.

The elements of the sequence of canonical polynomials

associated with D, given by 1.5.1, in the basis

Q={ T*(x) }, neN,.are given by

6

n+h • c T*(x) n

n,+h-l Z h* Q.(x) l/h*

•j=0 J ' J

J n + h

it S

1.5.3

where

hn

=

fn

=

«n , n i + ng r

&

r

P d. I A J

k=0

-k b

k C(n,j, k), C(n,j, k) =

/2k 2k

n-j+ky Yn+j+k

n g; = d . 2 4-r 2

k=0 •

-(k-l) akF(n,j,k), F(n,j,k) = C(r,j,k),

r-0

= i

U c

if t=0 i otherwise , indicates that r takes only

r-0 .

values 0 < r < i such that r+i is an odd number. The

- U -

numbers a^jb^ are the coefficients of A(x),B(x) respectively.

The x-terms in the recursive tau approximate solution of

the given problem 1.5.1,1 • 5.2,-given by equation 2.1.14.

which is expressed in terms of the Q ^ x ) , i. e of 1.5.3,are

fixed by a linear system of algebraic equations. From 1.5.3

therefore, such a system is never singular -which implies the

exis-tence"of. y(x), and which also proves, the existence of a

recursive tau approximate solution for an infinite number

of indicies n .

The convergence of such sequences to y(x) is established

by using different techniques depending on the relative

degrees of A(x) and B(x). If thq degree of A(x) is the

degree of B(x), then from 1.5.3 ,it follows that

I! T ( n ) | | = m a x | 4n )

| = Q(t~—/n! ) i

where n is an algebraic function of n . Otherwise

ii x( n )

n = o( | z r * )

where Z = max | 2(X. + (.X, (A,-l) - 1 | A, eR

K K

k

and R = { Ak: A(A

k) = 0 }, R / [ a , b ]

. These estimates hold for small values of n, for example,

A(x)=l, B(x)=l and n=5 the estimate of the last theorem is

0(-10"^) and the e-xac t:,re:sult-.is - 0 0 -).;, - f o r..- - a g;

a in if

A(x)= 1+10X , B(x)=l , the estimate and the exact result

both give 0(l0~2

).lf we take A(x) = x2

- 4- and B(x)=l with

n=10 the two results give 0(l0~"^).

CHAPTER TWO

INITIAL VALUE PROBLEMS FOR NONLINEAR ORDINARY

DIFFERENTIAL EQUATIONS.

2.1 Method for linear problems.

Let

v L = Z P,(x)d£. 2.1.0

j =0 J

dxJ

be a linear differential operator of order v with

polynomial coefficients P.(x), j = 0(l)v ,where v J

is a. positive integer. We wish to solve a given problem

Lu(x) = f(x) , xe[a,b] 2.1.1

u(xQ) = o

Q 2.1.2

u( ; i )

( xo) = a. j=l(l)v-l 2.1.3

where

cr . , j = 0(l)v-l are the given conditions at one point j

xqe [ a , b j , | a | <0°, | b | < o o

^ (xQ) denotes d

J

'u

dxJ

' x=x

o

and

F f (x) = Z f.x

J

. 2.1.4. j=0

J

-16-

RftWflT»V P.I .1

The left hand side of 2.1.2 and 2.1.3 may also be a

linear Combination of u ^ ' s , j = o(l)v

-l, all given at the

same Point x=x . o

We wish to determine a polynomial approximation u (x) of the N

unique solution u(x) of 2*1*1,. 2.1..2 and 2.1.3, in the form

. .N «> U

M( x ) = Z a* Qy(x) as Z .,a Q (x) =

u(x) 2.1.5 i V

j=0 J J

j=0 J J

and such that it satisfies the given initial conditions

2.1*2 and 2.1-3 exactly,, where Qj (x) is defined as in chapter one.

The basic idea of the Tau method, as originally

conceived by C.. Lanczos 1938 is to add to.equa-tion 2.1.1

a 1

small Perturbation term1

(x

) and solve the following

associated problem

LU^(xJ = f(x) +• ••• 2.1.6

called the perturbed problem with U^ satisfying 2.,1.2 ,-2.1.3.

P^MA-RLR P . 1 _ P

M(N) denotes the degree of the Perturbation term.

It depends o n N a n d on the operator L» M(N) may be

less than, equal to or greater than Without any

ambiguity . we shall henceforth write H^(x) to mean

H

M ( N )( X )

'

If f(x) is not a polynomial, the Tau method may be N

used to approximate f(x) by F,T(x) = Z.-f*x^ an w

j =Q J algebraic Polynomial.

contd.....

-17-

The original Problem is then replaced by .

L UN( X ) = F J Y ( X ) + H

N( X ) , xe[a,b] with defined in

[a,b]- By this two stage approximations of differential

equations, a direct interpolation to f(x) in [a,b] by a

Polynomial is avoided- For a general Purpose Program

it may be necessary to have access to the

coefficients of the basic mathematical functions of the

computer library subroutines as well as values of these

functions for given values of the argument.

The 'exact Polynomial' solution U^ of 2-1.6 satisfying

2-1.2 and 2-1-3 for a particular choice H^(x), clearly,is

not necessarily the 'best' uniform Polynomial approximation

of degree N of the function Even if LU^(x) displays

the equioscillatory behaviour characteristic of best

approximations, this does not imply U^(x) is a 'best'

approximation.

CHOICE OF HN(x):

The closeness of U^ to the 1

best' depends on the choice of

H„(x-) since the error function _ . " N - \ -

eN(x) - V x > . - - u(x) • - • " •• -^,1.7

satisfies the. same equation as u (x) . However, the connec-

tion between H^(x) and the error function is through the

unknown inverse operator L-

^". We will return to this

Problem in chapter seven.

-18-

With the choice

H J J ( X ) = Z T XN

" J ,

j=0 the Power series method is recovered. Comparing the

< 2.1:. 8"

following two choices,

2.1^9

r 2.1.10

we see that 2*1.9 is more economical from the point of

view of computer storage , whereas 2*1*10 is,, in general

closer to zero, as it follows from the expansion of 2.1.9*

The replacement of Tg(x) in 2.1*9 or 2*1.10 by Legendre

polynomials Pj|(x) causes improvements at the endpoint in

the absence of singularities in u(x) f x.e [a, b] , including

singularities at the endpoints a and b.

Let S be an index set such that for jeS,Q.(x) is not

defined. Let t be the number of polynomials u.(x) which

satisfy the homogeneous equation.

where L is the. differential. opera.tor of 2.1.0. Then t free

parameters b. would be available. <j

x. ,j=0(l)r are, -r+1 parameters in the expression .of H„(x)

THE LINEAR SYSTEM OF ALGEBRAIC EQUATIONS

From 2 -1.6, 2.1.2 and 2*.l . 3, the following system

of linear algebraic equations must be solved to determine

the T.'S , j=0(l)r and the b.'s, j=l(l)t.

Lu.(x) = 0 , j=l(l)t , xe[a,b] 2 . 1 * 1 1

to be fixed using 2*1*2 , 2v 1^3 > 2*1.11 and Card(S).

-19-

L e t { V J J ( X ) } NeN

b e a

Po l

yn o m i a l

basis, such that for any N: o

Vjj(x) = I 2.1.12 111=0

and let

H

N( X )

= -l0 • 2 -

1

- " J

Then , according to the definition of the canonical polyno-

mials LQ.(x) = xJ

+ Residual.

J

r

(N i) 1 F

U.T(x) = E T. ( E d

U

^ "j ;

Q (x) + S b.u. + Z f.Q. 0

. .. N

j =0 J m=o m m

3 = l J J 3=0 J J

and because-of-2vl»2;K

and^-2^1*-3»" we-jnusi ;&lso-have that

UN( x

o) = a

Q 2.1.15

U ^)

( xo) = o^ , j=l(l')v-l . 2.1.16

Therefore the number of free parameters T. must be J

r = card(S) + v ' 2.1.17

Thus we have

A T = B , • 2.1.18

an (r+l+t)system which though is generally a full matrix

has the advantage of being of a very small order, as card(S)

is finite and. small (or zero) for the given L in 2.1.0.

It also has. the important advantage, that the size r+l-fc.t

the system 2.1.15 and 2.1.16 is independent of the degree

N of the tau. approximation U^(x). Having determined

j = ( TQ, T^, ..., X^ B^, . . • ' ) from 2.1.18, then

2.1.14- is available explicitly.

-20-

2.2 NONLINEAR PROBLEMS

Consider the. following general class.of nonlinear

ordinary differential equations (Nonlinearity. of polynomial

form) of order v*

v+l m 2 U A « .

k=l j=l ^ ?

k

P , ( x , u , u ' , . . . , u( v

-k + 1

' ) ( u( v

-k + l )

)V i + 1

= f (X) 2 . 2 . 1

where P^. are polynomials or rational functions of their

arguments* Any member of 2*2.1 such as for example,

2 2

2uu" - u' + 4u = 0, v=2,m1=l=m

2=m^ »

P

n= 2 u

>P

2 1 = - u

' '

P3 1= 4 u , U

( 0 )

= u ,

may be linearized directly so that the method of the last

section can be applied to each of the sequence of linear

problems.. Other forms of nonlinearity, for example,

u,f

- e11

= 0 (exponential form), (u+u' )log(u+uf

)+u=0

(logarithmic form),, and xu" + (l+ur 2

)uf

- xu(l+u f 2

) - ^2

= 0

(algebraic form) all require first stage approximation to

reduce them to the form 2.2.1 before linearizing so that

the method of the 1-ast section can-- be applied.

2.3 METHOD FOR NONLINEAR PROBLEMS

Let

G(u,u',.-..,u( v )

) = f(x), xe[a,bj 2.3*1

be a given nonlinear equation of the form 2.2.1 above,

I1

? Pk 1

(X. u , u . , . . .

(u ( ^ ) ) ( u ( — ) A -

3 + 1

=f(

x) 2.3.2

-21-

The generation of {Q,-(x

)},-pM

i s

based on the linearity j J

e i M

o

of L of 2.1.0 and therefore the recursive formulation of

the T-method should be applied to a sequence of linear

problems converging in some sense to 2.3.2.

Let IT,, v(x) be the T-method solutions of each of the sequence « f fx

of linear problems with variable coefficients leading to

2.3.2. The Newton-Raphson-Kantorovich lin-ea-ri-z^tion process

from the Taylor, series:, expatsian?:'in-u',several.: variables--of G,

G + Au. 9G +. Au1

_9G + .. . + A u ^ 9 G . = f(x) , 3u 3u<

3 t iv

for the T-method leads to the use of the following iteration

u & U ( 3 c ) ) ^ T i ? f ( x ) - Gk + V x >

J=0 V 3 uu y

where

GK= G(uk , -U£, . ) •. V 2 . 3 . 5

The -process is repeated until

* . - - ' max. , - u£ . j} <"'Tol , 2.3.6

0<j<N '

K + 1

'«3 K

'J

with

N ... N * ,

j ~y - j i).

Tol is a given convergence tolerance parameter such that

if 2.3 . 6 is not satisfied in K = Nter (a preset number)

iterations then the whole computation will be made to stop.

A system of linear algebraic equations.2.1.18 is solved for

each K . Here, r may no longer be small because card(s)

depends on the degree N of U^

-22-

The initial approximation is usually chosen to be a polynomial

satisfying the given conditions 2.1*2 and 2.1.3. The

linearization process 2.3-3 has a quadratic convergence as

evident in the numerical results- In Ortiz and Pham 1980-

for a particular Ricatti type equation, a proof is given for

its convergence and a quadratic rate is established.

Therefore the iterative process 2-3.4- is characterized by the

following Property

x s f a ^1

^ '5

^ ^ " ^ ' 'k ( X ) | } 1

° x e [ a V| U N

'K ( X )

"U K

'K

-l | 2 }

where C is a constant which depends on the operator 6.

2.4. aTEE-BY-STEP TAtT METHOD'

The application'of ..Ortiz ts - recursive formulation of the Tau

method to the. construction, of Piecewi'se polynomial approxima-

tions to the solution of linear initial value Problems for

ordinary differential equations was reported in Ortiz 1975 and

extended t&. some•initial value problems for. nonlinear ordinary

differential equations, Onumanyi 1978.. Recently& Shimauchi 1979

applied the technique to the • approximation of a Bes-sel-function an

very large intervals a ^linear initial value Problem.

In this section we illustrate with some numerical results the

merits of this technique compared to the .

approximations for nonlinear Problems. Consider a partition

of the interval [a,b] into P (P; >2) subintervals (not

necessarily of equal length).

Ii = Q a ^ b J

p i = 1(1 )p such that

a = a-, <bn = a

0< b

0 = a

Q< ... < a^ < b - = a < b = b 2.4.1

1 1 2 2 3 p-1 p-1 P P

-23-

The initial conditions are assumed given at XQ= a and the

method of the last section is first applied to the subinterval

k J * subinterval [a^ b^ 1 makes use of the explicit

values of the .previous ''subinterval* at b^ as the initial

conditions for the current computation in b^] ..... Since

the condition at the matching points are known explicitly from

the previous subinterval. we refer to this step-by-step

approach as an explicit matching technique. (In chapter four,

an implicit matching technique will be described which is

applicable to. initial value Problems).. In the nonlinear

cases, all iterations may be completed before moving to

the next subinterval. This successive Tau method when compared

with the Global Tau method (single Tau.approximation) gives an

improved accuracy only at the expense of computer time. N is

chosen to be. the same for all subinterval.s, but could be different.

We now illustrate the pl

ower of the technique with three examples

of practical applications.

Example 2.4.1

u" + 2u!

- u - u2

= 0 , xe[0,oo J 2.4.2 . x

u(0) = -4.191691

ut(o)= 0 2.4*4

This singular nonlinear initial value Problem over an infinite

interval is taken from Synge 1961 and is related to Problems

connected with nuclear core structure. The symmetry condition

u' (0)-=Q replaces the. asymptotic condition u' (°°) = 0 . Thus

the original nonlinear Touhdary value problem is solved as an > \

initial value problem in a finite interval

[0,A] , A<co .

-24-

Table 2.4.1^

Method A • 7T C 4. 5) ' 1

u1

(4 • 5)

Synge l96l(h=0.3) 4.5 -0.0392613 0.0474951

Gear method (with treatment of singularity)

9 -0*0392519 0.0475002

Global Tau (N=20) 4.5 -0.0392520 0.0475021

Step-by-step Tau (h=0.3, N=9) 9 -0.0392 519 0.0475022

Table 2 s h o w s that for comparable figures of a c c u r a c y of

the different numerical methods,, global Tau requires degree 20

for A=4»-5 compared to the step-by-step Tau (of degree 9) for

step length h=Q-3 with A = 9* F~or the-:global Tau solution to;

reach A=9"-with compar-abla'-ae-cura-cy . w-iJki^equire -N>;>-201 *

Example 2.4.2

u" + 2u» + u x

u(0) = 1

u'(0)= 0

y _ = 0 , xe[0,l] 2.4.5

2.4.6

2.4.7

The Emdenf

s equation of astrophysics occurs;-in -the- -s-tudy of • stellar

structures, Davis 1962 . For y=5 it has the analytic solution

u(x) = (1 + ^) We compare it with our Tau approximations

for N=7.

Let

max max

X.£[0,1]

1 _<j<NPTS

(|UH(x ) u(x )|}

ll 2.4.8

-25-

where NPTS denotes the number of points of evaluation of the

continuous functions and u(x).

Table 2.1.2

p NPTS

Global Tau 3.89 X 1 0 "7

9

2 1.61-X 1 0 "9

18

5 2.52 X 10"*12

The results show consistent improved accuracy as P increases

(P is the number of equal subintervals) with N=7 in each

subinterval.

Example 2.4. 3

u» - X(l-u2

)u' + u = 0 , xe[0f2Trl, X >0 2.4.9

u(0) = 2.0 2.4.10 u

!

(0)=0 2.4.11

The van der Pol's equation originated in the study of vacuum

tube circuit theory with 'u(anode current) and x(grid voltage).

We wish to determine u and the period T for a given value of X.

Let

T = 2TT(1 + X2

- £x f + 0 ( x 6 ) ) 2.4.12

^ 3072

be taken here as the closed form of T for purpose of comparison

with some numerical methods-. This problem may be solved using

the apriori knowledge that u is periodic and formulate a new

problem with periodic boundary conditions as in Clenshaw 1966

using direct che-beyshev serie-s replacement1

of il or as in.-

Shinohara and.Xamamoto 1978. The approach used here is to apply

the Tau method globally or successively to the given initial

value nonlinear Problem over a sufficiently large interval to

enclose consecutive • (Positive) real zeroes of U^(x), xe[0,oo]«

-26-

Our approach is based in that of the paper of Bailey and

Waltman 1966, on- the distance between consecutive zeroes for

second, order differential equations.. The basic result there

is the estimate (a lower and an upper bound) of the least

distance h>0 for which the given equation has a nontrivial

solution vanishing at a point x and again at X + h . We P P

estimate the consecutive (Positive) real Zeroes of the Tau

polynomial approximations in the interval (or subintervals)

in which they are defined.

Let Z^ Z^ and Z^ be three consecutive (positive) real zeroes

obtained. Then

T = |Z3 - Z±\ 2.4.13

h = semi-period (T) = constant 2.4.14 2

= |Z2 - Z

x| = | Z

3 - Z

2| . 2.4.15

Since in this example (as in most cases) it is known that u

is Periodic, it is sufficient to estimate the first two zeroes,

Z^ Z^. But if u is not known to be periodic, this method

requires estimating several consecutive zeroes of the Tau

approximations to study their distances apart (in particular

for nonconstant period T).

In such a situation*, the approach here becomes more reasonable

than the Boundary value method with periodic boundary conditions.

The results given below were all obtained using N=9 (low

degree approximation).- To ensure convergence of the iteration,

the problem was solved in .£0,2tt3 . For larger interval and

larger values of A , use N much larger than 9 or very small

segmentation steps.

-27-

Table 2 . 1 . A

A=0.0 Semi-period

Global Tau

Step-by-step .Tau (P=4)

Clenshaw 1966

Exact

3.14188329

3.14159265

3.14159265

3.14159265

X=0.1

Global Tau

D . Greenspan 1972

Step-by-step Tau (P=4)

Exact

3.13627549

• 3.144

3.1435536

3.1435556

X=0»5

Global Tau

Clenshaw 1966

Step-by-step Tau (P=2)

iExact

3,-04

; 3-19

3.19

3.19

Results given here for the step-by-step Tau compare favourably

with other numerical methods reported in the literature and

for more accuracy in the case X > 0 i n c r e a s e P=2 to P_>4 and

also increase N to ensure convergence of the iterative

process 2,3.4.

-28-

CHAPTER THREE

APPT.TflA TTON OF THF1 01.0RAT. TAIT MF.THOD TO NONT.TNF.AR ROTTNDARY

VAT.TT-R PRORT.F.MS.

3.1 INTRODUCTION

The Tau method•has the important quality of being able to

solve boundary value problems (BVPS) for a fairly wide range

of complex' associated conditions without any extra effort if

compared with the case of an initial value problem (IVP).

Following Ortiz 1978,this chapter is concerned with the exten-

sion of the recursive formulation of the Tau method to the

nonlinear BVP

Gu(x) = f(x) , xe[a,b] 3.1.1

( g , u , u » , . . . ) = a. , j =0(l)v 3.1.2 j j

where G is a nonlinear differential operator of orderw » and

G belongs to

I Pk.(x,u,u',...,u

v

')(U ) = f ( x )

k=l j —1

with P? . polynomials or rational functions of their arguments,

/ - . N and g.'s are functionals such that (g., u, u

r

, . • u

) J *)

describe the associated conditions of the given problem and

f(x) is a polynomial. This notation gives much flexibility in

refering to^ any form of associated conditions. The method of

section 2.3 for nonlinear problems is used here, provided that

u exists without requiring uniqueness, as we shall show in

connection with a stiff nonlinear bifurcation problem later on

for the given conditions 3.1.2.

-29-

The linearization process

G + Au 9GJ- Auf

3G + . . . + A u ^ = f ( x ) 3 . 1 3u 3u» 3u

is used throughout this chapter and applied to several

numerical examples to illustrate the effectiveness of the

method.

3.2 INITIAL APPROXIMATIONS

From the linearization process 3-l*4-> the iterative scheme

j l ^ l l i M - = f ( x )

. G k +

hn(x) 3.2.1

requires a suitable choice of an initial approximation U^

for convergence and indeed for a rapid convergence. For most

problems U^ Q (x

) is taken to be the simplest polynomial

satisfying-the associated conditions 3.1.2. In certain cases,

it is possible to pick up a good choice U^ Q(X) from the

differential equation itself* a tangent to u(x), xs:£a,b]for a

second order case or a curve with a contact of order v-lin

the case of a differential equation of order v,are good examples.

This can be obtained in cases where u-V has a local maximum or

minimum in [a,b],by drawing a tangent U^ Q(x) to u(x) at such

a point.

Consider the following example

Example 3.2.1 ( see Conte and De Boor 1980 )

2uu!r

- u' + 4u2

= 0- xeTO,7r 1 3.2.2 2

u ( 0 ) = 0 . 0 U ( _ J T ) = 1 3 . 2 . 3

; 2

and choose U^ ^Cx) = 1 for the Newton iterative scheme

2 uku »

k + 1 - 2u-u|

+ 1 + (2u» + 8 u

k)

U k + 1 = ^ 2 _

u, 2

+ 2 u,,

-30-

then

u» 1( x ) + ^ ( x ) = 2 , xe[0, 7r ] 3.2.5

uN > 1

( 0 ) = 0.0 £ ) = 1 3.2.6

It was observed that 3.2*4 converged in one iteration to an

accuracy of 0*13 x l O "1

^ for N=14.- This is because 3.2.5 p

and 3-2.6 have the same closed form solution sin x as

3.2.2 and 3*2.3. The choice U^>p(x) =1 was suggested by the

differential equation 3-2..2, uJ

-'(0)-=- 0. In this example a good

choice U^ Q ( X ) converged after one iteration, but in other

cases, the choice U^' Q(X) may not converge after one itera-

tion but in a smaller-number than for any other choice.

Finally, the choice UJJ Q ( X ) may be crucial as to which

solution the iteration will converge to, when there are multiple

solutions (as in bifurcation problems).

-31-

3.3 NUMERICAL EXAMPLES

Example 3-3.1 (van der. Pol's equation)

u" - l( 1 - u2

)u' + lu = 0 , xs[-l,l] 3.3.1

2 4 u (—1) = 0 u(l) = 1 3.3.2

In table 3-3'*l the results obtained using N=15 is converted to

Ghebeyshev series and compared with Norton 1964 which used

direct Ghebeyshev series replacement as in Clenshaw 1957 with

aQ= ia

Q and U ^ ^ x ) = £(l+x) .

Table 3.3.1

N U (x) = I a T (x)

r=Q

r 1 Q1 0 a

r" Tau Norton 1964

0 +4841575989

1 +5095514886 +5095514886

2 + 172788627 + 172788627 3 - 95925858 - 95925858

U - 14830708 - 14830708

5 + 359122 + 359122

6 + 473599 + 473599

7 + 54566 + 54566

8 - 7368 - 7368

9 - 2786 - 2786

10 - 154 - 154

11 + 70 + 70

12 + 15 + 15

13 0' 0

U 0 1

15 0 0

-32-

Example 3*3.2 (Falkner- %kanT

s Equation)

Ufii +

a u u» u

»2

- 1 ) = 0 ,xe[0,oo] 3.3.3

u(0) = 0 = u'(.0) 3.3.4

u!

( » ) = 1 3.3.5

This problem arises in Falkner and Skan.f

s treatment of the

laminar boundary layer of an incompressible fluid with a,$

given constants.

For xe[0,A] , A< « „ n=20. , IL, n( x ) = xf,

. , u

2A

the Tau approximations to u-(x), uf

(x), u" (x) are given in

Table 3.3.2 I, II, III- The case a=£, 3=0 is the Blasiusf

equation. Table 3.3.2. I a=i 3=0.0 Tol = 10"° in 6 Newton iterations.

X

0.0 0.0 0.0 0.33205741 '

2.0 0.65002448 0.62976584 0.26675153

4.0 2.30574678 0.95551834 0.64233 825E-10

6.0 4.27962149 ; 0.99897287 0.24021947E-02

8.0 6.27921424 : 0.99999645 0.11905391E-04

9.0 7.27921329 1.00000000 ; 0.476~89513E-06

u»(0) = 0.33205741

-6 . Table 3.3.2 II a=l 3=1-0 Tol = 10 in 6 Newton iterations

u ' (x)

0.0 0.0 "0.0 1.232587527

2.0 1.361974127 0.973216879 +0.65826E-01

4.0 3.352109336 0.999958431 +0.16812E-03

6.0 5.352099533 0,999999988 -0.3546E -08

8.0 7.352099515 0.999999977 +0.2925E -06

9.0 8.352099519 1.0 ' -1.0597E -07

u"(0) = 1.232587527.

-33-

Table. III a = 1 3 = 2 > Q T q 1 = 1 Q

- 6 ± n Q N e w t o n

iterations.

X u(x) uf

( x ) u"(x)

0.0 4. 351932E-10 -3.126388E-11 1.68721601

2.0 1. 50518731 0.99144016 0.25948810E -1

4.0 3. 50241415 0.99982449 -0.13452410E -3

6.0 5. 50155310 0.99919837 -0.49660188E -3

8.0 7. 49893157 0.99836578 0.21774996E -3

9.0 8. 49781949 1.00000006 0.38390794E -2

u"(0) = 1.68721601

Results for a=1..0,3=1.6,3=1.8 were also obtained but have not

been reported here. The case a=1*0, 3=2.0 is a very difficult

problem from the point of view of numerical methods, see Aziz 1975.

Example 3.3*3 (with periodic boundary conditions)

2uu» - u'2

+ 4 - U2

= 0 XE£0,TT]

u(0) ~ u ( it) = Q- :

U 1

( 0) - U ' ( M = Q

/ \ . 2 u(xj = s m x.

Using N=10, UN^

0( x ) = 1,

e = max max • = 1.34 X 10

- 6

3':3'.6

3.3.7

3.3.8

3.3.9

m a xr _ u ( x . ) i >

Xj.eLQ^J J

^

for j = 1(1)NPTS.

The choice of U.T n

( x ) =1 is a tangent to u(x) at x = 3L 2

(u'tjr) = 0, U(TT) = 1 ) and with it 2 2

convergence to any tolerance was obtained in one iteration.

-34-

Example 3.3.4 (multiple solutions)

A model singular perturbation problem* Ortiz 1980, Kedem 1981.

eu» + (x2

- u2

)u» = 0 , xeC-1,0] 3.3.10

u(-l) = 0.96 u(0) = 0.001 3.3.11

e = 3.3.12 15

I. % , 0( x ) = 0 - 9 6

I I ..UN[

Q(x) = 0.96 - 0. 959(l+x)

II I Q

(x) = 0.001

(see fig. 3.3.1) max. { |U-n.(x) - u(x)|}* 1 0 "

6

-l<x<0 ± u

Example 3.3.5

u" - eu ( x )

= 0 * xe[0,ll 3.3.13

u( o ) = 0 = u(1) 3.3.14 u(x) = -log 2 + 21og { C sech(G(.2x-l)/4)} 3-3.15

8 6

C = 1.3360557 3.3.16

This equation erises in certain radiation problems and

problems of electrohydrodynamics,Bailey et al 1968,

and Aziz 1975. We consider first

the problem of representing the nonlinear term as a

polynomial in u(x).To this end we solve the linear differential

equation

Z'(x) - Z(x) = 0 , xe[0,l] 3.3.18

Z(0) = 1 3.3.19

by the Tau method.

Let

Z2(x) = 1 + + £

2x

2

^ Z(x) = e

x

be a quadratic Tau approximation, say, then we attempt to

solve

u*"(x)- { I+S

i U*

+£

2 U*

2

} = o , xe[0,lj 3.3.20

u*( 0)=0=u*(l) 3.3.21

-35-

H}

H*

0X5

VJJ

VO

H

-36-

W i t h the hope that the tau approximations of U'*(x)

will be 'close' to.u(x),xe£0,lj .

A p p l y i n g the method for the iterative scheme

< K+l

( x

> " 5

1°N,K+1 ^ . K ^ ^ w t ^ . - I " 5

2^

k( X ) t

-••' ' Hn( X ) 3.3.22

* H . : L( 0 )

= 0

= U

N , 1( 1 )

. 3-3-23

and choosing U

N , 0( x ) =

0-454.8x(X-1) , 3.3.24

gives

e

m a x = x

m

eX

r0, l J

{

'U

N , K + l( x

j > " u

( * j >| }

'f o r

j=l,...NPTS 3.3.

J

e max

= 0.64 X 1 0 ~5

, NPTS = 21 , N=6

in 3 successive iterations .

Zadunaisky 1979- reports an error of 0.5 x 10 -after using a

d i f f e r ential correction t e c h n i q u e .

t \

-37-

CHAPTER FOUR

SEGMENTED TAU METHOD FOR BOUNDARY VALUE PROBLEMS.

This chapter concerns the application of Ortiz' recursive

formulation of Lanczos1

- Tau.method^to the construction of

Piecewise polynomial approximations to the solution of

boundary value problems for ordinary differential equations.

In the recent literature collocation,Galerkin,.

Chebeyshev series replacement and splines have been used to

produce segmented polynomial approximations to the solution

of two point boundary value problems for ordinary

differential equations.. Here we discuss a technique of

implicit matching which makes it possible the solution of

boundary value problems using piecewise Tau approximation

and uses no shooting or related techniques. Boundary

conditions are satisfied exactly, and the same procedure is

applicable to mixed p r o b l e m s . The same ideas are used in

connection with integrated forms of a given differential

equation, the construction of the pieceyise Tau approx-

imation is related to that of the given differential

equation. We also give-a- p r a c t i c a l • loo.k a-t-the minimization

problem related to the optimum.position of the breakpoints

defining the segmentation^ The continuous transformation of

the breakpoints, which allows the maximum error to descend

to its minimum value is empirically treated in terms of a

tau corrector.

-38-

4.1 IMPLICIT MATCHING OF TAU APPROXIMATIONS

Theoretical results given in Hart 1968 and stated in

theorem 4»1*1 below indicates the power of range reduction.

Theorem 4.1.1

Let d£ be the deviation from u(x) of the best approximation

of degree N on an interval with u(x) having N+l

continuous derivatives.

Then

[ < y < K£

n + 1

4.1.1

where

K = u( N + l )

( 0 ) 4.1.2

2N

(N+1)» '

This result says that if £a,bj is halved, the error is

reduced by a factor of about

In the recursive formulation of the Tau method, the sequence {Q.(x)}. « of canonical polynomials is independent of the

J J £ 0

interval on which the solution is sought. Therefore we can

use the same sequence for all subintervals of a given

partition of £a,bj.

Theorem 4.1.2

The sequence {Q^ (x)) i s

independent of the interval

£a,bj on which u(x) is sought.

Proof:

LxJ

' = £ P (x) (h(j,k))x; 5

"k

4.1.3 k=

0

k

h(j,0) = 1, h(j,l) = j, h(j,2) = j(j-l),...,

h(j,k) = j(j-l)...(j-k+l) 4.1.4

Therefore n

k LxJ = I (h(j,k))( Z P

m kx

m + j

"k

) 4.1.5 k=0- m=0

n

k = Z p . x

m

, P , are constants. 4.1.6 m=0

m

-39-

L e t

n = max ( nv) 1.6

0<k<v K

and if n > j

then

n = j + H 4.1.7

w h e r e H is a n o n n e g a t i v e i n t e g e r .

Call H the height of L a n d w r i t e

L*J = T 1 a M x " + , 4 . 1 . 8 m n ra=0

a

F r o m the d e f i n i t i o n L Q . ( x ) = xJ

4.1.10 j

i + H - 1

i p r y L( *3

- z" ai^QjtfC*)) = L Q

n( x )

• A.1.11 a. ^ i = 0 n J

u

T h e r e f o r e , from the l i n e a r i t y of L

j + H - 1 Q .

+ H( x ) = l ( - I (x).) 4.1.12

J+ H

" 0 7 m=0 m m

j+H w h i c h gives explicitly the sequence{ Q . ( x ) } .

M w i t h o u t a n y

0 £

o consideration to [a,.bl . In a similar fashion we find the residuals if n < j .

R e m a r k 4.1.1

The above p r o o f also is a n existence proof for (Q.(x)}. „ . J j e N

Q

Consider a p a r t i t i o n of the interval [a,b]into P subintervals

(P 2) Ii + 1

, i= 0 ( l ) p - l ( n o t n e c e s s a r i l y of equal l e n g t h s )

such that u is a p p r o x i m a t e d by a function

U0 > N

( x ) x e l1 = [ a

1, b

1]

Ul f N

( x ) x e l2 = [ a

2, b

2] ^ = U

N( x )

4.1.13

U _ M

( x ) xel = £a ,b J p - 1 , N P P P

w i t h each IK a

p o l y n o m i a l of degree N of the form

TT / N N

0 0

i,N = Z . a * Q . , ( x ) . * a . . Q . . ( x ) , 4.1.14 j=0 J

1

" j =0 1 J J 1

i' = 0 (l)p-l, x e [ a1 + 1

» bi + 1

]

-40-

and

a = a1< b

1= a

2< . . . < a

p - 1< b

p - 1= a

p< b

p= b 4.1:15

P > 2 is an integer.

We attempt to solve the. following P perturbed problems

L Ui > N

( x ) = f(x) + Hi > N

( x ) , x e [ ai + 1

, bi + 1

] , i = 0 ( l ) p - l 4.1.16

( S y Ui f N t

U lf N

, . ^tU

(

i^

l )

) = a. , j =0(l)v-1 for some i- 4-1.17

U

i - l , N( b

i)

" U

i ^a

i + 1} = 0

' is

0 ( l ) v - l , i=l(l)p-l 4.1.18

where 4.1.18 are the matching equations of the Tau

approximations at the interior matching points

b

i = a

i + l ' i=i(l)p-l 4.1.19

of[a,bJ. These are based on the continuity of the solution

of the given p r o b l e m , u e GV

[a, b] . Other constraints

appropriate to a particular case may be imposed instead, for

example, symmetry, periodicity or jump discontinuity and so

on. The solution of 4.1.16 - 4*1.18 proceeds as follows:

Let

H

i , N( x ) =

J0

T

i jV

N - j , i( x )

' X £

ta

i+r

b

i + J 4.1.20

i = 0(l)p-l

r = card(S)+v-l 4-1.21

and v,T . .(x) is a shifted polynomial basis appropriate to

[ ai + 1

. bi + 1

l •

T.., i=0(l)p-l ,j =0(1)r are J

(r+l)P free parameters to be fixed with 4.1.17 and 4.1.18.

From theorem 4.1.2, we can write N

U

i , N( x ) =

* a

i jQ

j( x ) S

- f0

a i

jQ

J( X )

' ^ ^ i + l ^ i + J

Therefore, if

V

W <(*> = 2 D

IN

,-X I

° 1.23 f I

'1

m=0 m

'1

-41-

then

H . 1vr(x)= Z T . . N zJ

D ( n : j)

XR A

1

»N

3=0 m=0 m

'1

4.1.24

So

r N-j /„ . \ F U . N(x) = Z T

± 1 Z Q (x) + Z f,Q,(x) 4.1.25

j=0 m=0 m , : L 111

j=0 J J

Using the equations 4.1.1V, 4.1.18,we solve the following

enlarged system of (r+l)P linear algebraic equations to

fix the (r+l)P free parameters x...

The resulting (r+l)P system is of block structure.

We give a few illustrations for case card(S)=0:

v = 2 BVP Bandwidth = 3(r+l)-l = 5 for r=v-l

1

-x

V = = 3 BVP Bandwidth r * *

* * * * * * * * *

* * * * * * * * *

* * * *

Bandwidth = 3(r+l)-l = 8 for r=v-l

* *

* * -X

-x * - I M *

*

-42-

B

o

" B

1

B. l

B P - 1

B P

V

i=2(l)p-2

V t = B 4.1.26

The NAG library subroutines F01BMA and F04AUF for real band

matrix v, Bandwidth = 3 ( r + l ) - l ,

r =card(S)+v-l ,

were used in all the computations of this chapter with

computer time proportional to (P x Bandwith x r+1 ). A

subroutine for almost block tridiagonal arising in 'colloca-

tion with splines' is reported in De Boor 1978^

which is more efficient f.rom the computational, point of view

of solving 4.1.26. There is much saving in storage require-

ments and in computer t i m e , but accuracy is the same.

Theorem 4.1.3

Only one matrix and one sequence Q of canonical polynomials

are required to generate an N-th order Tau p i e c e w i s e

approximation to u(x), given either in differential

or integrated form.

-43-

4.2 CHOICE OF PERTURBATION AND IMPROVED ACCURACY

In 4*1.20, choosing Vw . (x) jn, I

as a shifted Legendre p o l y n o m i a l basis appropriate

to [ a

i + l ' ^ i + l ^ P^-a c e

Chebeyshev polynomial always

gives an improved accuracy of the piecewise Tau approxima-

tions except in the p r e s e n c e of singularities in [a,bj.

This fact was pointed out in Lanczos 1956. The use of the

endpoint improved accuracy for further computation with the

Tau method was reported in Ortiz 1975 and showed some improved

accuracy in the. Tau approximations over Chebeyshev perturba-

tions. Here, several n u m e r i c a l examples confirm agreement

of the boundary value proble ms with the implicit matching

technique of the last s e c t i o n .

4.3 ERROR ESTIMATES AND DIFFERENTIAL CORRECTION

OF PIECEWISE TAU APPROXIMATIONS

W e extend results of chapter seven on global error

estimations of the single Tau approximation to piecewise

Tau approximations.

Let r U

Q > N( x ) - u(x) , x s l

1 = [a^b-j]

* n( x ) e

j : ! ) 4.3.1

U M( x ) - u ( x ) , x

eI = fa , b l

i-l,N v

' p L

P PJ

§i,N( x )

= U

i , N( x )

" u ( x )

' x e

Ca

i + i 'b

i + i 3 i=0(l)p-l 4.3.2

Apply the analysis of chapter seven to give the following

P perturbed error equations N M

e, N M

( x ) z

(a* -a..)Q (x) - Z

a Q (x) ,M>N 4.3.3 j

=q

1

J J j=N+l J J

L e

i , H , M( x ) = H

i , N( x )

" S

i , M( x )

' x e

t ai + 1

, b .+ 1

] 4.3.4

< 8 i '9

i . H . M 'e

' i . H . M ' - - ;e

i r ^ M)

= 0

' A.3.5

e

i - l . H , M( b

i) ) e

i , N , H( a

i + l ) ) = 0 . j =0(1)v-l 4.3.6

-44-

Theorem 4-.3.1 (A Differential Correction)

The perturbed problems 4-. 3.4, 4-. 3.5, 4-. 3.6 yield

e

i , N , M( x )

= U

i , N( x )

" U

i , M( x )

' i=°(l>P-l

Proof: see chapter seven.

4.4- A MINIMIZATION PROBLEM FOR ADAPTIVE CONTROL

Let u e Cv

f a , b ] . We wish to determine P , specifying the

number of subintervals into which ;[a,lD] is- to b.e. partitioned,

in such a way that

A = max { max ' U . M( x ) - u(x) } < T01 4.4.1

• i = i d ) p t . _1 £ x l t i

a = t "<t. <t«<. . .<t <t =b . 4-.4-.2 O- — <•—• ~ P - JL P

The minimization of this function A o-f- P--1 variables or

equivalently, the optimum location of the points of matching

t^,i=l(i)P-l has been described for the case of rational function

approximation, Lawson 1963 . The problem may be viewed

as the minimization of a real valued function of P-l

variables t^ i=l(l) P-l subject to the constraint 4.4.2.

We attempt to balance the extrema of error functions

{U . ^(x) - u(x)} or their estimates, by shortening the

subintervals having the larger values and lengthening those

with smaller ones.

Other suggestions in the literature include the use of :

( i) Zeroes of T^" (x), x e [ a , b j given by t

± = ((a+b) - (a-b)cost(2i-l) /(2N)))/2 4.4.3

-45-

(ii ) Extremas of Tg(x), xe£a,bl given by the zeroes

of the first derivative of T£(x), with N-1=P

including endpoints a and b.

(iii) Variable knots based on some error minimization

criterion. The criterion depends on the

description of | U± > N

( x ) - u(x)| i = 0,l,...,p-l

either as a theoretical bound or based on a

computational global error estimation. To be able

to compare fairly w i t h single Tau approximation,

the same degree is used in each subinterval

though more flexibility may be achieved by

specifying different degrees for each subinterval.

Let us consider the singular perturbation boundary value

problem with a boundary l a y e r at x=l defined by the

differential equation.

Example 4.4*1

Lu(x) 5 -u"(x) + d_(p(l-cx)u(x)) = 0 , xe[0,l] 4-4-4 dx

u(0) = 1 u'(l) = 0 4.4.5

where c=0.98 and p=1000> and apply to it our considerations.

The graph of the solution is a spike symmetric about x=l

extending from u=l upto a b o u t u=50 when x runs in Q0,2j .

Problem 4-4-4 and 4-4-5 has been discussed numerically by

several authors in the recent literature ,(see Barrett and

Morton 1978 and the references given there).

We have constructed for this problem a piecewise . Tau

approximation of degree N = 6 over four equal and consecutive

subintervals of LO.ll . Then the Tau estimator ( e

i N( x ) )

M

was used to detect the region of rapid variation and

modifified the segmentation accordingly. With a new non-

uniform segmentation

-46-

(tQ=O.Oj t

1= 0 . 7 ;

:

, t2= 0 . 8 ; t

3= 0 . 9 5 t ^ l . O )

a new piecewise Tau approximation (of degree 7) is

constructed; it agrees with the anal-ytic solution u

up to two decimal p l a c e s , particularly in the proximity of

x=l. In Table 4-4*1.1 we report the results obtained.

Table 4.4.1.1

Use of the Tau estimator in segmentation control.

Type of approximatiorN^:

= 0. 5 0 .6 0 .7 0 .8 0 .9 1.0

Global Tau, N=27 1, 98 2, .43 3 .20 4. .73 9 .32 49.99

Global Tau, N=28 1. • 96 2, • 44 3 • 21 4 .73 9 .32 49.94

Piecewise Tau with )

uniform segmentationj

and N=7 )

Piecewise Tau with )

uniform segmentationj

and N=7 )

1 . 96 2 .44 3 .22 4< ,48 8 .40 48.21

Piecewise Tau with )

uniform segmentationj

and N=7 )

Piecewise Tau with )

non-uniform controlled^

segmentation and N=7 )

Piecewise Tau with )

non-uniform controlled^

segmentation and N=7 )

1 . .96 2. 44 3. ,21 4« 73 9. 31 49.95

Piecewise Tau with )

non-uniform controlled^

segmentation and N=7 )

Exact results 3. • 21 4. 73 9. 31 49.95

Barrett and Morton 1978 5. 26 7. 35 47.83

An automation with a general purpose software will follow an

algorithm of this type:

(i) Start with a few subintervals of equal lengths and

construct N

U ) , u

i > N + 1(

x

) f o r e a c h

subinterval

Ii + 1

, i = 0, 1,...,p-l

(ii) Construct Tau estimator

(e

i , N( x )

W l = U

i . N( x )

* U

i , H+l

( j t )

of degree M=N+1

-47-

(iii) Obtain d± = max l\(e

± N( Z

±) )

M^

N + 1I }

where Z . . = real zeroes of d ((e. A T

(X

)) M-MXI • ^ 1 ,N M=N+1

(iv) choose t. = Z . . the zero at which d. is obtained 1

J m 'x 1

as a new matching p o i n t .

(v) Repeat i,ii,iii and i v in that order until the

required user's accuracy is satisfied or halted.

4.5 The Integrated system with Implicit Matching.

In this section,associated conditions are satisfied

exactly as in all the problems discussed in- this work.

Case v=2:

Let

Lu(x) = P 2 ( x ) u » ( x ) + P 1 ( x ) u ' ( x ) + P Q ( X ) U ( X ) = f ( x ) , x e [ a , b j

4.5'.'l

Integration by parts of 4-5.1 gives

P2(xju

!

-/+" ( P1 - P £ ) u - + J ( "

P

1 + P

o ^u d x =

/f

Wd x +

4.5.2

and again integration by parts of equation 4.5.2 gives

P2( x ) u +/(?-,_ - 2P£ )udx +ff ( P» - Pj + P

Q) u d x d x =

f f f(x)dxdx + Cxx + C

2 4.5.3

where C^ and C2 are arbitrary constants.

In 0 r t i z l 9 7 4 » a direct recursive generation of the solution

of 4.5.3 was reported. Here , it is given as a simple

relation between V Q . ( X ) (. w of the original differential

u J N

equation (or system).

-48-Definition 4.5.1:

Let

IL s S S . - - I S v

L

4.5.4

denote v repeated integrations of the differential operator L.

Definition 4.5.2:

The sequence of canonical polynomials {q.(x) }. J J

E J N

o

uniquely associated with 5-.3 is defined by

IL( q j ( x ) ) = x

J

' , j = 0 , 1 , 2 , . . . 4.5.5

Theorem 4.5.1

•Given L in 2.1.0, v

i L = Z P . (x)dv. , 4.5.6

j =0 J

dxJ

v - 1 q.(x) = ( n ( j-k ) )Q. (x) , 4.5.7 J k=0

J

"v

Proof:

Assume { Q.(x) }. „ is already generated, J J £

0

then 4.1.12 of theorem 4.1.2 gives

X

L V H( X ) =

I T . . . a L ) Qj + H

(x

) 4.5.8

= - F LV

) L

V H( X )

4.5.-9

= 4.5.10

x J+ H + v

4.5.11

(j+H+1)(j+H+2)...(j+H+v)

Therefore, -replacing j w i t h j-H-v gives

-49-

^ j - v ^ ) = j (j-1)(j-2)...(j-(v-l))

4.5.12

W x )

3(j-i)(3-2).. . ( j - ( v - D )

4.5.13

Therefore, from the linearity of

q3( x ) = j ( j - l ) ( j - 2 ) . . . ( j - ( v - l ) ) Q .

- v( x ) . 4.5.14

The proof is complete if w e

take into account that the

first v-1 canonical p o l y n o m i a l s remain undefined .

Let

I N

U . „(x) = 2 a..q.(x) , i=0,1,...,p-1 j _ Q J-J J 4.5.15

then from the last result 4-5.7

j N v-1 U^ „(x) = S a . . ( n (j-k))Q. A x ) .

j=0 k=0 J

4.5.16

and

xe I1 = [ a

1, b

1]

xe I = fa ,b ] p L

p pJ

4.5.17

-50-

N u m e r i c a l results show that. 4.5.17, 5.16 improves the

accuracy of a single p o l y n o m i a l of degree N over [a,b]

for the integrated s y s t e m .

4.6 NUMERICAL EXAMPLES

Example 4.6.1

Consider the problem

Lu(x) = u"(x) - 4u(x) = 4cosh(l) , xe[0,l] 4,6.1

ii( 0) = 0 = u (1) 4-6.2

This problem has been u s e d as a test problem i n - O l i v e i r a 1 9 8 0 .

For both the d i f f e r e n t i a l and the corresponding integrated

forms, we compare the f i r s t Tau correction, defined by

(e„(x))M W J

., » for N = 3 ( 2 ) 9 , with the exact error. The N M=N+1

integrated form is never constructed directly, its Tau

approximation is computed by u s i n g the result of theorem 4.5.1

Table 4.6.1.1

Global Tau approximations and their Differential

Corrections

Degree Differential form Integrated form

F i r s t Tau E x a c t First Tau Exact estimator error estimator . error

3 7.369X10"2

7 . 4 1 5 x l 0 "2

9 . 8 6 6 x 1 0 "3

9 . 9 4 0 x l 0 ~3

5 4 . 5 7 6 x 1 0 " ^ 4 • 5 8 9 x 1 0 " ^ 7 . 3 5 8 x l 0 ~5

7 . 3 9 0 x l 0 ~5

7 1 . 2 7 5 x l 0 ~6

1 . 2 7 8 x l 0 "6

3 . 1 7 2 x l 0 "7

3 . 1 8 1 x l 0 "7

9 3 . 3 7 8 x l 0 "9

3 . 3 8 4 x l 0 ~9

9 . 0 4 8 x l 0 ~1 0

9 . 0 6 4 x l 0 ~1 0

The integrated form shows consistent improved accuracy over

the differential f o r m . In both forms, first Tau estimators

show close agreement w i t h their corresponding exact error not

only in its order of magnitude and sign but even in at l e a s t

two significant figures.

-51-

In Table 4-6..1. II we p r e s e n t the same information when

segmentation is used: four Tau approximations are constructed

over subintervals of Co,l] of equal lengths. We remark

that for. n > 4. the approximation of the differential form is

more accurate than that of the integrated form.

Table 4.6.1. II

Four piecewise Tau .approximations over equal subintervals of L0,ilj

Differential form Integrated form

Degree First Tau estimator .

E x a c t error

First Tau estimator

Exact error

3 2 . 0 7 2 x 1 0 "3

2 . 0 7 3 x l 0 ~3

2.0l6xl0"3

2 . 0 3 2 x l 0 "3

4 1 . 1 4 1 x 1 0 "5

1 - 1 9 6 x 1 0 ' 5 1 . 4 0 5 x l 0 "5

1 . 6 4 4 x l 0 "5

5 5 . 4 4 8 x l 0 ~7

5 . 4 7 0 x 1 0 "7

2 . 3 7 3 x l 0 ~6

2 . 3 8 6 x l 0 "6

6 2 . 1 1 5 x 1 0 "9

2 - 2 3 5 x 1 0 "9

I . l 6 3 x l 0 "8

1 . 2 8 2 x 1 0 "8

7 1 . 1 9 2 x l 0 ~1 0

1 . 1 9 6 x l 0 "1 0

1 . 1 8 1 x 1 0 "9

1 . 1 8 6 x l 0 ~9

8 3 * 7 6 6 x l 0 "1 3

3 . 9 0 8 x l 0 "1 3

4 - 4 4 0 x l 0 "1 2

4 . 7 5 0 x l 0 '1 2

In both forms, first Tau e s t i m a t o r s show close agreement with

their corresponding exact error.

In Table 4*6.1 III an attempt is made to r a n k , in terms of

accuracy , the global and segmented Tau approximations

before and after the Tau c o r r e c t i o n ( e ^ ( x ) i s applied

to them.

-52-

Table 4.6.1. Ill

A hierarchy of global versus segmented approximations of the boundary value problem 4.6.1 and 4.6.2.

Numerical technique Max. Abs Errors

Type of approximation

Corrected Piecewise cubic

differential form for Tau,

with the use of the technique

of increased accuracy at

matching points 5.8xl0"7

Segmented

Corrected Piecewise cubic,

differential form for Tau l.lxlO"5

Segmented

Corrected Piecewise cubic,

integrated form for Tau 1.4x10"5

Segmented

Corrected global cubic,

integrated form for Tau 7.4x10"5

Non*segmented

Piecewise cubic, differ-

ential form for Tau, w i t h

the use of the technique of

the increased accuracy at

matching points • . " 1.6x10"*

Segmented

Corrected global cubic,

differential form for Tau 4.6x10"* Non-segmented

Finite differences -

correction of collocation

with 4 cubic splinesioiiveira 1 980.

3.6X10"3 Segmented

Piecewise cubic, differen-

tial form for Tau 4.0xl0"3 Segmented

contd....

-53-

Global cubic, Integ-

rated form for Tau l . O x l O "2

Non-segmented

Collocation with cubic

splines>01iveira 1 9 8 0 . 1 . 5 x l 0 ~2

Segmented

Global cubic, differential

form for Tau 7.4-xlO"2

Non-segmented

We see that after correction, a global Tau approximation

of the given problem 4-.6.1 and 4.6.2 produces a better

result (4.6x10""^) than the segmented approximation obtained

by using finite differences correction of the collocation

- 3 with 4 cubic splines ( 3.6 x 10 By simply switching

to Legendre polynomial perturbation terms, the advantage

over the cubic splines collocation solution with finite

differences correction is now measured by a factor of

1.619x10"^.

It is interesting to remark that the accuracy attained with _ n

that Tau approximation (5.827x10 ) is almost identical

to the upper bound obtainable through

the use of Lagrange's error estimation formula (see

Meinardus 1967)for the best uniform segmented approximation

of the exact solution u(x) by.means of four algebraic

polynomials of the same degree as the Tau pieces. This

suggests the possibility of using these Tau approximations

as starting segmented approximations in a process for the

numerical segmented approximation of differentiable

functions. Such class could be widened by a judicious

choice of the segmentation p o i n t s .

-54-

Example 4.6.2

A similar linear BVP, now with a Neumann condition at

one end and over a large interval [0,20].

Lu(x) = u»(x) - u(x) = 1 , xe[0,20] 4.6.3

u(0) = 0 u1

(20) = 1 . Scott (see Aziz 1975)

Table 4*6..2.x displays the maximum of the value of the

first Tau error estimate and of the exact absolute error

for some of the Tau a p p r o x i m a t i o n s . They have been .

constructed over either [0,20] or over 4 equal and consecu-

tive subintervals of [0,20]. In this, as in the previous

example and in several other problems, the piecewise Tau

approximation, obtained from the differential form gives

consistently better results than the integrated form,

even for small values of N . For this problem, as in many

other Neumann p r o b l e m s , the global Tau approximation

obtained fro.m the differential form gives consistently

better results than the integrated form, even for small

values of N .

-55-

Table 4.6.2.1•

Tau approximation

interval [0,20].

of a N e u m a n n problem over a large

N = 7 N = 8

Numerical technique

First Tau E s t i m a t o r

Exact Error

First Tau Estimator

E x a c t Error

Global integra-

ted Tau 0.21 0.21 0 . 5 3 X 1 0 '

1

0.54X10"1

Global differ-

ential Tau 0.13 0.11 0 . 2 3 x l 0_ 1

0.21xl0"2

4 piecewise

integrated Tau 0 . 8 7 x 1 0 "3

0.88xl0~3

0.14xl0*3

0.15xl0"3

4 piece^ise

differential Tau 0 . 1 3 x l 0 "3

0.14xl0~3

0.24x10"^ 0.24x10"^

Performance:

Error in global differential Tau(N=20)=6.11 x 1 0 "8

Time = 1.789 cp sees.

- 8

Scott ,reported .in Aziz 1975 =6.79 x 10

Time = 1.532 sees.

Example 4.6.3 see £ a d u n a i s k y 1979, Cohen and Jones 1974

(l+x2

)u"(x) + 4xu'(x) + 2u(x) = 0, xe[0,2] 4.6.5

u(0) r 1 u(2) = 0.2 4.6.6

2 -1 u(x) = (1+x -) ^ (the Runge function)

Table 4.6.3. I

3 4 5 6 7 8

E r r o r in Global

Tau for differ-

ential form

0.26 0.36x

1 0 "1

0.12x

1 0 "1

0.64x

1 0 "2

0.16x

1 0 "2

0.15x

1 0 -3

Operational

a p p r o a c h of the

Tau method,Ortiz and Samara 1 9 7 8 .

0.26 0.36X

1 0 "1

0.12x

1 0 "1

0.64x

1 0 "2

0.l6x„

1 0 "2

0.15x

io"3

E r r o r in two

pieces of equal

subintervals

differential

form

0.36X

l O "1

0.82x

1 0 "2

0.12x

-2 10 *

O.llx

lo-3

0.62x

10"*

0.14x

10"*

Error in four

pieces of equal

subintervals,

differential

form

0.85x

1 0 "2

0.12x

10 *

0.16x

10"*

0.71x

1 0 "5

0.36X

1 0 -6

0.73x

1 0 "7

Error in sixteen

pieces of equal

subintervals

differential

form

0.54-x

1 0 "3

0.31x

1 0 "5

Fyfe 1969 using

an adaptive

control for

0.5x10"*, took

sixteen subinter-

vals for cubic

splines

0.78x

10"*

Table continued next page

Collocation with

cubic splines in

16 places with

one deferred

correction

Zadunaisky-1979

Error in sixteen

pieces of equal

subintervals,

differential

form. LEGENDRE

perturbation

0.4 x.

10 -4

0.14x

lO"4

"

0.79x

to'

-58-

Example 4.6.4

A stiff boundary value problem with non-polynomial right

hand side.

Just as the evaluation of a mathematical function in a

computer requires the existence in its storage of a

polynomial or rational approximation to such function, the

presence of functions in differential equation (coefficients,

right hand side or a functional nonlinear term as in

example 3.3.5) to be treated with the recursive formulation

of the Tau method requires access to the cofficients of

such polynomial or rational approximation.

In the following singular perturbation boundary value

problem

Lu(x) = u" (x) - Ku(x) = cos(x) , xe[0,7r/2] 4-6.8

u ( 0 ) = 1 = U(TT/2) 4 . 6 . 9

U(x) = + c?e

(

~/ K

- cos(x) 4-6.10 1

^ K+l

where K is the stiffness parameter and c^, c^ are

constants (-see Guderley 1 9 7 5 ) .

Table 4.6.4. I

K

Degree of approximation

Perturbation term used

N o . of equal subintervals

Max. Abs. Error

10* 27 Chebeyshev :;No segmental 1 . 0 x 1 0 ~7

tion 2 . 0 x l 0 "

1 0

1 03

14 Chebeyshev 8 2 . 0 x l 0 "1 0

1 03

14 Legendre 8 l . l x l O "1 0

1 04

28 Chebeyshev No segmenta- l . O x l O "2

) tion n 10 14 Chebeyshe v 12 7.8x10" '

10* 14 Legendre 12 4 . 4 x 1 0 "7

-59-

The graph of the solution u(x) is a curve which, for large

values of K, is close to u=0 inside the interval

and jumps to u=l when x approaches either x=0. or x ^ / 2

(see fig. 4*6.1).

¥ e have computed the solution for K=10,000, with a global

Tau approximation of a moderately high degree (N=27,28) and

also with piecewise Tau approximations of about half of that

degree. The ability of the piecewise Tau approximations

to follow the rapid variation of u(x) near x=0 and x=7r/2

is shown from the results given in Table 4. 6.4- I, for both j

Ghebeyshev and Legendre perturbation terms. In the flat

region of u(x), where the matching points lay, both types

of perturbation terms provide an equally good accuracy,

slightly better in the Legendre case, as is to be expected,

see Lanczos 1 9 7 3 .

U

1--

1 L •A J.

K=10^ t t / 2

fig. 4.6.1

Graph of the Tau approximation and u(x) conincide with

_ 7 deviation 10

Example 4.6.5 ( s e e

Conte 1966, Davey 1980 )

A stiff fourth order linear boundary, value problem

u ^ ( x ) - 3 6 0 1 U " ( X ) + 3 6 0 0 U ( X ) = -1 + 1 8 0 0 x2

, x e [ 0 , l ] 4 . 6 . 1 1

u(0) = 1 = u'(0)

u(l) = 1.5 + sinh(l), uf

( l ) = 1.0 + cosh(l)

4.6.12

4.6.13

-60-

This example was given in Conte 1966 to illustrate u. l i m i t a t i o n

of the shooting methods-. Delves—(^see-Hall and Watts 1976) gave

this example as a case w h e r e p i e c e w i s e approximation may n o t

improve the a c c u r a c y o v e r a single a p p r o x i m a t i o n . This is

n o t the case for this p r o b l e m with the implicit matching

for the recursive f o r m u l a t i o n of the Tau method, see

Table 4-.6.5. I .

Table 4 . 6 . 5 . I

Technique e

m a x for N=8

A r i t h m e t i c

Global Re-cursive Tau method

w i t h H N ( X ) = ( T 1 + T 2 X ) T | _ > 1 ( X ) 2 6 . 2 x l 0 "8 Double

precision

Global a p p r o x i m a t i o n

Delves( H a l l and W a t t s 1 9 7 6 ) 1 . 2 x l 0 "8

Global a p p r o x i m a t i o n

l . O x l O "8

Barrodale and Y o u n g C Hall& W a t t s )

Global a p p r o x i m a t i o n

l . O x l O "8

recursive Tau method w i t h

% W =T

1T

N ( X ) + V N . ^ x ) 0.46X10"9 Single

precision

Global a p p r o x i m a t i o n

operational a p p r o a c h to the,

Tau method (Ortiz&Samara 1 9 ^ 0.46X10"9 Single

p r e c i s i o n

D a v e y i 9 8 0 2.1 x l O "1 1

I m p l i c i t Matching at

x=4 Recursive tau m e t h o d . 0 . 4 7 X 1 0 "1 1

Single :precision

M a t c h i n g at one p o i n t x = i , the two Tau pieces give 0.47x10"''''''

compared to one Tau 0 . 4 6 x l 0 ~9

where a single Tau is even

more accurate than D e l v e s 's

r e s u l t s .

W h i l e it is p o s s i b l e t h a t for a particular example,

piecewise- p o l y n o m i a l a p p r o x i m a t i o n may n o t improve over a

single p o l y n o m i a l a p p r o x i m a t i o n , this example is not the

case with our m e t h o d .

A g a i n , the operational a p p r o a c h and the recursive

formulation both give i d e n t i c a l results up to machine

accuracy for the tau m e t h o d .

Davey 1980 recently r e p o r t e d ..this example with the

Compound matrix method and his. result compares well w i t h

those obtained from the tau method using only one

matching p o i n t at x = £ .

E x a m p l e 4»6.6

u"(x) +(2/x )u'(x) + u5

( x ) = 0 , xeC0,l] 4.6.14

u(0) = 1 u(1) = / 0 . 7 5 4.6.1$

u (x) = ( 1 +X2

/ 3 ) "4

4.6.16

This problem has been u s e d as a test problem for several

other numerical methods ( see De Hoog and Weiss 1978).

Table 4 . 6 . 6 . I

Degree of the Tau

approximation

Global approxima-

tion Tol = 1 0 "6

Segmented approximation

Tol = 1 0 "6

M a x . abs

E r r o r

n o . of cycles required

M a x . a b s . N o . of cycles

required

4 8.7x10""* 3 7 . 8 x l 0 "5

3 , 3

5 6 . 0 x l 0 ~5

3 1 . 6 x l 0 "6

3 , 3

6 6 . 6 x l 0 "6

3 l . O x l O "6

3 , 3

-62-

The segmentation is over [0, . The Tau error estimator

(e^(x)) " = 8.069 x 10"^ compared to the exact = 8.7 x 10"^.

For the segmented Tau approximation (e. , (x))- r =7.60X 10"^

compared with exact = 7.76 X 10"^.

The error estimator in the nonlinear problems is based on

the last iteration when the required convergence criterion

is satisfied.

-63-

CHAPTER FIVE

METHOD OF LINES BASED TAU METHOD FOR ELLIPTIC

PARTIAL DIFFERENTIAL E Q U A T I O N S .

As an application of our results on boundary value problems,

we discuss the application of the recursive tau method to

the method of lines (MOL) for the case of elliptic partial

differential equations, a n d also assess the results of

Wright 1976 obtained by using the low order central difference

formular with the method of l i n e s . We obtain better results

than those reported by W r i g h t . However, the introduction of

finite differences, as we will see, reduces considerably the

accuracy obtainable with this type of semi-discrete methods.

The MOL is simple in concept and consists of discretizing

any system of partial differential equations(Pdes) in all

but one of the independent variables.. This semi-discrete

procedure yields a coupled system of ordinary differential

equations(Odes) which are then numerically integrated with

any numerical method. This reduction to Odes can be

realized in many other ways.' An interesting approach has been

recently suggested by Mason 1979.- who bases, his results on

Lanczos' original formulation of the tau method. Here we

p

report . straightforward central difference formulae of 0(h )

to the given P d e . The r e s u l t i n g system of second order

ordinary differential equations is solved by the recursive

formulation of the tau m e t h o d . 'Two forms of approach have

been used and compared in this chapter, a direct and a

Gauss-Seidel iterative a p p r o a c h .

In both forms of approach all the boundary conditions are

satisfied simultaneously.

-64-

The MOL with the Tau method was first reported in Wragg 1966

applied to a stefan p r o b l e m . Most of the applications of

MOL since Rothe 1930 has been to parabolic pdes reduced

to a coupled system of initial value problems in odes.

There is a wealth of p a p e r s , particularly in the Russian

literature,, on the theory and applications of the method

of lines. These works had been reviewed by Liskovets 1965*

For elliptic pdes the w o r k s of Guderley-Nikolai 1966 ,

and Jones et al 1972 are among many recent publications on

this method. ' - -

5.1 A Direct Formulation

Consider the following Poisson's equation with

Dirichlet boundary conditions:

d 2 u (x .y ) + 3 2 u ( x , y ) = f ( x , y ) , in S ,- 5.1.1

3x' ay

s = { (x, y) : 0< I x I , | y | < 1 }

u(0,y)

u(l,y)

u(x,0)

u (x, 1)

uQ( y )

u1(y)

UQ( x )

u1( x )

on r of S 5.1.2

y

X

3 u is discretized by using central differences. If (p-l)

3 y2

interior lines are involved, then on each of them

-65-

d2

u1 :( x ) + h ~

2

( ur + 1

( x ) - 2 up( x ) + u

r - 1( x ) ) + 0 ( h

2

) = f ( x , yr) .

d x

f o r r=l(l)i>-l v

5.1.4'

where the discretization step is

h = Ay 5.1.4

For r r = 0 uQ( x ) = u ( x , 0 ) 5.1.5

C: p u

p(

x

) = U

(X

> 1 ) 5.1.6

The problem (5.2.1) with (5.2.2) becomes the approximated

problem

UQ(x) = u(x,0) 5.1.7

d2

ur( x ) - 2 h "

2

ur( x ) = f ( x , y

r) - h "

2

( ur + 1

( x ) + u ^ ( x ) ) 5 . 1 . 8

d x2

ur( 0 ) = u ( 0 , y

r) , u

r( l ) = u ( l , y

r) 5.1.9

r = 1,2,...,p-l 5.1.10

u (X)E u(x,l) 5.1.11 p

Problem (5.1.7 - 5.1.11) consists of a system of linear

second-order two-point BVPS which may be reduced to first

order system and solved by shooting methods or similar

techniques. The choice yr is commonly taken to be equal

spacing but, again, they can be selected to be zero of

T*(y) or extremas of T * ( y ) .

Applying the method of section 2.1 directly to this system,

th indicates an N degree tau approximation to u along

r r

each horizontal line r=l,2,....p-1.

Then

UQ(x) = u(x,0) 5.1.12

d2

u * ( x ) - 2 h "2

u ^ ( x ) = f ( x , yr) - h ~

2

( u J+ 1+ u ^ _

1( x ) ) + H

N > r( x ) 5.1.13

d x2

u*(0) = u(0,y ) u*(l) = u(1,y ) , r = l,2,...,p-l 5.1.14 r r r r

u*(x) = u(x,1) 5.1.15 P

where (5.1.12) is a perturbed system of (5.1.8) and

(5.1.12),(5.1.14)9 (5.1.15) are attempts to satisfy exactly

and simultaneously the boundary conditions. We then assume

a polynomial approximation of degree N , u* to u of the form r r

u*(x) = ? a *( r )

Q .( r )

( x ) « Z a .( r )

Q .( r )

( x ) = u (x) 5.1.16 r

j=0 J J

j=0 J J r

r = 1,2, ...,P-1," 5.1.17

with Q ^ ( x ) defined by. J

L Q ( x ) = xJ

' 5.1.18 j

L = df - 2 h "2

. 5.1.19

^ 2 dx

Then generate

Q |r )

( x ) = -ih2( x j - j ( j - l ) Q ^r

^ ( x ) } 5.1.20 J J

r = 1,2,...,p-l 5.1.21

In a more general partial differential equation, L is

(v) different for each line r and thus {Q; (x)} is a different

sequence for each l i n e .

The choice of perturbation

= jn

T

kr ) T

H - k) ( x )

• 5.1.22 K~ U

r = 1,2,...,p-l 5.1.23

is used throughout this chapter. For a rectangular region

H „ (x) is the same for each line r but different in the JN , r

non-rectangular regions.

Let

( r )

= ( a *( r )

, a *( r )

a *( r )

) ( 1 x N+l ) 5.1.2*

X

n(r)

-67-

( l , x , x2

, . . . , xN

)T

( N+l x 1 ) 5.1.25

coefficient-matrix of the canonical

p o l y n o m i a l s . ( N+l x N+l ) 5.1.26

u*(x) = a *( r )

£( r )

X 5.1.27

T ( R ) = ( T I ( R ) ' T 2 ( R ) ) ( 1 X 2 ) 5 ' 1 ' 2 8

(r)

G = coefficient matrix of the Chebyshev polynomials

used to perturb the differential system along

each line r . (2 x N+l ) 5.1.29

F f X = Z f . xJ

5.1.30 yr j=o

Then from (5.1.12) (5.1.15) , the following system of

linear algebraic equations is solved.

a *( r )

X + h "2

( a *( r + 1 )

+ a ' ^ ^ ^ a ^ ' x = lr

^ 5.1.31

a *( r )

+ h "2

( a *( r + l )

+ a *( r

-1 ,

) a( r )

= f V £( r )

+ t( r )

C( r )

2( r )

.

r = 1,2,...,P-1 5.1.32

W.e put _5.1.32 in one enlarged system of algebraic equations of the form :

A = f + CT

TT

. 5.1.33

where a = ( a , a ',..., a ^ ) .

From the boundary conditions

B a* [u ( 0 , yp)

5.1.34

where B i s ^ a n (2(p-l) x (N+l)(P-l) ) matrix .

From (5.'l.33) and (5.1.34) we solve for 2(p-l) + (N+l) (p-l)

parameters by using 2(f>-l) +(N+l) (p-l) algebraic equations.

-68-

From (5.1.33) assuming A is invertible

a*T

= A "1

f + A ' V 5.1.35

substituting (5.1.35) into 5.1.34-) gives

(B A " V ) TT

= - B A "1

f +

j a d » yr)

5.1.36

T T solving (5.1.36) for T leads to the determination of a*

from (5.1.35) and hence the solution ,r=i,2,.... p-1.

The work cost is in the inversion of matrix A and solving

a small system in (5.1.36). For l a r g e p the size of A

can be very large^. though block structured.

It is an•intermediate stage . The approach-described here

contrasts with the-1

"- uatrix lines-tau method"; of W r i g h t M 9 ? ^

in the type of algebraic equations solved.

Wright 1976 Present Method

(i) Generates N sequences of

canonical polynomials

for each r=l,2,.... p-1

against (storage)

( i i ) M

NX N I = 1 > 2 > - - - £ + 1

against(work done)

(iii) O(h^) finite difference-s

used.

Generates only one

sequence of canonical

polynomials for each

r=l,2,....,p-l. for storage

A"1

;N(p+l)xN(p+l) block

structure and inversion is

done once. for (work done)

Q(h ) finite differences

used.

The problem of inherent instability normally associated with

the MOL for elliptic pdes using shooting methods to solve

the coupled system of second order two-point BVPS is absent

in our method., which we illustrate in the next table for

Laplace's equation on a unit square taken from Jones et all972.

-69-

Jones et al 1-97-2 Present method

p=12 p=l6;

Ax = 1/4 ,1/8 ,1/12 , 1/16 N = 7 degree- of approxim^

ation along line r . 2

°(A

y) was u s e d . P

0(Ay) was u s e d .

Result = failure due to lar-ge p=12 but very satisfactory when p=12,54,57 for

Result = very satisfactory.

0 ( A y ) * .

The satisfactory results of Wright 1976 were obtained

using (Ay)* with his 'matrix lines - Tau method' while his

o b('Ay) difference formulae gave unsatisfactory results.

5.2 AN ITERATIVE-APPROACH

From (5.1-12) - (5.1.15) set up a Gauss-Seidel type of

iterative process

u g> k + 1

( x ) = u(x,0) 5.2.1

M ? , k + l( x )

- 2 h_ 2

u * v +

, ( x ) = f(x,y ) dx r,Jc+.L r

+ u

? - l , k + l( x

^ +

% , r , k+l

( x ) 5

-2

'2

U

? , k+1

( 0 )

= u

(°»yp> ; u

? , k + i( 1 )

= u (

W 5

-2

-3

r = 1,2,...,p-l 5.2.4

u* V + 1

( x ) = u(x,1) 5.2.5

k = 0,1,... 5.2.6

where

u* n

(x) are supplied • r, 0

r=l,2,....,p-l with' all boundary conditions satisfied by the

choice of that initial g u e s s .

-70-

With this approach, (5.2.2) - (5.2.6) is solved as follows

for the ( k + l )s t

stage:

The right hand side of (5.2.2) is known explicitly from the

k stage for all r = l , 2 , . . . p - 1 and (5.2.1).

Therefore, starting with r = l , (5.2.1), (5.2.2) and (5.2.3)

give a scalar second order two-point BVP along each line r .

We no longer solve a coupled system of second order

two-point BVPS as in section 5.1. Therefore, the method

of section 2.1 is here applied directly along each line

and repeated for each k until successive k values give

tolerable agreement in solution.

This approach avoids the problem of inverting a large matrix.

In fact there is only one system solved for the T parameters

for each l i n e . The computer time is higher as to be

expected for iterative p r o c e s s e s . This limitation may

be improved using successive over relaxation parameter with

the Gauss-Seidel p r o c e s s .

This approach has a l o t more to offer particularly when

general pdes have to be solved and for irregular regions.

The two forms of approach-may be applied to nonlinear elliptic *

pdes in the manner of chapter three.

5.3 NUMERICAL EXAMPLES

Example 5.1

V2

u ( x , y ) = 0 0 < x , y < 1

u(0,y) = 0

u(l,y) = sin(iry) .(also used, cos(7ry);l)

u(x,0) = 0

u (x, 1) = 0

u(x,y) = ( sin(Try) sinh(irx) ) ( sinh(Tr))

-71-

Table 5.1-1

1 . 0 0

o;73

0.5-

0.25-

0.0534

0.0534

0.0533

0.0532

(1)

(2)

(3)

(4)

0.0755

0.0755

0.0753

0.0752 *

0.0534

0.0533

0.0533

0.0532 #

0.1412 0.1412

0.1411

0.1409

0.1997

0.1997

0.1993

0.1993 z

0.1412

0.1412

0.1411

0.1409 *

0.3205

0.3205

0.3203

0.3201

0.4532

0.4532

0.4527

0.4527 &

0.3205

0.3205

0.3203

0.3201

sm7ry

0.00 0.25 0.50 0.75 1.00

(1) Ay = 1/16 N = 7 0 ( A y ) iterative present method

(2) Ay = 1/16 N=7 0 ( A y )2

direct present method

(3) Ay = 1/4 N=7 O(Ay)4

- direct Wright 1976

(4) Ay = 1/8 N=7 0(Ay)^ direct Wright =exact .

Max. error for the present method = 5.5 x 10 -4

-72-

Table 5.1.II

1.00,

0.75;

0.5CM-

0.25*

0.0532 ( 1 )

0.005827

0.005827

0 . 0 0 7 3 ^ *

0.0000

0.000000

0.000000

Q.0000 *

0.0532

0.005827

0.005827

0.0073 *

(2)

(3)

0.1409

0.039020

0.039020

0.0366

0.0000

0.0.00000

0.000000

0.0000 *

0.14.09

0.039020

0.039020

0 . 0 3 6 6

0.3201

0.172467

0.172467

0.1704 a

0.0000

0.000000

0.000000

0 .0000 #

0.3201

0.172467

0.172467

0.1704 *

cosiry

0.00 0.25 0.50 0.75 1.00

(1) Ay = 1/8 0 (Ay)

(2) Ay = 1/16 g ( A y )2

(3) Ay = 1/16 P ( A y )2

(4) Exact

N = 7 direct method Wright 1976

N = 7 iterative present Method

N = 7 direct method*present method.

W i t h 0(Ay) the results here show a ^bet%©r-accuracy than those of

W r i g h t1

s o(Ay)^" of the same degree.

-73-

Table 5.1.Ill

1.00'

0.7 5"

0.50-

0.25"

0 . 0 7 5 3 ^

0 . 0 6 6 4( 2 )

0 . 0 6 8 2 ^ - *

0.0752

0.0955

0.0829 *

0.0753

0.0664

0.0682 *

0.1993

0.1839

0.1828

0.1992

0.2494

0.2511 *

0.1993

0.1839

0.1828 *

0.4527

0.4292

0.4349 *

0.4527

0.5397

0.5426 *

0.4527

0.4292

0.4349

0.00 0.25 0 . 5 0 0.75 1.00

(1) Ay = 1/16 N=7 (X Ay)4 -

direct method Wright 1976

2 (2); Ay = l/l6 N.=7 0( Ay) direct method present method

(3) Ay = 1/16 Finite Element Method with o(Ay)2

(private communication with Mr•Colin Mason,

a colleague).

Ignoring the effect of discontinuity in the boundary data

at point(l,l), the results obtained are very satisfactory.

For improved accuracy a treatment of the point singularity

at (1,1) is necessary,for example choosing fcrr.u(l,l) an

intermediate value between 0 and 1 .

-74-

Example 5.2 Elastic Torsion problem on a rectangular

,.o region. 7 u(x,y) = -2

u(-l,y) = 0

u(l,y) = 0

u(x,-l) = 0

u(x,l) = 0

In Table 5.2.1, the symmetry of the problem is used and .

only results in the first quadrant is given.

The results show that the recursive formulation of the Tau

method with MOL gives very accurate approximations for

0 (Ay ) central difference formulae , - We have attempted:'in

cases to satisfy all boundary conditions simultaneously as

in Mason 1965.-. The method of Mason 1979- howeyery-- se-ems to be

more attractive from "the p o i n t of view of computer storage and

speed. Nevertheless-, this method--of--reduction based'1

on MOL

may become attractive for elliptic Pdes whose solutions are

non-separable or for. nonlinear- P d e s . .

- 1 , 1

I Table 5.2.1

-1,0.

0 . 3 U( 1 )

0 . 3 5 3 ^

0 . 3 5 3( 3

^

0 . 3 6 3U )

0.562 0.438

0.575 0.448

0.575 0.448

0.589 0.459

o', 0 0#5,0

L,1

-1,0 1,0

-75-

(1) Ay = i ...Ax = i .0 (Ay)

(2) Ay = i N = 20 0(Ay)

(3) Ay = i N = 20 o(Ay)

U ) Ay = 1/8 N=20 o(Ay)

0.(A?) Finite difference method

iterative present method

direct present method

direct present method = exact

.Jt_is. interesting to remark that when the solution of the given

problem is a polynomial, the dire.ct or iterative approach

of this chapter gives the exact polynomial solution. As

an illustration, the following example is approximated

by the iterative approach

2 2 The exact polynomial - y ^ + x along each line y ^ .

The direct or iterative formulation of the recursive

tau method described here may be used with higher order

finite difference formulae_ Q ( A y ) s a y , to improve the

accuracy of results with fewer interior lines.

-76-

CHAPTER SIX

SINGULARITY TREATMENT IN THE RECURSIVE FORMULATION

OF THE TAU METHOD

The numerical solution of a singular boundary value problem

is usually a function w h i c h experiences rapid changes in

magnitude. The knowledge of the behaviour of a physical

system governed by such type of equations is frequently

required at the neighbourhood of singular points. For

example, in a physical problem geometrical singularities

such as corners or sharp edges lead to singular points in the

corresponding differential equation. Mathematically, these

singular points are usually few in number but they determine

the principal features of the solution. Thus, while one

might want to avoid the few points where a differential

equation is singular, it is precisely at these points that

it is often necessary to study the solution more carefully.

Using a more or less standard numerical procedure which is

the expansion about the singular point and the solution of

a regular problem over a reduced interval which excludes

the singularity, followed by the matching of the two pieces

of solution, usually gives a satisfactory results.

In this chapter we report ways of solving singular problems

with the recursive formulation of the tau method using the

implicit matching technique of section 4-.1, singular functions

or transformation of variables.

-77-

6.1 IMPLICIT MATCHING

The implicit matching technique described in section

4.1 may be applied directly to singular problems. To

illustrate the technique we consider the following singular

linear two point boundary value problem,Cohen and Jones 1974.

2x(l+x)u»(x) + (l+5x)u'(x) +. u(x) = 0 , xe£o,1.5] 6.1.1

u(0) = 1 u(l.5) = 1 6.1.2

u(x) = ( l+/x )( 1+x J "1

6.1.3

The coefficient of the h i g h e s t derivative u" vanishes at

x=0 and the first derivative uf

becomes unbounded there.

The direct application o f numerical methods fails if we

ignore the presence of a singularity at x = G .

We compare for this p r o b l e m the implicit matching with

uniform segmentation over segmented interval,..-with that of

clustered matching points a-raund x.=(T', see .table 6.1.1.

This technique may be a p p l i e d directly to any form of

singularity in linear o r nonlinear problems and so has

this quality of being widely applicable but can be

computationally expensive from the point of view of computer"'

storage. The procedure we will now adopt is the one

suggested in Cohen and Jones 1974.

(i) The exponents of singularity are obtained using the

generating polynomial Lx^:

LxJ

' = j( 2j - 1 )x; i

~1

+ ( )( 2j'+ 1 )xJ

' 6.1.4

the cancellation of the coefficient of the lowest power

gives

j(2j-l) = 0 6.1.5

-78-

then = j = 0, =

1 / 2 a r e exponents of singularity.

From 6.1.4- and the definition of Q.(x), J

Q (x.) = { x3

- j( 2j - 1 )Q. ,(x) }( 2 j2

+ 3j + 1 ) ' X 6.1.6 J J

u(x) = ( A+Bx* )P(x) = ( A + B x ^ )( I a.Q.(x) ) 6.1.7

Let

* A N

UM( x ) = (A + Bx ) P

w( x ) = ( A+Bx )( Z a*Q.(x) ) 6.1.3

in 1M j _ Q j j

We attempt to .-solve the following perturbed problem

2x( 1 + x )Pg(x) + ( l+5x )P^(x) + PN( x ) = T T * ( X ) , 6 . 1 . 9

PN( 0 ) = 1 6.1.10

P(x) = ( 1+x )_ 1

, x e [ 0 , 1 . 5 ] 6.1.11

The reason for using one boundary condition 6.1/10 for a

second order differential equation is explained in Lanczos 1961.

The second boundary condition is implicitly enforced by

demanding a finite behaviour (but not vanishing) of solution

at a point where the derivative is infinite (see also Picken

1 9 7 0 ) . Observe that 6.1.11is valid for 0<x<l and is slowly

convergent for x close to 1 . It is therefore recommended

in. Cohen and J-ones 1974- that 6,1.11 be approximated by an

economized expansion for 0<x<l and matched with a finite

difference method with deffered correction elsewhere.

For consistency in method it is better to match at x=l with

another economized expansion valid for l£x£l.5 as follows

-79-

p

O , N( x

't

i o)

- X £ =

M

Pn( x ) = s 6.1.11

P1 > n

( X , T2 0, T

2 1) xeI-

2 = [1,1.5]

from the perturbed p r o b l e m s

2 X ( 1 + X ) P ») N + ( 1 + 5 X ) P J > N + P 0 > N = x

1 0T g ( x ) , 6.1.12

xe[0,lj

2 x ( l+x ) P

i > N + ( l + 5 x ) P

i > N + P

1 > 1 T = T

2 0T | ( X ) + T

2 1T J

- ; L( X ) 6 . 1 . 1 3

xe[l,1.5]

P

O ; N( 0

'T

I O) = 1 6.1.14

P

0 , N( 1 , T

1 0)

" P

1 , N( 1 , T

2 0, T

2 1) = 0 6.1.15

" T

2 0 ' T

2 1 ^ = 0 6.1.16

Since in'general P ( x ) is n o t knowrT^fii' closed form, we cannot

Impose on it a condition a t x=l»5» We attempt."'"to .

fix the constants A and B u s i n g the given boundary

conditions.

u(0) = A P N ( 0 ) = 1 6.1.17

which implies that A = 1

U1 > n

( 1 . 5 , T2 0, T

2 1) ( l+B/1.5 ) = 1 6.1.18

which, implies that

B = (1.5) "4

( ( t

2 0 't

2 1}

" 1 }

6.1.19

Again P 0 , N( x , t

1 0 , T 1 1 ) ' x e I

l =

' w i t h t

1 0T

N( x ) + t

1 1T

N - 1( x )

plus an extra constraint imposed at x=l by the continuity

condition in the second derivative of the solution

can be used instead,in a similar fashion as described a b o v e .

Table 6.1.1

NPTS = 1 5 Method e

m a x( x ;

max 0<x.<1.5 j

(1 U M(::,)-u(x.)|} *J J

max 0<x.<1.5 j

for j =l(l)'NPTS

Global Tau method 0.79

Implicit matching 15 Tau pieces 0.28 X 1 0 "

1

Present technique

u s i n g [ T 1 0 ] C T 2 0 * T21-^ i;85 X x o -6

Results of Cohen & Jones 1974. 1.27 X 1 0 -6

Present technique u u s i n g C T

1 0, T

1 33 [T

2 0,..T

2 13 1.20 X 1 0 -

6

o "1 J o If the series r(x) £ s(x) (where u(x) = Ax r(x) + Bx s(x))

d2 it is necessary to remove the x for negative or fractional

3 2

by substitution u(x) = x v(x) in the given differential

equation and to carry out a similar analysis and procedure

for the differential equation in v .

-81-

6.2 A LOGARITHM DIFFERENTIATION" SFQTIENfiE OF POT.VNnMT A T.S

Definition 6.2.1

Q . Let us define a sequence of polynomials (Q.(x)}. ,, which

J «J Q

is generated by L( xJ

l o g (x) ) and such that 6

L( Q?(x) ) = x ^ l o g ( x ) j = 0,1,2,... 6.2.1 J

9

We call it a "Logarithm differentiation sequence"

or canonical sequence (Q-log) .

This choice of name will be clear from, theorem 6.2.3.

We restate here theorem 2. , of chapter 1,section 3.

Theorem 6.2.1

{ Q.(x) }. N exists and is unique for a given L of the

J J 0

form 2.1.0 r with LQ.(x) = x

J

and generated by LxJ

. J

Theorem 6.2.2

e / \ { Q - ( x ) }. „ exists and is unique for a given L of the

J J efl 0

form 2.1.0 .

Proof: Existence

We shall follow the proof of theorem 4-.1.2 to

*1eH o

© / \ explicitly generate

J J ^ . v

L( x log (x) ) = E x^log (x) } . 6.2.2 e k=0 K dx e

v . , . , = Z P v ( x ) { g ( j , + h ( j ,k )x^ log (x)} 6 . 2 . 3

k=0 K e

where

h (j , 0) = 1

h (j , 1) = j

h (j,2) = j(j-l)

h(j,3) = j(j-l)(j-2)

-82-

h(j,k) = j(j-l)...(j-k+l),... 6.2.4

and

g(j,0) = 0

g(jfl) = 1

g(j,2) = 2j - 1

g(j > 3) = 3 j2

-6j + 2

a. . .

g(j.k) = f jh ( 3

'k ) 6

'2

'5

Therefore

v . , L( x

J

l o g (x) ) = Z Pl r(x)h(j.,k)x

J

"J c

log (x) + e

k=0 K e

v . , Z P, (x) _d__h( j, k)x^ 6.2.6

k=0 dj

Let

n

k P j x ) = Z p

, p are constants 6.2.7 * m=0

m

k m

k

then

n

k L( x

3

log (x) ) = Z h( j, k) Z V m

xm +

^ -k

log (x) + e

k=0 m=0 m

k e

v n

k 2 d_h( j , k) Z p

m x

m +

j "k

6.2.8 k=0 dj m=0 k

L e t

n = max { n, } 6.2.9 0<k<v

K

Therefore

L( xJ

l o gQ( x ) ) = r

1

am( j ) ( l o g

e( x ) ) x

m

+ an( j ) ( l o g

e( x ) ) x

n

n - 1

+ 2 x l D +

£ a (j)x1 1

6.2.10 . m=0 dj

m

dj n

where a (j) = p h(j,k) , 6.2.11 k

Therefore using the definition LQ.(x) = x^ and J

6.2.1,6.2.11 becomes

• n-1 L( x

J

l o gQ( x ) ) = Z a

m.(j)LQ®(x) +• a

Q( j )LQ®(x) +

m = 0 n - 1 2 -da

B I(J>LQ

T n(x) + jla (j)LQ (x)6.2.12

m=0 m

dj n n

and from the linearity of L ,

n 1 = 1 { . x J l o g

e(

X) - a

f f l(j)Q*(x)}

a

nU J m=0

n - 1 - ( 2 d a (j)Q (x) } • 6.2.13

anU ) - O d j

m

From 6.2.. 9 we know that

n > j 6.2.14.

and write

n = j +..H 6.2.15

where H 0- is an integer . 6.2.16

Therefore

j+H-1

Q ^ + h W = •{• xj

l o geC x ) - E a

m( j ) Q ® ( x ) }

•j+H

j+H-1

f 2 ( da (j) )Q (x) } 6.2.17 a

j + H( j ) m=0 d j

m

From the existence of { Q.(x) } . „ and 6.2.17 it follows

0 . the existence of { •

-84-

Theorem 6,2.3

9/ N * - , { Q

J( X )

> J « H0

s

< j f ~ > l - »0

Proof :

Let us state here equation 6.2.17 with H=0:

Q?(x) = _ 1 J xJ

'log (x) - V a (j)Q®(x)}

' a.(j) m = 0

j

j - l t

z

d a C.1)Q (x) } 6.2.18 a . (j.) m=0 dj j

From theorem U.Z.2

mv

m

i j -1

Q<(x) = 1 ( xJ

- Z am(j)Q

m(x) } 6.2.19 ' i — — — m

w

m r,-\ m=0

j

Differentiate 6.2.19 w . r . t . j

i J-l dSLS(x).. = 1 ( x

J

l o go( x ) - I ( d a (j)Q (x) +

..3

/.\ m=0 dj m m

3j a.(j) J

J

am(.i) .3Q

m(x) )} 6.2.20

3j

Therefore

j "1

a_Q. (x) = { xJ

l o go( x ) - E a (j) 3Q (x) }

3 j a.(j) m

- ° 3J

i J'"1

" -J

7 . . ( E } 6.2.21 a.(j) n-0

d.

From 6.2.18 and 6.2.21 the proof is complete.

The proof is unaltered if H ^ O . 6.2.22

-85-We can now prove the uniqueness of Theorem 6.2.2 easily.

To prove that

a . { Q.(x) }. „ is unique w i t h respect to the given

J J e l

o

differential operator L .

Proof:

From the uniqueness of

we have that

r 3Q.(x) \ is u n i q u e . 1

3jJ

'jeNd

From theorem 6.2.3 it follows that

{ Q?(x) }. N is unique.

J J n

APPLICATION:

xu" + u?

+ U = 0

• A

Q.(x) = xJ

- j Q . _1( x ) , j = 0(1)N is- generated directly ;

Q®(x) = x^ log (x) - j2

Q ® -(x) - 2 j Q . A x ) , j=0(l)N

J e J —-J- J ~-L

is generated directly, but differentiation of Q.(x) w.r.t. j gives J 3Q. (x)

=

x ^ l o g (x) - j2

3Q.(x) 2jQ. -,(x), j=0(l)N .

Therefore theorem 6.2.3 hol^s here.

Assume a solution of the form

oo u(x) = Z a . ( Q . (x) + _§a,(x) ). 6.2.31

j =0 J J

3j

We seek a polynomial function U ^ s u c h

that u is approximated by

N

U„(x) = Z a*( Q.(x) ) * Z a.Q.(x) = u(x) 6.2.31

where

Q.(x) = Q.(x) + £_Q.(x) ,j eN 6.2.32 J J

3j J °

and satisfies the .following boundary conditions

Un( 6 ) = U

6 , U

N(1) = u (1) 6.2.33

for

0 < 6 << 1

and 6 is taken as.small as desired provided it is

larger than the machine zero to be used.. Our numerical

experiments in single precision allows us to choose

-12

5=10 with extreme su.ccess and very rapid convergence

rate. We attempt to salve the following perturbed problem

L U N ( X ) = H N ( X ) , xefr.l] 6 . 2 . 3 4

UN( 6 )= U

6 II^(l) = u(1) 6.2.35

{ or the initial, value problem UN(l)=u(l) j

T

J^(l) =u'(l) }

Choose

v-1 N-k H N ( X ) = Z T , Z ( T * I _ ( x ) + , ( x ) ) 6 . 2 . 3 6 w

k=0 K

k=0 3j N

'K

w r i t e T*(x) = Z a(

.m )

x* 6

'2

'3 7

3=0 3

then

V - 1 N-k / vr vn

Uw(x) = Z t , Z a J

N

"k ;

( 5,(x) I 6 . 2 . 3 8

* k=0 j=0 3 J

This method has shown .high success on. problems with

indicial roots both equal to zero as well as

when they.are complex c o n j u g a t e s in which case

N

{Q.( ylog (x) ) }. „ w i t h U (x) = Z a*Q.(ulog (x) ), J e jEiN

Q i* .

=q j j e

and H N ( X ) = T 0 T | ( u l o gQ( x ) ) + T 1 T * _ 1 ( U L O G E ( X . ) ) ,

were u s e d .

Example: x u " ( x ) + u ' ( x ) - u ( x ) = 0

Let

N U,

T(x) = Z c,.( Q . ( x ) + 3 Q . ( x ) ) a

j=0 J J

9jJ

then

N N L U „ ( x ) = Z c.LQ.(x) + Z c.L3_Q (x) N

j = 0 3 J

j=0 J 3j J

N , N . = Z c.x

J

+ Z c . xJ

l o g (x) j=0

J

j =0 J e

= Hn( X ) .

It is therefore sound from. the. above analysis to choose

HN(X) = Tq{ T*(x) + ( l o g

e( x ) ) T | ( x ) } +

T

l{ T

N - l( x ) +

(l o

Se

( x ) ) T

N - l( x ) }

'

Observe that the basic a s s u m p t i o n for the form of

solution is the of p r i n c i p l e of s u p e r p o s i t i o n .

6.3 TRANSFORMATION. OF VARIABLES;

The problem we wish to solve is known to have a singularity,

say at x=:0^ - the- nature*..o-£-- which -follows, from. Lx^ ;

dur problem is to find a suitable, transformation t=s(x)

which removes such a s i n g u l a r i t y .

Transformation of independent variable;

We shall consider very briefly the possibility of using

a simple transformation of the independent variable x to

a new variable £ . Suppose we have a transformation

defined implicitly by

x = s(€) . 6.3.1

Given

du = f(x,u) 6.3.2 dx

we define a new unknown u(£) by

u(S) = u(x) - u(s(C)) . 6.3.3

Then we derive a differential equation for u(£),namely

du = ds ' f ( s U ) , u ( 0 ) . 6.3.4 dC dC

By a suitable choice of the transformation 6.3.1,the

equation 6.3.4 may be easier to solve than that of 6.3.2.

This is a standard procedure and was applied'

to the recursive formulation of the tau method using

£= ax + 3 where in this case s(£) = - 3) .

This was not successful because under any linear transform-

ation of this form the distance of the new singular

point remains the same. It is therefore necessary to choose

a suitable nonlinear transformation or resort to a

transformation of the dependent variable.

-89-

Transformation by series. Application to

a nonlinear BVP.

Given

u" (x) - 2 u3

( x ) = 0 , xe[6 ,2] see Cohen and Jones 1974.; 6.3.5

u(5) = 6"1

u(2) + u ' ( 2 ) = i 6.3.6

we adopt the method of section 6.1 and 6.2 to find the

behaviour of the solution about the singular point x=0 .

The general theory of regular singularity in the linear case

no longer holds h e r e .

Our method can detect any possible solution of the form Ax*'

and it is reasonable- to' apply, L( Ax^ ) = 0 to get

J (J -1)AxJ

' ~2

F 2A 3 X 3 J ' 6 . 3 . 7

Equating exponents of x

j-2 = 3j

giving

checking with the coefficients that

2A = 2 A3

yields A = + l .

So u(x) ~ x "1

at the singular point x=0 .

choosing

0 < 6 << i 6.3.8

the Newton scheme failed to converge with the recursive

Tau method.

So let

i 0 0

u(x) = x_ J

- + L a.Q. (x) 6.3.9 j = 0 J J

then use the following transformation

- -1 u(x) = u(x) - X

to give

x2

u»(x) - (2x2

u3

(x) + 6 X U 2 ( X ) + 6u(x)) = 0,xe[6,2] 6.3.10

u(6) = 0 u(2) + u'(2) = i 6.3.11

00

u(x) = I a.Q.(x) 6.3.12 j =0 J J

(6-3.10), 6.3.11) is now a regular nonlinear BVP and for any

choice of <$<4 may be solved by Newton scheme with the

recursive Tau method.

Therefore

u(x) = u(x)+x~^ solves the original problem.

The recursive Tau method with six Newton iterations

solved the given problem (6.3.5), (6.3.6) without any

transformation for 6=£. See table 6*3.I .

Table 6.3. I case 6=0.5 N=10

x

/ Method standard global tau Cohen&Jones 1974-

0.50 2.000000 2.000000

0.75 1.337678 1.33767

1.00 1.011456 1.0114.6 .

1.25 0.822925 0.82292

1.50 0.706696 0.70671

1.75 0.635709 0.63570

2.00 0 .59734-6 0.59733

6 Newton iterations

without any transf-

ormation

Table 6;.3 II case 6*0.1 N=9

x / Method standard global tau Transformation b.^.y applied •

0.10 10.0 ' 10.0

0.15 6.52762061 6.66670381

0.20 4-. 70730764. 5.00009800

0.25 3.78317063 4.00001942

0.30 3.29231816 3.33336696

0.4-0 2.69829691 2.50079856

0.50 2.1324674.9 2.00156155

1.00 1.10858508 1.01249955

2.00 0.59681340 0.59999997

Observe that u(x)' behaves like: near x=0,. more closely in

the transformed problem.

-12

For the case 5—10 , N —9 > »the untransformed problem

gave a divergent approximation while the transformed problem

showed - " the required behaviour.

6.4 NUMERICAL EXAMPLES

Example 6.4.1

Consider a singular linear initial value problem

xu"(x) + u»(x) + u(x) = 0 , x e [ l 0 "1 2

, l j

u (1) = 0.89635374.6

u'(l)=-7.570406420

u(x) = ( l+£log (x))P(x) - R(x) 6

P(x) = 1 - X + x' X" + . . . (2l)

2

(31) n *

R(x) = - x + 3 x _ o ~ 11 x

o + ... 2 u r j

2

T W )2

Table 6.4.1 I absolute. relative errors

6.4.1

6.4.2

6.4.3

6.4.4

6.4.5

6.4.6

Method Global Tau N=7 Gear Tol=10 .-10

Global Tau N=7

1.0E-12

1.0E-08

0.1

0.2

0.3

0.4

0.5

0.989 x 10

0.991 x 10

0.303 x 10-

0.311 x 10"

0.167 x 10

0.329 x 10

0

0

1

0

0.180 x 10 0

* * * * * * *

0.173 x 10

0.463 x 10

0.428 x 10

0.207 x 10

0.130 x 10

0.860 x 10

-9

- 8

-9

-9

-9

-10

0.592'.x-10

0.599 x 10

0.472 x 10

0.397 x 10

0.112 x 10

0.874 x 10

0.623 x 10

-9

-9

- 8

-10

-9.

-10

-10

Effort

1.422 cp sees arithmetic = double preci precision

3.043 cp sees single precision

0.819 cp' sees single precision

v-1 N-k ^ H,

T(x) = Z T-i Z (1+log (x))T* ,(x). , xe p.0" ,lj6.4.7

k=0 ^ j=0 e

The results of Table 6 . 4 - 1 . I show rapid convergence of

the improved method and for the same accuracy with Gear's

method used less than 1/3 the computer time used by'Gear'.

This illustrates that the use of appropriate singular function

detectable within the context of the numerical procedure

is more efficient than local mesh refinements for singular

initial value p r o b l e m s . This is also true for singular

boundary value p r o b l e m s . The improved method can

- 1 2

approximate as close to x = 0 as 1.0 x. 10 and even more

provided we avoid stepping at the Pole (i.e. machine zero)

in single, p r e c i s i o n . If there are more than one singular

points in [0,l] we- use piecewise Tau method with this

approach. The comparison with'Gear'is fair because in this

example, the singularity is near the end of the range of

integration and !

Gear'uses local mesh refinement around the

singular point,which is a form of singularity treatment.

If the singularity lies within the range of integration then

the program o fr

G e a r ' s h o u l d be made aware of the existence of

such point for a fair c o m p a r i s o n .

F.yample 6.4.2

Consider a singular l i n e a r boundary value problem slightly

similar t-o the last example . The' particular solution required

is known a p r i o r i (or by failure of .the standard method) to

have a singular b e h a v i o u r .

-94-xu" (x) + u' (x) - u(x) = 0 , x e [ i 0 ~

1 2

, l ] 6.4.8

u ( 1 0 "1 2

) = 0.5E+08 u(l)=0.8498791 6.4.9

u(x) = (1+ilog (x))P(x) - R(x) 6.4.10 e

P(x) = 1 + x + x2

+ x3

+ ... 6.4.11

(21)2

( 3 ! )2

R(x) = x + 3 x2

+ 1 1 y? +

6.4.12

2(21)2

(3 I )2

A singular point at x = 0 . L e t NPTS =11,the number of output

points, e

m a x( x ) = m a x

( | UN( x . ) - u(x)|} =0.38 x 1 0 "

8

, fo r i=l(l)NPTS

6<x. <1 J

J

-12

6= 10 ,N=7 . A comparison with the global standard tau

method is out of question as far as accuracy is concerned, gg

the latter leads to a divergent approximation.

C H A P T E R S E V E N

A POSTERIORI ERROR ESTIMATION AND A DIFFERENTIAL

CORRECTION IN. THE TAU METHOD

Theoretical results on the convergence of the Tau method

are not easy to apply in day to day computation. For this

reason, a practical approach is followed which enables us

to provide the user of our program with an estimate of the

order of accuracy of the numbers turned out by the computer

This approach is then related to correction techniques

for differential equations and as described in section 4.4

to the problem of adaptive control of the segmented process

7.1 ERROR ESTIMATE AND A DIFFERENTIAL CORRECTION

Given that N

00

n ( x ) = Z a*Q.(x) * I a Q.(x) = u(x) 7.1.1 N j _ Q J J J - Q J J

Let eN( x ) = U

N( x ) - u(x) , xe[a,b] 7.1.2

Definition 7.1.1 t h

Call e (x) the "N order Tau error function" which is

represented by an ".nfinite series 7.1.3.

From 7.1.1, 7.1.2 N

00

eM( x ) = Z ( a* - a. )Q.(x) - Z a.Q.(x) . 7.1.3

N j=0 J J J

j =N+1 J J

and from applying L to 7.1.2

L eN( x ) = H N ( X ) , x

£C a , b ] 7.1.4

( g y ^ . . . . . ^ ) = 0, j =0(l)v-1 . 7.1.5

Like the original problem we proceed in the same manner to

estimate with the method the function e^(x) which is

given by 7.1.3.

An analytic approach to the problem 7.1.4, 7.1.5 will give

algebraic error estimates or bounds on the T 'S and

hence on the error in the approximation^ ( see Lanczos 1 9 3 8 , 1 9 5 6 ) .

For a general purpose program we follow a practical

approach. We seek a polynomial, approximation e^

such that M > N and

M 00

ew M

( x ) = Z bfQ.(x) * Z b Q (x) = e„(x) 7.1.6 i N

'1 1

j = 0 2

j = o J

Using 7 .1.3

M N M

e¥ (x) = Z b*Q,(x) « Z (a* - a.)Q.(x) - Z a,Q,(x) 7.1.7

J = 0 J J j = 0 2

j = N + 1 2 2

writing N M

e„ M( x ) = Z (a* - a.)Q.(x) - Z a.Q.(x) , xe[a,b] 7.1.8

1=0 J J J

i=N+l J 2

gives an "error estimate whichr includes round-off errors

and the dominant, terms of the. truncation. Numerical

results show that, it is a very accurate estimate of e^(x),

particularly when M>>N .

To relate this error estimate, to a correction result, write

7.1.8 as N M

e (x) = Z a * Q . ( x ) - Z a.Q.(x), M>N, xe[a,b] 7.1.9 ' j=0

J J

j=0 J J

Then 7.1.9 gives

L E N > M ( X ) = ( f ( x ) + H N ( X ) ) - ( f ( x ) + H N ( X )

= H N ( X ) - flM(x) . 7 . 1 . 1 0

so we solve the perturbed problem

L E N Y M ( X ) = H N ( X ) - H M ( X ) , x e JTa, B] 7 . 1 . 1 1

(

' e

N , M ' e

N , M * * * * ' M^" ^ = 0 ' 7 . 1 . 1 2

Until now no attempt has been made with the tau method to

satisfy a user prescribed accuracy. To do this, one

computes U (x) and eN M

( x ) for a suitable M > N . Though

only a little extra work is done in generating . . . . . x ) ,

still one has to invert another matrix for e^ •

-97-

A more efficient approach, is obtained using the following

new result.

Theorem.7.1>1 (A differential correction)

The solution of the p e r t u r b e d problem 7.1.11, 7.1.12, gives

e

N , M ^ =

" »X £

Ca

»b

] . 7.1.13

We will need the fallowing Theore.m for the proof of

theorem 7.1.1.

Theorem 7.1.2 (Uniqueness theorem)

Let u(x) be the unique solution, of a given, differential equation

for xeT-R , +R 1 : Then <» a a

u(x) = £ a.Q.(x) - 7.1.14. j =0

J

3 C- ' is the only representation of it in -R < x <R ; . a a

Proof: Assume for contradiction that

00

u(x) = Z b.Q.(x) -R, < X < R, 7.1.15 j =0 i b b

then

u(x) = Z a.Q.(x) = Z b.Q.(x) -R < x < R 7.1.16 j-0 J J j=o J J

where R is the smaller of R and R, containing the origin. a D

CO . 00 Writing Z a x

J

= Z S.xJ

, -R < x < R 7.1.17 j = 0

J

j=0 J

and setting x = 0 to eliminate all but the constant terms,

we get aQ = B

q 7.1.18

For differential equations, R = = R^.

Differentiating once gives CO _ oo

Z j a.x^ ~ = Z j b . x ^ '1

7.1.19 j =1

J

j =1 J

and seting x=0 gives

a

l = b

l 7.1.20

-98-

By repeating this process N times, gives

a., = 15 vc 7.1.21 N N

Therefore N

- i N

- 1 IL

T(X) = Z a . X

J

= Z b.xJ

, -R • < x < R 7.1.22 N .

= Q J j=0 J

a a

Proof of. theorem 7.1.1

L( UN( x ) - U

M( x ) ) = H

N( X ) - H

M( X ) , xe[a,b] 7.1.23

Applying theorem 7.1.2, and choosing H^(x) and H^(x) from

the same class of p o l y n o m i a l s , gives that

e

N , M( x ) E { U

N( x )

" Vx ) }

' X £

£a

'b

3 ' 7.1.25

since we are attempting to solve the same problem {7.1.11,

7.1.12} and{ 7.1.23, 7.1.24-} • The proof is complete.

The result 7.1.25 means that solving an error differential

equation for estimating e^(x) may be b y - p a s s e d . —

Every time., two successive Tan approximations U^Cx), U^(x)

are .computed an estimate of the p r e v i o u s tau..approximation U^(x)

with 7.1.25 .is obtained.. We call e^ "the first tau estimator

of and. in all cases of numerical experimentation

with u(x) sufficiently smooth in [a,b], the order of

accuracy and sign of e^(x) were always ensured.

To satisfy a user's accuracy requirement, the program

produces an approximation of order N,say,and produces an

error estimate of this approximation. If the requirement is

not satisfied then increases the value of N for a higher

order approximation. At this stage you have solved the diff-

erential equation twice and an error equation once. Instead

use 7.1.25 after two''successive approximations with the given

differential equations and repeat until the requirement is sati-

sfied.

7.2 NUMERICAL EXAMPLES

Example 7.2.1

uI V

( x ) + u" (x) = 0 x e [ 0 , JirJ 7.2.1

u ( 0 ) = 0 U'(JTT) = 0 7 . 2 . 2

u" (0) - 5 u ' ( 0 ) = 0 u» 1 f

(4TT) - 50U(4TT) = - 0 . 2 5 7 . 2 . 3 1

"1

u ( x ) = ( 444 - IOOTT )~±

( 1 - x - e o s (x) - 1 . 2 s i n ( x ) ) 7.2.4-

Describes a specific case of a Bernoulli column on an elastic

foundation where u(x) measures the lateral deflection.

The error estimators are shown in Table 7.2.1 I .

Table 7.2.1 I

N M • Tau Estimator: Tau Exact Error

5 6 0.249 x l O "5

7 0.246 x l O "5

8 0.2472 x l O "5

20 0.247054 5 1 9 3 3 7 x l 0 "5

0..247054 519337 5250x10 -5

10 11 0.314 x l O "1 1

12 0.3130 x l O "1 1

13 0.312936 m - H xlO

22 0.312937 1 6 0 2 9 9 X 1 0 "1 1

0.312937 160299 5814x10 -11

12 13 0.306

14 0.3047

15 0.30466

22 0.304466612558 x l O "1

^ 0.304466 612558 5986x10 -14

100-

20 21 O . U x l O "2 7

22 O . U x l O "2 7

23 O . U x l O "2 7

25 O . U x l O "2 7

O . U x 1 0 "2 7

Succes&ive estimations confirm the accuracy of the

sign and order of the error from the first Tau estimator.

(x)

Also the first Tau estimator e^ jj-j-2. gives an accuracy

upto the first two digits of the significant numbers.

We will now consider Falkner-Skan equation Example 3*3.2

to illustrate here the p o w e r of our error estimation.

For a nonlinear problem we solve a sequence of approximate

linear problem converging to the original problem. The

error analysis is performed for each iterate and the

results reported are for the last iterate of the sequence.

Example 7.2.2 (Falkner-Skan equation)

In Table 7.2.2 I, we give the error of Tau approximations

of degree N=15,17,19 of the second derivative at x= 0

using the Tau estimator e" M

( 0 ) , with 3=2 (see Na 1979).

Tab] Le 7.2.2 I

N Tau Exact error M Tau estimator e" M

( 0 )

15 6.4- x 1 0 "5

i 2 0

6.5 x 1 0 ~5

17 1.6 x 1 0 "5 ; 2 0 1.7 x 1 0 "

5

19 3.2 x 1 0 "6

: 2 0 4.1 x . 1 0 "6

-101-

-6 Note that ej^ 2 0 ^ ^

=

4*-lxl0~ already gives the order of

- 6 - 6 accuracy 10" of the true Tau error 3.2 x 10" = e!j^(o)»

This result is extremely successful for 3=2 of this problem.

In Table 7.2.2 II we r e p o r t values of u"(o) obtained and

compared with the method of Invariant Imbedding given in Na 1 9 7 9 ,

for 3=1.0,1.6,1.8, 2 . 0 .

Table 7.2.2. II

3

Invariant Imbedd-ing reported irj. Na 1 9 7 9

Standard values ' reported in

1 9 7 9 .

Tau of degree

20

1.0

1.6

1.8-

2.0

- • 1.2.42.3400

1.5327980

1.2325870

1.5215120

1.2325875

1.5215139

1.6064802

1.6872160

1.0

1.6

1.8-

2.0 1.6995770 1.6872170

1.2325875

1.5215139

1.6064802

1.6872160

For 3=2, most initial value methods (shooting methods)

would fail as the problem is now unstable and therefore

numerically difficult. Therefore the result for invariant

Imbedding reported in Na 1979 and quoted here is quite good

for boundary valu e p r o b l e m s . Our results for all cases of

3 , and in particular 3 = 2 are very accurate. In Aziz 19'75,

Pereyra reported his results and remarked that H.B. Keller

succeeded only upto 3=1*8 for his shooting method. In

conclusion therefore, the recursive, formulation of the Tau

method with the error estimation of this chapter is a

US91 li 1 method for solving with highly accurate error

estimates nonlinear and linear boundary value problems in

ordinary differential equations • . - -,

-102-

The error estimates and differential correction result

apply equally to initial value problems and piecewise

polynomial Tau approximations in the differential or

Integrated form... In the n e x t example taken from Oliver 19&9

we compare our error estimates with those of Oliver who

based his.own estimation only on. the dominant terms of

the truncation of the direct Cheheyshev series replacement.

Example 7.2.3 (see Oliver 1969)

uI V

( x ) - 401u"(x) + 4-OOu(x) = -1 + 2 0 0 x2

, xe[0,l]

u(0) = u' (0) = u" (0) = u " 1

(0) = 1

u(x) = 1 + 4(x* + ex

- e ~x

)

7.2.5

7.2.6

7.2.7

The coefficients b* in lJ~ (x) = Z b*T*(x) r=0 r r

7.2.8

conversion from Tau approximation and those obtained by

direct Ghebeyshev.series replacement are given in Table 7.2.3 I .

r

Converted Tau Oliver 196-9

0 1.741704 1.741704

1 0.831660 0.831660

2 0.957713 X 1 0 "1

0.957713 X 1 0 -1

3 0.596699 X -2

10 * 0.596699 X 10 *

4 0.168802 X L O -3

0.168802 X 1 0 "3

5 0.169074- X 10"4

- 0.169074 X 10"^

T = 0(10' - 3 )

-103-

The two methods give the same accuracy.in the approximation

U j j(X) of u(x), for N = 5 . In the next table we compare the

error estimates given, by our method and that of. Oliver which

show that our error estimates are 'more- accurate.

Table 7.2.3. II error estimates

N M Converted. Tau Oliver 1969 I

5 6

7

0.8361 x 10"4

-

0.8727 x 10-4

1.46 x 10"4

-

1.18 x 1 0 " ^

True Errors 0.8722 x 10"4

- 0.872 x l O " ^

T

= o ( i o- 4

- )

The error estimates given here. and';

the correction results

yield unsatisfactory convergence in the neighbourhood of

th

a singularity because the N order Tau error function

e^(x) has the same singular behaviour as the solution

function u(x), this becomes obvious to the user after a

check on the. size of the T - t e r m s . However, when the

singularity in the solution u(x) is treated (as in chapter

six) and the approximation, has. a rapid, convergence, then

the Tau error estimates are also rapid and satisfactory as

the correction r e s u l t s . F o r rapid convergence, it is

therefore necessary that both the T'S and the coefficients of

the Chebeyshev series representation tend to zero fast.

We illustrate this p o i n t w i t h the following first order

initial value problem also taken from Oliver 1969.

-104.-

Example 7 . 2 . 4

3 ( l - x ) u ' ( x ) +u(x) = 0, x e [ - l , l ] 7.2.9

u ( 0 ) = 1 7.2.10

u ( x ) = ( l - x )1 / 3

7.2.11

The p r o b l e m has a s i n g u l a r d e r i v a t i v e at x = l . The error

estimates of O l i v e r 1 9 6 9 arid the present, a p p r o a c h were

both u n s a t i s f a c t o r y even for N=10,. M = l 6 . In our a p p r o a c h

N = 1 0 , M = 2 6 still .unsatisfactory. Observe that in the two

Tables 7.2.-4rI>H» the c o e f f i c i e n t s of the Chebeyshev series

r e p r e s e n t a t i o n s do n o t go d o w n to zero, fast and the tf

s

too do n o t go.to zero f a s t even for M = l 6 .

Table 7.2.4 I

r Oliver 1969

Converted Tau

0 0.8990357 0.8990357

1 -0.4495179 -0.4495179

2 -0.1284337 -0.1284337

3 -O.6421684 X 1 0 -1

-0.6421684 X 1 0 -1

4 -0.3951805 X 1 0 -1

-0.3951805 X 1 0 "1

5 -0.2716866 X 1 0 -1

-0.2716866 X I D "1

6 -0.2001901 X. 1 0 -1

-0.2001901 X I D "1

7 -0.1546924 X 1 0 -1

-0.1546924 X I D '1

8 -0.1237539 X 1 0 -1

-0.1237539 X I D '1

9 -0.1016550 X 10""1

- 0 . 1 0 1 6 5 5 0 X l O "1

10 -0.4405050 X 1 0 "1

-0.4405050 X 1 0- 2

Table 7.2,.4. II Error estimates

N M Converted Tau Oliver 1969 .

10 12 1.473 x 1 0 ~2

1.23 x 1 0 "2

16 3..4U x 1 0 ~2

2.43 x 1 0 ~2

20 4.733 x 1 0 "2

26 6.023 x 1 0 ~2

True Errors 1 .277 x 1 0 " 1 1.28 x 1 0 "1

For M=l6

^ = 0 ( 1 0 "1

)

-106-

C H A P T E R E I G H T

A V C O L L Q C A - T I O I Y . ---• TAU- M E T H O D

A new approach to the Lanczos T-method is presented here,

based on a collocation of the perturbed equation for an

approximation of the solution of ordinary differential

equations. An error estimation of the method is given with

a differential correction result. Numerical examples are

used to illustrate the method.

1 I N T R O D U C T I O N

The method presented here attempts to generalise the Lanczos

x-method to problems with general coefficients and right

hand side. The T-method of Lanczos requires that the

coefficients and the right hand side of the differential

equation be polynomials or rational functions .The method

of selected points (orthogonal collocation)on the other

hand removes this restriction but ignores the knowledge

of the T's entirely (see Lanczos 1938,1956). - For purpose

of a reliable and accurate g l o b a l error, estimation the

simultaneous determination of-the coefficients of the

expansion a n d the Tl

Sr is required and this is the reason

for the work of this phapter.

The T-method:A c o l l o c a t i o n a p p r o a c h

Consider a 'linear d i f f e r e n t i a l operator L of order v ,

v *

L = l P . ( x ) d H _ • / j=0 ^

d xj <

Given

L u ( x ) = f(x) , xe[a,b] ( l )

W i t h the initial or b o u n d a r y conditions represented for

convenience in the form

V .I B.(u,u»,...,u =

a. , j = 0 , l , . v - 1 . ( 2 )

J J

A s s u m e

N U,

T(x) = Z a*x

J

- £ a.xJ

= u(x) . ( 3) N

j =0 J

j=0 J

Then

L UN( X ) = F ( X ) + H

N( X ) , x e [ a , b ] ( O

= a , j = 0 , l , . . . , v - i ( 5)

w h e r e

r n EL,(x) = Z T

T T* Ax) , x e [ a , b ] . ( 6)

k=0 k

Collocating (4-) at (N+l) e q u a l l y spaced interior points

of [a,b]I in addition to the v conditions (5) to be

satisfied by U ^ will lead to (N+l+v) algebraic equations

for the unique d e t e r m i n a t i o n of the (N+l+v) parameters

aJ , xk , j=0,l,...,N; k = 0 , 1 , . . . , v - 1

The p e r t u r b i n g o r t h o g o n a l p o l y n o m i a l s are here taken to

be the shifted Chebyshev p o l y n o m i a l TjJ^(x) appropriate to

the interval £a,b] .

-108-

Thus

= -- =

-

where

A=

T

T B = ( . . . , a

v^

1, f ( x

1) , . . . , f ( x

N + 1) )

and A is given by

NN . */__ \ *

(L 1

) ! x = Xh + 1

( L x )

I X = Xn + 1

( L x N )

I x = Xn +"

T

H( x

N + 1)

"T

N - v+l W

+ 1)

* • * . . . - * 0 0

whereyv rows denoted by * are associated with the initial or

boundary conditions of the problem to be solved.-

-109-

3 Error Estimation

L e t

eN( x ) = U

N( x ) - u(x) , xe[a,b] ( 7)

N CO Z (a* - a.)xJ

- Z a.xJ

( 8) j = 0

J J

j =N+1 3

and from the linearity of the operator L applied to (7)

L eN( x ) = H

N( X ) , xs[a,b} ( 9)

B

j ^e

N, e

N ", , , e

N "1

^ = 0

' j=0,l, .. .,v-l (10)

Assume a polynomial approximation E^ ^(x) of degree M>N

such that

EN M

( X ) = Z b*x3

- Z B.xl = eN( x ) (11)

JN,M j = 0 J

j =0 J

and writing (ll) as

N M E m w ( x ) = Z (a* - a.)x

J

- Z a.xJ

, xe[a,bj (12) j=0 J

J

j =N+1 J

gives an error estimate w h i c h includes round-off errors

and the dominant terms of the truncation.

From (12),

N , M . E

w M( x ) = Z afx

J

- Z a.xJ

, xe[a,b] (13) j =0 J j=0 J

Applying the operator L to (13) gives

A

L EN > M

( X ) = HN( X ) + H

M( X ) , x e [ A , B ] ( U )

^ ^ N . M ' ^ . M " - - ' ^ " ^ = 0

J "0

'1 ( 1 5 )

Collocating the perturbed error equation (14) as before

and satisfying (15) gives a polynomial approximation

% ^r u e e r r o r

function e^(x) of UN( x ) .

-110-

Definition

+ V» ^N ^ " Estimator

a n <

*

is the first tau E s t i m a t o r ,

As a consequence of the a b o v e analysis of the error,the

following result is o b t a i n e d w h i c h has been reported for

the recursive tau method ,in the-last -chapter which is equally

true for the p r e s e n t a p p r o a c h based on collocation p r i n c i p l e .

Theorem

U

N + p( x ) =

Vx )

" E

N , N + p( x )

- ' *e

& 'b

l ( l 6 )

p>0 is an i n t e g e r .

Proof: r

L | ^ Un( X ) - U

N + p( x ) J = H

N( X ) - H

M( X ) , xe[a,b] (17)

whetfe M=N+p .

O b s e r v i n g that«juN(x) - satisfies the same homogeneous

conditions (15) and choosing HM.(x)= -H

M(.x) gives the

result UN( x ) - U

M( x ) =

E

N > M(

X

) > a n d

(16) follows.

This result may be used in an attempt to satisfy a user's

accuracy requirement as d e s c i b e d in the last chapter.

W e refer to (16) as a d i f f e r e n t i a l correction .in the context

of the tau method,see S t e t t e r ^978 and Z a d u n a i s k y1

* ^ .

-111-

4 CANONICAL P O L Y N O M I A L S

W h e n P . ( x ) , j = 0 » l » . . . a r e all polynomials and f(x) is J

n o t , the canonical p o l y n o m i a l s A n p a r t i c u l a r its recursive

generation (see Ortiz 1969,1974.) is suitable to this a p p r o a c h .

L e t

= A 5

jo- jio A w 0 = u ( x > where

LQ (x) = xJ

, j = 0 , 1 , 2 , ... j

Then a p p l y i n g collocation to the perturbed equation as

before yields

A C = B

and in the matrix A r e p l a c e Lx^ w i t h L Q . ( x ) , j = 0/l,...,N

w h i c h by d e f i n i t i o n is the same as x3

.

If one or more P . ( x ) is n o t a rational function then the J

use of Canonical p o l y n o m i a l s is d r o p p e d .

5 NONLINEAR Problem

For n o n l i n e a r problems a l i n e a r i z a t i o n scheme is used w i t h

this a p p r o a c h . N e w t o n method is a very effective choice

and is used for all the n o n l i n e a r problems of the next

t i o n . W r i t i n g U ^ = d£u ; ^ ^ ( x , ! ^ , ! ^ , . . . ,

Let

S S C ^ w m •• j. -L. is — ^ w f r g —

dxJ

'

$ ( x , u , uT

, . . . , uV

) = f(x) x e [ a , b ] (19)

t h be a given n o n l i n e a r v o r d e r equation, then Newton scheme is

- = - \ + V x ) (20) j a u ^

;

5' NUMERICAL RESULTS

NPTS denotes the number of output points,of

=

a < xX

< b ( | U

N( X

J)

" U ( X

J) | }

- J-1< j <NPTS

E1

= max {IE (x)I} m a x

a<x.<b ' % N + l l

!<•'• j <NPTS

= max (|U„(x.) -,.U 7

( x . ) | } a < x . <b "

J r ± J

1< i <NPTS

E2

= x m a x

a"x*<b {

l % , N + 2 (x

) l> - J-

1< j <NPTS

1< jJ

< N P T S

All computations reported here were done on the

Imperial College Computer CDC 6500/ Cyber 17b, in single

precision.

N denotes the degree of the approximation U^(x). The time

is given in cp sees.

\ U ) = ( V Tkx

k

) T g _p( x ) , p=v-l

2

Vx ) 5

2 oT

kT

S - k( x )

-113-

P r o b l e m 1 : Fox and P a r k e r 1968

u»(x) ' + (sin(x) )u(x) = ex

, XE[0,1]

u(0) = 1 u(1) = 0

Table 1.1

Coefficients of UM( x ) = Z a T*(x) ; N=2,3,4

* r=0 r r

•: r - Reference a

r p r e s e n t method Zrr / \

a-,.

0 0.4.06 0.3949 1 -0.500 -0.5000 2 0.094. 0.1051

0 0.395 0.3945 1 -0.509 -0.5092 2 0.105 0.1055 3 o . 009 0.0092

0 0.402 0.4013 1 -0.509 - 0 . 5 0 9 2 2 0.098 0.0977 3 0.009 0.0092

4 0.001 0.0009

There is a good a g r e e m e n t between the p r e s e n t method and

the method of selected p o i n t s given in the r e f e r e n c e .

The coefficients for N = 9 , 1 0 are'given-irr table 1 . 2 . to show

the rapid convergence of the solution and the T'S.

The direct or recursive L a n c z o s ' T-method w i l l require the

a p p r o x i m a t i o n of the c o e f f i c i e n t to be able to solve this

p r o b l e m , see P r u e s s 1 9 7 3 .

-114-

Table 1 .2 Mx

) = 2 a T*(x) N

r= 0

r

r N = 9 a r

N=10 a r

0 0 . 4-0134.111176826 0 . 40134111177812

1 - 0 .50945523095494 - 0 . 50945523095484

2 0 .9772050822262E -01 0 . 977205082472782 -01

3 0' .9525457082227E -02 0 . 95254570830197E -02

4 0 .9419617362967E -03 0 . 94196175922048E -03

5 - 0 .7060151206413E -04 - 0 . 70601512995942E -04

6 - 0 .3636540913098E -05 - 0 . 36372058850010E -05

7 0 .3751970452435E -06 0 . 37519759052290E -06

8 0 •5481373407307E -07 0 . 55730498334590E -07

9 0 .1877366852998E -09 0 . 18722690026250E -09

10 - 0 . 30923622144760E -09 T = 1.3611673E-0

08 T c

= -2.8558536E- 09

T 1 = 4.5711283E-07 ,

T

1 = 1.4001376E- 08

The computation of approximations from N=2 up to N=10

required with the present method 0.747cp sees, in all.

Table 1.3

Coefficients of EL (x) ; N = 2, M = 3,4 ; NPTS=11 M „ '

Y M ( x ) = sn v ; ( ' x )

r=0

r b r

of E 2, 3 ( x )

br of E

2 a( x ) e

max

0 0.0004 -0.0064 1 0.0092 0.0092 2 -0.0004 -0.0074 3 -0.0092 -0.0092 4 -0.0009

E1

= max

1.36 x 10 * E

2

=2.37 x 1 0 "2

max 2.42 x 1 0 ~

2

-115-

P R O B L E M 2 : O l i v e r 1969

u( I V )

( x ) - 401u»(x) + 4 0 0 u ( x ) = -1 + 2 0 0 x2

, xe[0,l]

u(O) = 1 = u1

( 0 ) = u»(0) = u ' " ( 0)

u(x) = 1 + i ( x2

+ ex

- e "x

)

NPTS = 11 for tables .2.2 a n d 2.3

1

Hn( X ) = ( T

O+ T

I X+ T

2X

2

+ T3X

3

) T * _3( X )

2

Hn( X ) = T

qT * ( X ) + ^ T j ^ f x ) + T

2T *

- 2( X ) + T

3T * _

3( X )

Table 2 . 1 N

Coefficients of UN( x ) N=5; U (x) I a T*(x)

r O l i v e r1 9 6 9 R e c u r s i v e and

P r e s e n t M e t h o d Recursive and P r e s e n t Method

Truncated Cheb.series

V x ) 2 y * >

0 1 .741704 1 .741704 1 .7416762 1 .741676

1 0 .831660 0 .831660 0.8316163 0 .831616

2 0 .095771 0 .095771 0 .0957518 0.095752

3 0 .005967 0 .005967 0 .0059654 0.005965

4 0 .000169 0 .000169 0.0001720 0.000172

5 0 .000017 0 .000017 0 .0000186 0 .000018

'Time 0 .13 0 .14

Table 2, .2

E r r o r estimates for U ^ C x ) using ^H^Cx)

Recursive form P r e s e n t method O l i v e r1 9 6 9

E 1 max

E 2 max

emax

0 . 8 3 6 1 x 10'4"

0 .8727 x lO"4

"

0.8722 x 10"4"

0 . 8 3 6 1 x lO"4

"

0 .8727 x lO"4

"

0.8722 x 10"4"

1 . 4 6 x lO"4

"

1 . 1 8 x 10~ 4

0.872x 10"4"

-116-

Table 2.3 p

Error estimates for U,(x) using HM( x )

Recursive Tau Present Method

E1

max 0.1735 x 10"4

- 0.1730 x 1 0_

4

E2

max 0.1729 x 10"4

- 0.1729 x 10"4

-

fi

max 0.1730 x 10"* 0.1730 x 10"*'

Table 2 . 1 shows that the p r e s e n t approach and the recursive

m e t h o d , C0 r t i z

1969,1974] g i y e i d e n

t i c a l results. With the

choice 1

Hn( X ) the two forms of approach give identical results

with those obtained in O l i v e r 1969 . However the error

estimates are more accurate than Oliver's estimates,see

o

Table. 2 . 2 . Table 2.3 shows an improved accuracy when H^(x)

is used instead of "^ILT(x) .

-117-

P r o b l e m 3 : K.Wright 1964

u'(x) = (1 - u2

( x ) )i

, x e C 0 , l ]

u(0) = 0

u(x) = sin(x)

S t a r t i n g with UQ( x ) = 0 ; N P T S = 1 1 ; N = 6

th Table 3.1 gives the results at .the-"end'of. the-5 i t e r a t i o n .

Table 3.1

N TJ„(x) = I a*T*(x) N

r=0 r r

r

Method of selected points using zeros of

T*(x);a* N r

P r e s e n t method u s i n g

r

0 0 . U 9 9 2 6 3 9 0.44992639

1 0.42522115 0.42522115

2 -0.02934470 -0.02934470

3 • -0.00449977 -0.00449977

4 0.00015412 0.00015412

5 0.00001411 0.00001411

6 -0.00000032 -0.00000032

Time=0.46; e = 0 . 5 2 2 x l 0 "7

max Time=0.50; e

m f l = 0 . 4 2 7 x l 0 "

7

max ~

E ^a x= 0 . U 9 x l 0 -

7

Tq = 0 . 5 6 2 X 1 0 "°

For N = 7 , the same a g r e e m e n t holds as in Table 3*1. For

N = 8 , the coefficients agree upto 11 decimal places in the two

m e t h o d s . In this result and several other numerical exper-

iments, the two methods give identical results for first

order equations involving only one perturbation term.

Therefore, for first order systems of ordinary different-

ial equations the p r e s e n t method and the method of selected

points will give p r a c t i c a l l y identical results.

-118-

Problem 4 : Scott 1975

u» - eu

= 0 , xe[0,l]

u(0) = 0 = u(l)

u(x) = -ln2 + ln(asec(£a(x-£)))

a = 1.33605569490610

Initial Guess: x(x-l)

With N=12; NPTS=11 and a f t e r 3 Newton iterations:

E1

= 2.2200 x 1 0 ~1 3

in 0.840 cp sees max

v

E 2

= 1.4566 x 1 0 "1 3

in 0.944 cp sees max

e max

= 1.6520 x 1 0 "1 3

in 0.748 cp sees

Scott = 1 . 5 x 1 0 "1 1

in 0.77 cp sees

The first global error estimate E"*" is very accurate in an max

overall time of ( 0.840 + 0.748 ) 1.588 cp sees.

R E F E R E N C E S - 1 1 9 -

1 Albasiny, E . L . and H o s k i n s , W . D . 1969 Cubic splines

solutions to two-point boundary value problems,Corap. J .

12,151-153.

2 Ascher,U., Christiansen, J . and Russell, R. D . 1979 A

collocation solver for mixed order systems of boundary value

problems, Maths. C o m p . 33, 659-679*

3 A z i z , A . K . (ed.) 1975 Numerical solutions of, boundary value

problems for ordinary differential equations, Academic press.

4 Bailey,P. ,Shampine,L.F.. and W a l t m a n , P 1 9 6 8 Nonlinear

two point boundary value problem, Academic press.

5 Bailey,P. and W a l t m a n , P . 1966 On the distance between

consecutive zeroes for second order differential equations,

J . Math.. A n a l . A p p l . ,14,23-30.

6 Barrett,J.W. and Morton,K.W. 1978 Optimal finite element

solutions to diffusion - convection problems in one

dimension, University of Reading,Numerical Analysis Rep.3/78.

7 Birkhoff,G. and Saunders Maclane 1966 A survey-of Modern Algebra.

8 Brabston,D.C. and K e l l e r , H . B . 1977 A numerical method for

singular two point boundary value problems, SIAM J .

N u m e r . Anal.,14,779-791.

9 Crisci,M.R. and Ortiz,E.L. 1981 Existence and convergence

results for the numerical solution of differential equations

with the tau method, Imperial College Res. R e p . 1 - 1 6 .

10 Clenshaw,C.W. 1957 The numerical solution of linear

differntial equations in Chebyshev series, Proc. Camb.

P h i l . Soc. 53,134-149.

-120-

11 Clenshaw,C.W. 1964 A comparison of the 'best' polynomial

approximations with the truncated Chebeyshev series

expansions,SAIM J . for Numer.Anal. series B,1,26-37.

12 Clenshaw,C.W. 1966 The solution of Van der Pol's

equation in Chebyshev series, in numerical solution of

nonlinear differential equations ,Greenspan,D.(ed.)

Wiley and sons, 55-63.

13 Clenshaw,C.W. 1962 Chebyshev series for Mathematical

functions,National P h y s i c a l Laboratory Mathematical

Tables, 5,London:H.M.S.O.

14 Clenshaw,C.W. and N o r t o n , H . J . 1963 The solution of Non-

linear Ordinary differential equations in Chebyshev

series,Comp. J . 6,88-92.

15 Cohen,A.M. and J o n e s , D . E . 1974 A note on the numerical

solution of some singular second order differential

equations,J.Inst.Maths. Applies.,13,379-384.

16. Coleman,J.P. 1976 The Lanczos Tau-Method,J.Inst.Maths.

Applies. 17,85-97.

17 Collatz,L. 1966 The numerical treatment of differetial

equations,Springer-Verlag.

18 Conte,S.D. 1966 The numerical solution of linear

boundary value p r o b l e m s , SIAM Review,8,309-321.

19 Conte,S.D. and C.De.Boor,. 1980,Elementary numerical

analysis an algorithmic a p p r o a c h , e d . Mcgraw-Hill.

20 Courant,R. and H i l b e r t » D . 1 9 3 1 Methoden- .der Mathematisce

Phys-lk., Sp ringer-Verlag., Be-rlin.

21 Dahlquist,G. and Ake Bjorck 1974(translated by Ned

Anderson) Numerical Methods,Prentice-Hall.

22 Daniel,J.W. and Moore,R.E. 1970 Computation and thoery

in ordinary differential equations, Freeman and Company.

23 Davey,A. 1980 On the numerical solution of difficult

boundary value problems,J.Comp.Phys. 35,36-47.

-121-

25 Davis,H.T. 1962 Introduction to nonlinear differential

and integral equations, Dover Publications,Kew Y o r k .

26 De.Boor,C.1978 A p r a c t i c a l guide to splines, Applied

mathematical sciences 27, Springer-Verlag.

27 D e . Boor,C.and W e i s s , R . Algorithm 54-6 S0LVEBL0K(F4.),

Collected.'.Algorithms from A C M .

28 De.Hoog,F.R.. and Weiss R . 1978 Collocation methods for

singular boundary value problems,SIAM J . Numer.Anal. 15,

198-217.

29 Douglas,J.JR. and D u p o n t , T . 1973 Galerkin approximations

for the two-point boundary problem using continuous,

piecewise polynomial spaces,Topics in Numerical Analysis,

J.H.Miller,ed. Academic press,London,89-92.

30 Elliott,D. 1961 A method for the numerical integration

of one-dimensional heat equation using Chebeyshev series,

P r o c . C a m b . P h i l . S o c . 57, 823-832.

31 Fox,L. 1957 The numerical solution of two-point boundary

problems in ordinary differential equations in ordinary

differential equations, Clarendon press,Oxford.

32 Fox,L. 1962 Chebyshev methods for ordinary differential

equations, Computer J .4-, 318-331.

33 Fox,L.(ed.) 1962 Numerical solution of ordinary and

partial differential equations,Pergamon.

34- Fox,L. and Parker.I.B. 1968 Chebyshev-polynomials in

numerical.analysis,Oxford University press.

35 Fyfe,D.J. 1969 The use of Cubic Splines in the solution

of two point boundary value problems, Computer J . 12,

188-192.

36 Frazer,R.A.,Jones,W.P. and Skan,S.W.1937 Approximations

to functions and to the solutions of differential equations,

Report ARC technical report 1799.

-122-

37 ' Gawain,T.H. and B a l l , R . E . 1978 Improved finite difference

formulas for boundary value problems, Int.J.Num.Meth.

Engng. 12,1151-1160.

38 Greenspan,D. 1972 Numerical approximation of periodic

solutions of Van der p o l ' s equation,JMAA 39, 574-579.

39 Gladwell,I. and Wait,R.(ed.) 1979 A survey of Numerical

methods for partial differential equations, Clarendon

press, O x f o r d .

4-0 'Gladwell,I. and S a y e r s , D . K . (eds)l980 Computational

techniques for ordinary differential equations,Academic

p r e s s .

41 Guderley,K.G. 1975 A unified view of some methods for

stiff two-point boundary value problems,SIAM Review 1 7 ,

416-442.

42 Guderley,K.G. and N i k o l a i , P . J . 1966 Reduction of two point

boundary value problems in a vector space to initial value

problems by projection, Numer.Math. 8,270-289.

4-3 : - Gustaf sson, B . 1973 A numerical method for solving singular

boundary value problems, Numer.Math. 21,328-349.

44 Haidvogel,D.B. and Z a n g , T . 1979 The accurate solution of

poisson's equation by expansion in Chebyshev polynomials,

J . C o m p . P h y s . 30,167-180.

45 Hall,G. and Watt,J.M.(eds) 1976 Modern numerical methods

for ordinary differential equations,Clarendon press,Oxford.

46 Ince, E . L , 1926 Ordinary differential equations, New Y o r k ,

Dover.

47 J a m e t , P . 1970 On the convergence of finite difference

approximation fcjo one dimensional singular boundary value

problems, Numer. M a t h . 1 4 , 355-378.

48 Jones,C.W., Miller ,J. C .P . , Conn, J.F.C. Edinb. LXII, 187.-203.

-123-

4-9 Jones,D.J.,South,J.C.JR. and Klunker,E.B.1972 On the

numerical solution of elliptic partial differential

equations by the method of l i n e s , J . Comp. P h y s . 9,4-96-527,

5 0 K e d e m , G . 1981 A P o s t e r i o r i error bounds for two-point 1

boundary value p r o b l e m s , SIAM J . Numer. A n a l . 18, 4-31-4-4-8.

51"Keller,H.B. 1968 N u m e r i c a l methods for two-point

boundary - value p r o b l e m s , Blaisdell Publishing company.

52 Keller,H.B. 1976 N u m e r i c a l solution of two point boundary

value problems, Regional conference series in applied

mathematics, 24-.

53 Knibb,D. 1973 The numerical solution of Parabolic Partial

differential equations using the method of Lanczos,J.Inst.

Maths. Applies., 1 1 , 1 8 1 - 1 9 0 .

54"Lanczos,C. 1938 Trigonometric interpolation of empirical and

analytical functions, J.Math.Phys.,17,123-199.

55 Lanczos,C. 1956 Applied Analysis,Prentice Hall,New J e r s e y .

56-Lanczos,C. 1961 Linear differential operators,D.Van

Nostrand Company L t d .

57 Lanczos,C. 1973 Legendre versus Chebyshev polynomials,

in Topics in Numerical Analysis, edited by J.J.H. Miller,

Academic press for the Royal Irish A c a d e m y .

58 Lanczos,C. 1952 Tables of Chebyshev polynomials, National

Bureau of Standards A p p l . M a t h . Series,9» Washington:

Government Printing O f f i c e .

59 Lawson,C.L. 1963 Segmented Rational minmax approximation,

characteristic properties and computational methods,

Jet propulsion Lab.,Techn. report N o . 32-579, California

Institute of Technology Pasadena,California.

60-Liskovets,0.A. 1965 The Method of Lines,Differential

equations 1,12,1308-1323.

61 Luke,I.L. 1969 The special functions and their approximation,

New york,Vol I and II .

62 Mason,J.C. 1965 Some n e w approximations for the -±*4-

solution of differential equations, D . P h i l . Thesis, O x f o r d .

63 Mason, J . C . 1969 Chebyshev method for separable partial

differential equations, Information processing 68,

Amsterdam, North Holland C o .

64. Mason, J . C . 1979 The vector Chebyshev tau method for linear

partial differential equations, The Royal military College

of science,Dept. of M a t h s , and Ballistics report 7 9 / 3 .

65 Meinardus, G . 1967 Approximation of functions:

Theory and Numerical methods, Springer-Verlag, B e r l i n .

66 Miller, J . C.P. 194-6 Two numerical approximation of Chebyshev

polynomials, P r o c . R o y . S o c . Edinb. LXII, 204--210.

67 Miller,R.E. 1979 On consistent finite difference formulae

for ordinary differential equations, Int. J . N u m . M e t h .

E n g n g . 14-,1567-1573.

68 Minnick, R.C. 1957 Tchebysheff approximations of power

series, J . A s s . Comp. Mach.,4-, 4-87.

69 Murray, W.L . 1978 N u m e r i c a l schemes for diffusion-convection

in decelerating flows, N a t . Eng. Lab. report

70 N a , T . Y . 1979 Computational methods in engineering boundary

value problems, Academic press, New Y o r k .

71 Natterer,F. 1973 A generalized spline method for singular

boundary value problems of ordinary differential

equations, Linear Algebra and A p p l . 7, 189-216.

72 Necas,J. 1967 Les methodes directes en theorie des equations

elliptiques, Masson, P a r i s ,

73 Nehari,Z. 1962 On a nonlinear differential equation arising

in Nuclear physics, P r o c . R o y . Irish Academy 62,117-135.

74- Norton, H.J. 1964- The iterative solution of nonlinear

ordinary differential equations in Chebyshev series

Computer J . 7, 76-85.

75 Oliver, J . 1969 An error estimation technique for the solution

of ordinary differential equations in Chebyshev series,

Computer J . 12, 5 7 - 6 2 .

-125-76 01iveira,F.A. 1 9 8 0 , N u m e r . Math. 36,27-31.

77 Onumanyi,P. 1978 N u m e r i c a l experiments with some nonlinear

ordinary differential equations using the tau method,M.Sc.

Thesis,Imperial C o l l e g e , L o n d o n .

78 0numanyi,P.,Ortiz,E.L. and Samara,H. 1981 Software for a

method of finite approximations for the numerical solution

of differential equations, Applied Math. Modelling,5,282-286.

79 Ortiz,E.L. 1969 The Tau method, SIAM J . N u m e r . A n a l .

6,4-80-492.

80 Ortiz,E.L.,Purser,W.F.C. and Rodriguez Canizares,F.J. 1972

Automation of the tau method.Imperial College Res.Rep.

NAS 01-72,l-54.> (Presented to the Conference on numerical

Analysis organized by the Royal Irish Academy,Dublin,1972)

81 .Ortiz,E.L. 1975 Step-by-Step tau method,Part I: Piecewise

polynomial approximations, Comp. and Math, w.ith A p p l i . -1,

381-392.

82 Ortiz,E.L. and S a m a r a , H . 1978 A new operational approach

to the numerical solution of differetial equations in terms

of polynomials, Innovative Numerical Analysis for the Eng-

ineering Science,University press of Virginia.

83 "Ortiz,E.L. 1978 On the numerical solution of nonlinear and

functional differential equations with the tau method, in

" Numerical Treatment of differential equations in

Applications",R.Ansorge and W.Tornig,Eds. Springer-Verlag,

Berlin,127-139.

84. "Ortiz,E.L. and P h a m , A . Nonlinear differential equations:

Quadratic convergence of the iterative formulation of the

tau method,Imperial College Res.Rep.

85 'Ortiz,E.L. 1980 Polynomial and Rational approximation of

boundary layer problems with the tau method, in Boundary

and Interior Layers-Computational and Asymptotic Methods,

J.J.H.Miller,Ed., Boole Press,Dublin,387-391.

-126-

86 O r t i z , E . L . 1974- C a n o n i c a l polynomials in Lanczos tau

method, in Studies in N u m e r i c a l A n a l y s i s , B . P . K . S c a i f e , E d .

A c a d e m i c - P r e s s , New Y o r k , 73-93*

87 O r t i z , E . L . 1964. On the Canonical polynomials associated

with certain linear d i f f e r e n t i a l operators, Imperial College

Res Rep, 1 - 2 4 .

8 8 ' O r t i z , E . L . 1966 Polynomlostfngen von Differentialglei-

chungen, Z . A . M . M ,4 6 , 3 9 4 - 3 9 5 .

8.9 O r t i z , E . L . and C h a v e s , T . 1968 On the numerical solution

of two-point boundary value problems for differential

equation, ZAMMM 4 8 , 4 1 5 - 4 1 8 .

90 , O r t i z , E . L . and L l o r e n t e , P . 1968 Sur guelques aspects

algegriques d'une methode de M,Lanczos, M a t h . Notae,

21, 1 7 - 2 3 .

91 O r t i z , E . L . and F r e i l i c h J . H . 1975 Simultaneous approx-

imation of a function and its derivative with the tau

method, Imperial College R e s . R e p . , 1 - 4 5 .

92 .. O s b o r n e , M . R . and W a t s o n , G . A . 1968 Note on two methods of

solving ordinary l i n e a r differential equations, Computer J.

1 0 , 3 8 3 - 3 8 4 .

93 Picken,S.M.I970 A l g o r i t h m s for the solution of differential

equations in Chebyshev Series by the selected points

method,NPL M a t h s . Report,94«

94 ' P r u e s s , S . A . 1973 S o l v i n g linear boundary value problems

by approximating the c o e f f i c i e n t s , M a t h . Comp. 2 7 , 5 5 1 - 5 6 1 .

95 Reddien,G.W. 1973 P r o j e c t i o n methods and singular two

point boundary value p r o b l e m s , N u m e r . M a t h . 2 1 , 1 9 3 - 2 0 5 .

96 . Rentrop,P. 1979 A Taylor Series method for the numerical

-127-

solution of two-point boundary value problems, Numer. M a t h .

31,359-375.

97 Rosenblat,S. and S z e t o , R . 1980 Multiple solutions of nonlinear

boundary value p r o b l e m s , Studies in Applied Mathematics

vol LXIII No.2,99-117.

98 Rothe,E. 1930 Zweidimensionale Parabolische Randwertanfgabenals

Grenzfall Eindimensionalle Ranswertaufgaben,Math. A n n .

102,650-670.

99 Russell,R.D. and S h a m p i n e , L . F . 1975 Numerical Methods for

singular boundary value problems,SIAM J . Numer. A n a l . 1 2 ,

13-36.

100 " Schreiber,R. 1980 Finite element methods of high-order

accuracy for singular two-point boundary value problems

with Nonsmooth solutions,SIAM J . N u m e r . A n a l . 17,54-7-566. •

101 Shampine,L. 1966 Monotone iterations and two-sided converg-

ence, SIAM J . N u m e r . A n a l . 3,607-615.

102 ' Shimauchi,H. 1979 On an approximation of Bessel function by

the tau method, Memoir of the Faculty of science,Kyushu

University ser.A, v o l . 3 3 , n 0 . 1, 173-178.

103 Shinohara,Y.and Y a m a m o t o , N . 1978 Galerkin approximations

of periodic solution and its period to Van der Bol

equation, J . Math. Tokushima Univ. vol. 12,19-4-2.

104-. Sommerfield,A. 1935 Partielle Differentialgleichungen,

Springer-Verlag,Berlin.

105 Stetter,H.J. 1978 The defect correction principle and

discretization methods, N u m e r . Math. 29, 4-25-4-4-3.

106 Synge,J.L.1961-62 On a certain nonlinear differential

equation, P r o c . Roy. Irish A c a d . 62,17-4-1.

107 Todd,J. 1962 Survey of numerical analysis,Mcgraw H i l l . ' '

108 Van der P o l 1935 Tchebysheff polynomials,Physica,

218-236.

-128-

109 W r a g g , A . 1966 The use of Lanczos-tau methods in the

solution of a S t e f a n p r o b l e m s , Computer J . 9> 1 0 6 - 1 0 9 .

110 W r i g h t , C . 1976 N u m e r i c a l solution of Differential

j, e q u a t i o n s , M . P h i l . T h e s i s . I m p e r i a l C o l l e g e , L o n d o n .

I l l , Wright,K.1964. C h e b y s h e v Collocation Methods for ordinary

d i f f e r e n t i a l e q u a t i o n s , C o m p u t e r J . 6 , 3 5 8 - 3 6 3 .

112 Wright,K.1962 The N u m e r i c a l solution of ordinary N o n - l i n e a r

d i f f e r e n t i a l e q u a t i o n s , T h e s i s - O x f o r d .

113 Z a d u n a i s k y , P . E . 1 9 7 9 On the implementation of a method

for the estimation of global errors, in Lecture notes in

Computer S c i e n c e , 7 6 , 2 8 7 - 2 9 3 , S p r i n g e r - V e r l a g .

114- Zadunaisky,P .E. 1 9 7 6 On the estimation of errors

p r o p a g a t e d in the n u m e r i c a l integration of ordinary

d i f f e r e n t i a l e q u a t i o n s , N u m e r . M a t h . 2 7 , 2 1 - 3 9 .

115 H a r t , J . F . 1968 C o m p u t e r A p p r o x i m a t i o n s , W i l e y and s o n s .

APPENDIX A

Software for a method of finite approximations for the numerical solution of differential equations P. Onumanyi, E. L. Ortiz and H. Samara Department of Mathematics, Imperial College, London SW7, UK (Received January 1981)

Introduction In this paper we discuss software for the computer imple-

mentation of a method of finite approximations for the

numerical solution of ordinary differential equations, which

is based on the Tau method.

A polynomial approximation of high accuracy to the

function y(x), solution of the given nonlinear differential

equation, is obtained by means of software of a fairly

simple structure. Problems with complex initial, boundary

or mixed conditions, involving combinations of function

and derivative values, can be dealt with by means of our

program. Encouraging results have been obtained in the

solution of equations with regions of rapid variation, oscilla-

tory behaviour and in the presence of stiffness. As well as

in the treatment of nonlinear boundary value problems

where the solution is not unique. Examples are given in the

last section of the paper. Applications to eigenvalue

problems and to partial differential equations are reported

elsewhere.

Finite approximations method based on the Tau method: different approaches in the design of numerical algorithms The Tau method

A brief account is given here of some recent develop-

ments in the design of numerical algorithms for the

approximate solution of differential equations by means of

the Tau method. Equivalence results for these different

approaches are discussed elsewhere.1

The Tau method applies directly to the construction

of numerical solutions of linear differential equations

of an arbitrary order v, with polynomial coefficients Pi(x)

and with a polynomial right-hand-side f(x) = /0 +f\X + ...

+/fxf

:

i = 0 dx (1)

C?j,y) = Oj j=\(\)v x E [a, b]

\al,\b\<~ The gj are functional acting on >>(*) and (gj,y) = Oj,

j = 1(1) v, stands for the initial, boundary or mixed condi-

tions imposed on y(x); we refer to them as supplementary

conditions. One of the interesting features of the Tau

method is its ability to deal with supplementary conditions

of a wide variety of forms. The requirement on the pi and

/ , to be polynomials, is not as restrictive as it looks at first

sight, since polynomial approximations p,-,/of general pt, J

can easily be generated to great accuracy by using the Tau

method itself. The relation between solutions y and y

obtained respectively by taking the original p,-,/and their

approximations P/,/is discussed in Necas2 and, more

particularly, in Pruess.3

The basic idea of the Tau method, as conceived by

Lanczos4'5 is the addition to equation (1) of a small

perturbation term Hn(x) which would cause:

DyZ(x)=f(x)+Hn(x)

(gj,y%) = ohx(=[a,b)

to have an exact polynomial solution y%(x) which satisfies

exactly the given supplementary conditions.

The perturbation term Hn(x) is a linear combination of

polynomials of a polynomial basis <j> = {(pi(x)}, i = 0 , 1 , 2 , . . . .

If Hn takes the form Tqx" + ... + Trx"~ r, the classical

Taylor series method is recovered. A more balanced distri-

bution of the vaues of Hn(x) over [a, b] is obtained by

choosing a polynomial close to the best uniform approxima-

tion of zero by polynomials of degree n while, at the same

time, y% satisfies the supplementary conditions.

Such choice characterizes the Tau method. Variations of

this idea give rise to related methods such as collocation,

Chebyshev or Legendre series replacement techniques,

spectral methods, etc.

The minimization of the perturbation term Hn(x), added

to equation (1), is related to the reduction of the error en(x) — _>>(*) in the numerical solution, as en(x)

satisfies the differential equation Den(x) = Hn(x) with

homogeneous supplementary conditions; in fact Hn(x)

controls en{x) through/)"1: en{x)= D~ lHn(x). The former

equation is useful in the numerical estimation of the

accuracy of the Tau method, as discussed elsewhere.

Recursive formulation

A sequence of polynomials Q = {Qn(x)}, is called

canonical with respect to a differential operator D and a

282 Appl. Math. Modelling, 1981, Vol. 5, August 1981 0307-904X/81/040282-05/S02.00 © IPC Business Press

Numerical solution of differential equations: P. Onumanyi et al.

polynomial basis <f>3 {0*0:)}, if DQn(x) = 0„(x), n GIN. If

the Q„(x) do not exist for one or several (s) indices n GS,

then DQn(x) - 0n(x) + rn(x)> where rn(x) (the residual of

Qn) belongs to a subspace of dimension s, which is usually

very small. The existence of 'gaps' in Q is inherent in the

differential equations we are considering. For example, if

0 = {*"} = X, then for Dy(x): =y'(x) +x ry(x) there will be

no polynomial Qn(x) such that DQn(x) - x n for n = 0, 1,

. . . , r - l .

In the solution of the Tau problem the sequence Q plays

a decisive role: if Q is known, and Hn(x) = T0<pn(x) + ...

+ rr0n_r(x)» then:

Z Z / G / W + I r / Q n - / (3) 1=0 i=0

The number r of -parameters to be considered depends on

the number v of supplementary conditions to be satisfied.

It also depends on the number s of undefined canonical

polynomials: s parameters 7/ will be chosen so that the

coefficients of such Qt(x) are made equal to zero.

In the paper Ortiz,6 on the algebraic and algorithmic

theory of the Tau method, the author shows that the

sequence Q of canonical polynomials can always be con-

structed by means of a direct recursive procedure. He

introduces the generating polynomials:

Pm(x)=D<fim(x) = an<Pn(x)

+ an_ !0n _!(*) + ...+ oc0<p0(x) (4)

which describe the effect of D on a polynomial of 0, and

uses it to construct a recurrence relation for Qn(x). Such a

recursive process is self starting.

This recursive approach was followed by Ortiz et alP in

their paper on software for the Tau method. The computer

program described there has been in use at Imperial College

and other institutions since 1972.

Operational formulation

In Ortiz and Samara8-9 the authors develop an opera-

tional approach which by-passes the explicit construction

of the sequence Q, required to produce the Tau approximate

solution y„(x). They identified and studied the properties

of a group of simply structured interrelated matrices by

means of which it is possible to generate, with great sim-

plicity, a matrix tt^ which has the same effect as D on the

coefficients of a polynomial in the basis <f>. Such matrix ir

actually produces the vector generating polynomial D<p„,

Qn ~ (0o> 0i, • • • > <Pn)T. Incorporating to ir^ the supple-

mentary conditions an enlarged matrix, called T^, is

obtained. The coefficient vector an = (a0,..., an) of y*(x)

follows by inversion of a suitable truncation of the algebraic

system:

<?r0=K0 (5)

where V^ is a vector made up of the values taken by the

supplementary conditions and the coefficients of /in 0. The

truncation of equation (5) implies the assumption that the

coefficients a,- ofyn(x) in the basis 0 are negligible for i>n.

if 0 is chosen to be the Chebyshev polynomials, the

iChebyshev series replacement method is obtained as a

particular case of this operational approach. However, its

implementation does not involve the laborious algebraic

manipulation which often makes the application of

Chebyshev series expansion methods impractical.

Displacements approach

In a more recent paper,10 an alternative approach, which

shares with the recursive formulation6 the advantages of

operating exclusively in the basis X and deals only with

matrices of banded structure, has been discussed. In their

paper, these authors show that the effect of D on the coeffi-

cients of a polynomial can be described as a series of

displacements of the elements of matrices of a particularly

simple structure, weighted with the coefficients of the pt(x)

of equation (1). In such a way, a matrix n*, of banded

structure is constructed and augmenting it with the relevant

supplementary conditions, an enlarged matrix T* is

obtained. A very sparse correction matrix which plays

a role similar that of Hn(x) in Ortiz's recursive formulation

of the Tau method, is added to T* to produce a new matrix

f . Truncations of the algebraic system:

at = sx

give, by inversion, the coefficients an of y%(x). Software

based on this displacements approach is extremely compact.

The three approaches allow for the treatment of complex

initial boundary or mixed conditions for linear or nonlinear

differential equations over limited or extended domains, as

illustrated in the examples given in this paper.

Structure of the programs The main program

A main program MAIN TAU controls four main sub-

routines: (1), ODE which uses data directly or indirectly

supplied by the user to construct an approximate solution

of a given or auxiliary linear differential equation with

polynomial coefficients of the type of equation (1); (2),

ITERATE which allows for an iterative use of the Tau

method to solve nonlinear differential equations; (3), SEG-

MENT a subroutine which implements the step by step

formulation of the Tau method, and (4), PRINT which

produces a variety of possible outputs of the results

obtained by the program.

All three versions of this program are designed to deal

with nonlinear problems as a sequence of linear problems,

iterated by ITERATE and each one solved by ODE. If the

interval in which the solution is required is a long one in the

sense that the maximum degree admitted by ODE is insuffi-

cient to produce a finite approximation y„ (x) of the

required quality, the user is offered the use of a step by

step or segmentation technique.

The difference between our three programs is to be found

in the use of alternative subroutines ODE: ODE 1, ODE 4,

and ODE 7. Each one of them can, in turn, be used with

different iteration schemes, as will be explained later.

The subroutine ODE

The design of the subroutine ODE 1 is based in the

recursive formulation of the Tau method of Ortiz.6 It

controls a series of secondary subroutines designed to

generate the sequence of canonical polynomials associated

with the given differential operator D , their residuals and

the perturbation term Hn(x)\ to compute the values of the

r,-parameters according to the given supplementary condi-

tions and to make the coefficient of any undefined

canonical polynomial equal to zero. Finally, to generate

the finite Tau approximation y*(x) of the required degree n

as indicated in equation (3). This subroutine is an update of

the one described in Ortiz et al. 1

Appl. Math. Modelling, 1981, Vol. 5, August 1981 283

Numerical solution of differential equations: P. Onumanyi et al.

Data required for this subroutine is of the order v of D\

the nonzero coefficients of the polynomials P/(x), / 3 1(1) v\

the maximum exponent of the polynomial fix) and its

nonzero coefficients; the supplementary conditions

(Sj,y) - OjJ - 1(1) v\ the end points of the interval [a, b] in

which the solution is required; the order (or orders) n of

approximation required.

The subroutine ODE 4 implements the operational

approach of Ortiz and Samara.8'9 The secondary sub-

routines it controls are designed to generate the matrix

in a polynomial basis specified by the user (usually

Chebyshev, Legendre). By inversion of the coefficients

of an approximate solution y*(x) which satisfies the given

supplementary conditions are obtained. Data required to

operate ODE 4 is the same as for ODE 1.

Finally, subroutine ODE 7 implements the displacement

formulation of Ortiz and Samara,10 which operates exclu-

sively in the simpler X basis and only performs a basis

transformation in the process of constructing the correction

matrix It is the most compact of the three subroutines.

Details on the structure and performance of these sub-

routines, as well as operation counts and time comparisons

are given elsewhere.

Subroutine ITERATE

The subroutine ITERATE takes from ODE a finite Tau

approximate solution y^m(x) of a linearized form of a given

nonlinear differential eauation to produce, with ODE again,

an approximation y^S^t, which is the Tau solution of a

different linearized differential equation. These differential

equations have as solutions the elements of a sequence of

polynomials 0>n,m(x)}, k = 0,1, 2 , . . . , such that:

y[kn%] (*) = F[yW(x), 1 (x)) (6) where y stands for the vector (y , y', y",..., The fix

point of such sequence being the solution of the given non-

linear differential equation.

Different forms of F have been discussed by Ortiz11'12

and Onumanyi.13 For one of them, quadratic convergence

can be guaranteed if the appropriate conditions (see Ortiz

and Pham)14 are satisfied. The form of the scheme F is

selected by the user by fixing parameters in a general form.

The iterative scheme is initiated taking for the initial

guess yn%(x) a low order polynomial which satisfies the

given supplementary conditions of the problem, such

approximation is called the natural one. Other choices are

also possible.

The coefficient vector in the Chebyshev basis defined in

[a, b] of two successive iterates of equation (6) are com-

pared until the maximum difference — a\ k^\,

i = 0(l)/w, becomes smaller than a given tolerance para-

meter epsilon. A second parameter, called Iter, is fixed by

the user to specify the maximum number of iterations of

equation (6) which ITERATE is allowed to perform. If Iter

is reached, the program prints a message indicating that the

degree m of the approximations is being raised. When the

maximum N allowed by the program is reached, a message

notifies the user that the original segment [a, b] is being

halved.

Subroutine SEGMENT

Command is then passed to the subroutine SEGMENT

which constructs successive approximations over sub-

intervals of the given range [a, b] by using the step by step

approach developed in Ortiz.1S Unless the subroutine

SEGMENT is entered automatically, the user must supply

the segmentation points*/ E [a, 6] or the segmentation step

h, if it is to be constant.

Program output

The user can order the program to print the coefficients

(in powers of x or in the basis 0) of the approximate solu-

tion y%(x) (vector solution y*(x) if [a> b] is segmented); the

r/-terms, which indicate the error in the equation and the

value of or its derivatives at one or a set of points of

the interval [a, b].

Some model problems Example 1

The linejar singular boundary value problem:

y"+~y'(x)+y(x) = 0 x (7)

^(0)=X10TT) = 0 JCG[0, 10TT]

has an oscillatory solution in the interval [0, IOTT]. Figure 1

shows the graphs of yjfcx) for n = 25 and n - 30, they

coincide within the accuracy of the plot. For n = 30 the

maximum error is below 10~12.

Example 2

The differential equation16

-/(*) +[*O-«)X*)f = 0

X0)=Xi)=i *e[o,i]

for c = 2 and k = 20, has a solution in the form of a narrow

vertical spine defined in the interval 0 < 1 and extend-

ing in the j>-axis fromj' = 1 till about y = 150.

The results produced by our program give for x = 0.5 the

value y = 148.413159 ... Table 1 gives the order of the

coefficient a„ of the Chebyshev expansion of^(je) for

different values of n. Twelve figures of the value of _y*o(0.5)

remain unchanged when the order of the finite approxima-

tion is taken to n = 30.

Barrett and Morton11 report negative results on this

example when using their finite elements optimal technique.

Example 3

The fourth-order differential equation:

[EIy"{x)\' + Fy"(x) +Ky(x) = P

(gj,y) = o /= l(l)4

Appl. Math. Modelling, 1981, Vol. 5, August 1981 284

Numerical solution of differential equations: P. Onumanyi etal.

Table 1

Degree n an « 10*

8 10 15 20

- 2 - 3 - 6

- 1 0

Table 2 Relative errors

Relative errors

No. of points Deflection Bend. mom.

Miller's finite differences 5 0.161 X10"3

10 0.413 X10"4

20 0.962 X10"5

50 0.141 X10"5

100 0.330 X10"4

200 0.725 X10"7

0.110X10"* 0.924 X10"4

0.187 X10 - 4

0.367X10" 5

0.919 X10"6

0.247 X10"6

Our program, for n • 10 /» = 10 0.129X10"8 0.150X10"5

Table 3 Synge's nonlinear singular infinite boundary value problem

Method /(4.5) / { 4 .5 )

Synge Gear (with treatment of singularity) Our program (n =» 20, no segmentation) Our program (n 3 9, with segmenta-tion, h = 0.3)

-0.0392613 -0.0392519 -0.0392520 -0.0392519

0.047495 0.04750 0.047502 0.047502

Table 4 Falkner-Skan equation, selected values of y"(0)

Quoted by 0 Na" Na"

Our program (n - 20)

1.0 1.2423400 1.2325870 1.6 1.5327980 1.521520 2.0 1.6995770 1.6872170

1.2325875 1.5215139 1.6872160

describes a Bernoulli beam-column on an elastic foundation.

EI is the bending stiffness, F the axial compressive load, K

the elastic foundation modulus, P the distributed load, and

y(x) measures the lateral deflection.

This problem has recently been considered by Miller18

using finite difference formulae. The author reports results

obtained with his method in the case of a uniform beam

column with K=P=0, zero lateral deflection at the left end

and a linear spring resisting rotation with stiffness R ; zero

slope at the right-hand-end, a linear spring resisting deflec-

tion with stiffness S, and an applied lateral load of value Q.

The axial compressive load is equal to F. The problem can

be formulated by means of the differential equation:

EIy""{x) + Fy"{x) = Q

y(a) = 0

EIy"(a) — Ry'(a) = 0

/(*) = 0

EIy"\b) -Sy(b) +Q = 0 xG[a,b]

Miller's results correspond to a = 0, Fb 2!EI = 1r2/4, Rb/EI =

5tt/12, Sb 2/EI = 25 tt3/4, Qb 2(EI = tt2/16, Table 2 contains

the results given by the best of two finite difference

formulae proposed by Miller for the relative errors of

deflection and bending moment, for different values N of

the number of grid points. It also reports the results

obtained with our finite approximation method for approxi-

mations of degree n = 10. It is not a simple matter to make

a fair comparison between the two methods, but we could

say that Miller's result for 10 points (10-4 and 10"4) roughly

corresponds to ours for n = 10. Our method provides a

results 10"4 times more accurate for the deflection and 10"1

times more accurate for the bending moment (for more

details see Onumanyi and Ortiz19).

Nonlinear boundary value problems

Example 1. The nonlinear singular infinite boundary

value problem:

y"(x) + 2 y'(x)-y(x)=y\x) x

X 0 ) =-4.191691 y(°°) = 0

of Synge is related to problems connected with nuclear core

structure. Synge20 and Nehari21 have given numerical values

for.y(*) a n d.y'00 at x = 4.5.

A solution of this problem was obtained by using our

segmentation facility with h = 0.3. An accuracy of 16

decimal places is achieved after three iterations, starting

with a quadratic initial guess and then taking n - 8. In

subsequent steps (x > 0.3) a linear guess is chosen, then

n = 8, and three iterations as before. The accuracy is the

same as in [0,0.3]. For x>0,Xx) » 0.

The value proposed by Synge, for which Nehari showed

it can be accurate up to at most five decimal places, was

confirmed with an error of 0.94 x 10"5 (see Onumanyi).13

A computation of X * ) in [0,4.5] with a single approxima-

tion of degree 20 is reported in Table 3. Results forj>(4.5)

and .y'(4.5) are also computed as an initial value problem (it

follows easily that^'(O) = 0) by using Gear's method which

requires the treatment of the singularity. No treatment of

the singularity is required by the method of this paper.

Example 2. The Falkner-Skan equation of boundary layer

theory is a third order nonlinear boundary value problem

defined in the interval 0 < x < Written in terms of the

similarity variables it takes the form:

y'\x) +y(x)y"(x) + p[l -y'\x)} = 0

y(0)=y'(0) = 0 y'(°°)= 1

The quantity that is sought is the missing boundary

condition "(0).

Results obtained for several values of /3 by invariant

imbedding by Na22 and reported by him from the literature

on the Falkner-Skan equation are reproduced in Table 4,

together with results obtained with our program for n = 20,

with the infinity condition taken atx = 9.

Problems related to boundary layers are also discussed in

Ortiz.23

Peierences 1, Ortiz, E. L. and Samara, H. Imperial College Res. Rep. NASI 1-78,1978 2 Necas, J. 'Les methodes directes en theorie des equations . elliptiques', Masson, Paris, 1967 3 Pruess, S. A. Maths. Comp. 1973, 27, 551 4 Lanczos, C. J. Math. Phys. 1938,17,123

Appi. Math. Modelling, 1981, Vol. 5, August 1981 285

Numerical solution of differential equations: P. Onumanyi et al.

5 Lanczos. C. 'Applied analysis', Prentice Hall, New Jersey, 1956 6 Ortiz, E. L. SIAM J. Numcr. Anal 1969,6,480 7 Ortiz, E. L. et al Imperial College Res. Rep. NAS06-72. (Paper read at the Conference on Numerical Analysis, Royal Irish Academy, 1972) 8 Ortiz, E. L. and Samara, H. 'A new operational approach to the numerical solution of differential equations in terms of poly-nomials*, Innovative Numerical Analysis for the Engineering Science, University Press of Virginia, 1980a, 643 9 Ortiz, E. L. and Samara, H. Computing (in press) 10 Ortiz, E. L. and Samara, H. Imperial College Res. Rep. NASI 2-80, 1979 11 Ortiz, E. L. 'Sur quclqucs nouvelles applications de la mdthode Tau', Seminaire Lions, IRIA, Paris, 1975 12 Ortiz, E. L. In 'Numerical treatment of differential equations in applications' (ed. Ansorge, R. and Tornig, W.), Springer-Verlag, Berlin, 1978 13 Onumanyi, P. 'Numerical experiments with some nonlinear

ordinary differential equations using the Tau method', MSc Thesis, Imperial College, London, 1978 14 Ortiz, E. L. and Pham, A. Imperial College Res. Rep. NAS08-80, 1980 15 Ortiz, E. L. Comp. Maths. Applic. 1975,1,381 16 Murray, W. L. Numerical schemes for diffusion-convection in decelerating flows, Nat. Eng. Lab. Report, 1978 17 Barrett, J. W. and Morton, K. W. University of Reading. Numerical Analysis Rep. 3178,1978 18 Miller, R. E. Int. J. num. Meth. Eng. 1979, 14, 1567 19 Onumanyi, P. and Ortiz, E. L. Imperial College Res. Rep. NAS01-81,1981 20 Synge, J. L. Proc. Roy. Irish Assn. 1962,62,17 21 Nehari, Z. Proc. Roy. Irish Acad. 1962,62, 117 22 Na, T. Y. 'Computational methods in engineering boundary value problems', Academic Press, New York, 1979 23 Ortiz, E. L. In 'Computational and asymptotic methods', Boole Press, Dublin, 1980

Appl. Math. Modelling, 1981, Vol. 5, August 1981 286

APPENDIX B

International Journal for Numerical IViei:hods in Engineering A n International Journal published under the Wiley-lnterscience imprint by John Wiley &Sons,

of Chichester, New York, Brisbane and Toronto

CHIEF EDITOR: OLGIERD C. ZIENKIEWICZ, University of Wales, Swansea, Glamorgan, U.K.

AMERICAN EDITOR: RICHARD H. GALLAGHER, University of Arizona, U.S.A.

ASSOCIATE EDITOR: ROLAND W. LEWIS, University of Wales, Swansea, Glamorgan. UK.

From: R.H. G A L L A G H E R , . Dean, College of Engineering,

University of Arizona, Tucson, Az . 85721

Telephone: 602-884-2351

August 6, 1981

Dr. E. L. Ortiz Imperial College of Science & Tech. Dept. of Mathematics Huxley Building Queen's Gate, London SW7 2BZ

Reference paper: 26.A

Dear Professor Ortiz:

Enclosed you will find the review of your manuscript. It is positive and on that basis we are pleased to accept it for publication in IONME, as a Short Communication. Since no revisions are requested I am forwarding the manuscript directly to J. Wiley for publication.

Sincerely vours,

Richard H. Gallagher RHG/e

NUMERICAL SOLUTION OF HIGH ORDER BOUNDARY VALUE PROBLEMS' FOR

ORDINARY DIFFERENTIAL EQUATIONS' W I T H AN ESTIMATION OF THE ERROR

P . Onumanyi and E . L . Ortiz

Mathematics Department Imperial College University of London L o n d o n

1.- INTRODUCTION

1 2

In recent papers, Gawain and Ball , and Miller discussed finite difference

formulae for the numerical solution of two point boundary value problems-

with particular reference to fourth order ordinary differential equations*

In this note we show that results of remarkably high accuracy and compu

tational simplicity can be obtained by using Ortiz1

recursive formulation 3 of the Tau method * • Besides, an error estimate of the numbers presented ca

be produced at. low. CLomputational extHsL.cjist, as shown in an example taken 2

from Miller • Considerable experience has been gained recently in the use

of this polynomial technique which involves no quadratures or large matrix

inversions. An extensive list of linear and nonlinear examples with complex

mixed condi-^.ons is reported in Onumanyi^; one such example, the nonlinear

Falkner-Skan equation of fluid dynamics is briefly discussed at the e n . of

this paper. Theoretical error estimates are given in Crisci and Crtiz .

2.- THE RECURSIVE FORMULATION OF T'HS TAU METHOD

Let us consider the differential equation of order i> , defined for x •= f a ,

D y(x) := P y

( x ) y( % 0

( x ) + ... + p ^ x ) y( 1 )

( x ) + P Q ( X ) y(x) = F ( x ) ,

(r)

where y (x) stands for the derivative of order r of y(x); p^Cx), r = i!i)v

and f(x) are polynomials or polynomial approximations (immediately derivable

by using the method described in this paper) of given functions. The solutio

y(x) satisfies given initial, boundary or mixed conditions (v;e refer to ;

:.en

as supplementary conditons) given in general by

v - 1 0

ii y( i ) ( x

i 1 > = ' 3 = < i=0

J

J

where the points x ., xn ... ,x . , . , at which y(x) and itstf-1 derivstiv

* oj' lj1

v-1 3 are given need not be the same for each value of j.

3 k Ortiz

1

has shown that each differential operator of the type (l) is ur;

- 2 -

quely ' associated with a sequence Q of canonical polynomials Qn( x ) . These

polynomials Qn(x).are defined for an infinite set of indices : n 6 3N - S

where H = ( 0 , 1 , 2 , 3 » • • • ) and S is a small finite (or empty) set of indices

with a number s of elements at most equal to V + h.The number V is the order

of the differential equation ana h is the height of D , that is, the maxi-

mum difference between the exponent n.of xn

and the leading exponent of

Dx1 1

, for all n£]N.

Given a polynomial basis V = {v^(x)j , B , the canonical polynomials-

Qn satisfy the relation D Q

n( x ) = v

q( x ) + R

n( x ) , where R

nC x ) , called the

residual of Qn, is a polynomial given by a linear combination of the s

polynomials v^(x) with i £ S . They appear as a consequence of the impossibli-

lity of generating Qn'

s

with n e ' S . Their existence is not a weakness of

the representation capability of the sequence Q , but a consequence of struc-

tural properties of the differential equation we are considering.

We say that yn( x ) is a Tau approximant of order n of y(x) if

F s+V , v

1=0 1=1

where „ F

f(x) = Z f, v-(x) , i=0

1 1

and the s+v free parameters X are chosen in such a way that : 1) the / I

coefficient of any in the expression of yn( x ) , with i £ S ,be equal

to zero, and 2) the supplementary conditions (2) on y(x) be satisfied exac-

tly by yn( x ) .

Clearly Yn( x ) satisfies the same differential equations as y(x)»

but for a perturbation term added to f(x) and equal to \

S+V» { X H (x) = T~ T - • . (x) (V-n • l n + s + v - i

1

i=l

A useful choice of V is the set of Chebyshev polynomials

defined for x £ Q a , b 3 . This choice characterizes Lanczos' Tau method

An interesting point about the sequence Q is that its elements can toe

generated recursively by means of a recursive relation involving at most

v)+h terms, which requires no special starting. Such expression is constru

ted with the help of the generating polynomials p

n( x ) = E

a s d e s c r i

~ If

in Ortiz , Generating polynomials make also possible the detection of th:-

existence of exact polynomial solutions to (l), which may not satisfy the

supplementary conditions (2), but are equally useful from a numerical poin

of view: if u^(x), ... , u^Cx) are exact polynomial solutions of (l), ther.

(3) can be expressed as

- 3 -

F • t s+^-t , v

= f=Q W

x ) +

J^i u

x( x ) +

£ < WS +

Only s+v-t p a r a m e t e r s Z w i l l now determine It follows that in

the event of y(x) being a polynomial of degree n, this method will detect it » (

exactly. The expression for yQ(x) given by (-5) is automatically constructed

and its parameter adjusted to conditions (2) with the software described in 8 *

reference

3.- ERROR ESTIMATION

Let us call e (x) = y„(x) - y(x) the error function of the a pur oxi mailt n n y

n(x) to y(x). The relative error is given by Re

n(x) = e

n(x)/y

n(x). The

function y(x) satisfies the differential equation D y(x) = f(x), whereas

y (x) satisfies the differential equation D y„(x) = f(x) + H (x) , H (x) n n n n

given by both, y and y , satisfy the same supplementary conditions

(2). Therefore, the error function ©n(x) satisfies the differential equa-

tion D e (x) = H (x) with homogeneous supplementary conditions given by (2).

n

n

(n)

tf . = 0, j = 1(1)V* . All the free parameters in the expression of H (x' 3 are fixed in the process of construction of y

f l(x).

We proceed to approximate the ^-times differentiable function in

the same way as we did before for y(x).. Let (e (x)) be an m-th order Cm > " • -- n m Tau approximation of e

n(x), then

D (e (x)) = H (x) + H (x) , n ra n m

with (2), d = 0, j = 1(1)^. , implicitely defines

Only a rough approximation of required, as the order of it is

enough to estimate the error between y and y . In practice we use m n+l

for large n. Examples are given in the next section. It should be noted

that in order to construct the Tau approximant (e (x)) of e (x) only Q n m n n+l

...,Qra(x) need to be computed; all other items are available from the cons-

truction of y (x). n

-NUMERICAL RESULTS

2 We wish to compare our results with those of Miller . Miller considers 3

physical example provided by a Bernoulli beam-^column on an elastic foundatic

where the lateral deflection y(x) satisfies the differential equation

it D y(x) := Q EI y"(x) 2 + F

y " ^ + = p(x) (6;

on the interval 0 ^ x ^ L , with a distributed load p , an elastic foundation

modulus k , an axial compressive load F and a bending stiffness E I . Two boun-

dary conditions are given at each end of the beam. They 3re a linear combina-

tion of the values of y , y ' , y " , and ym

at either x = 0 or x = L .

Numerical results provided by Miller refer to the case when the lateral

deflection at x = 0 is equal to zero and a linear spring resists rotation

with stiffness R; the slope of y ( x ) is equal to zero at x = L and a linear-

spring resists rotation with a stiffness 3; an applied load is equal to an

axial load is equal to Fv and k = p = 0 . Under such conditions, equation (6)

reduces to

D y(x) := EI y""(x) + F y"(x) = 0 (7)

and the supplementary conditions are

/ y(0) = 0 , EI y"(0) - R y ' ( O ) = O

i y ' ( L ) = 0 , EI y»'(L) - S y ( L ) +- Q = 0 .

We take V = ( xn

J , n£3N ; the generating polynomial corresponding to ( 7 )

is

F (x) = D xn

= n(n-l)(n-2)(n-3)EI +-n(n.-l)F xn

~2

; n

therefore

0 (x) = - T xn + 2

- n ( n + 2 ) ( n2

- l ) EI <3 _(x) j ( G

*n

(n+l)(n+2)F L n~2 J

for all n€3N, then S is an empty set and s = 0 . For n = 0 , 1 , Pn( x ) = 0; thus

u^(x) = k ^ = constant,and u^(x) = x are exact polynomial solutions of ( 7 ) .

Let m f \ T ( n ) k T (x) = I c x n

k=0 *

be the expression of the Chebyshev polynomial of degree n defined for 0 x

L , as an algebraic polynomial in x . Clearly , for 0 £ x £ L ,

Q (x) = XI c <n )

Qk( x ) is such that D Q

n( x ) = T

n( x ) , n e i N , (!'

n

k=0

and then

yn( x ) = k

x + k

2x + T i

n )

Qn( x ) + Q

n - 1( x ) . (11

A.

The canonical polynomials Q (x) could have been defined directly, by using ^ A.

a generating polynomial of the form PR( x ) = D T

f l(x) . However, the approach

we have.followed offers several advantages: the simplicity of expression (9)

the possibility of using other choices of U , different from the Chebyshev

polynomials (Legendre polynomials, for example), without essential changes

in the procedure for the construction of yn( x ) , and its immediate applica-

bility to a segmented formulation of the Tau method

-5-

p T a k i n g , as in Miller , the dimensionaless values

*fFL2

= n2

E I ; 2RL = 57! EI; kSI? = 2 5 H3

E I ; l 6 0 L2

= T72

EI,

equation (7) takes the form

y"»(x) + y»(x) = 0 , 0 £ x ^ n / 2 ,

with the supplementary conditions

y(0) = 0

y"(o) - 5y'(o) = o c

y ' C n / 2 ) = 0

«vy»w(n/2) - 200y(T7/2) = - 1 .

Thus,from (9) Q

o( x ) = x

2

/ 2 ; O^Cx) = X3

/ 6 ; Q2( X ) = (x^ - 12x

2

)/12; ...

which replaced in (10) give the Tau approximant (11). The parameters k..,k_t (n) (n) A c

, and follow from (12). The adjustment of these parameters requires

the solution of a kxk linear system of algebraic equations irrespectively of

the order n of the approximation y ^ C x ) . This is the only matrix inversion

performed in the construction of the Tau approximant yn( x ) .

Table 1 shows the relative errors obtained for the deflection y and the oendi

moment M wi'th Tau approximants of orders n = 8,9,10,11, 20 and 2 1 , as well y

as Miller's reported results for 5 0 , 100, and 200 points with the best of

his two methods. The relative error obtained by Miller's finite difference

method by using 2C0 points compares with the Tau approximant of degree 11.

Eoth, deflection and bending. moment are approximated continuously over

when the method described in this paper is used.

TABLE 1 : Maximum relative errors for deflection and bending moment

TAU APPROXIMANTS FINITE DIFFERENCE

Degree n

Deflection

y .. . _ Bending

M

Moment .1

y. Number | Deflection of TDoints i y

Bending M

Moment y

8

9

10

11

20

21

0.212xl0~5

0.797xl0~8

O

0.129x10

0 . 4 1 2 X 1 0- 1 1

0.119x10 ^

0.258x10

0.653X10"5

0.480xl0~5

0.150xl0~5

0.526x10"^

0 . 3 7 ^ x l 0 "2 1

0.995x10*"2 5

50

100

200

O.HUxlO"*5

0 . 3 3 0 x 1 0 ^

' 0.725x10"7

I I

1

\ i 0.367x10" 1

0.919x10

0.2V7xl0~6

I

! Table 2 displays estimates of the maximum of the error function ®

n( x ) in

[]0,'n/2Uobtained with the estimator ( ©n( x ) )

m for n = 5, 10, 12, 20 , *nd

different values of m . Exact values of the maximum of the error function in

- 6 -

In all cases the estimator (e (x)) . gives n n+1

£otTT/2] are also given.

accur3tely;not only the order of the error, but its first two digits.

Error e s t i m a t i o n s T A B L E 2

ra Estimator: m a x (e (x)) dix^L

n m Exact value of the maximum error

10

12

20

6 7 8 20

11 12 13 22 13 Ik 15 22 21 22 23 25

0.2^9 0.2kG 0.2^72 0 . 2 ^ 7 0 5 ^ 5 1 9 3 3 7 5 2 5 O x I O "

5

xlO"^

xlO'C

0 . 2 ^ 7 0 5 ^ 5 1 9 3 3 7 5 2 5 0 x l 0 "5

xlO -1U

x i o i i i X l 0

- I 3 .

0.31^ 0.3130 0.312936 0 . 3 1 2 9 3 7 160299 58l

z

fxlb~J

"J

1'0.312937 160299 581^x10

— 0.306 x l O * 2 0.30^f7 X l 0 - .

X

J

0.30^66 x l O - ^ 0.30M*66 612558 5986x10

-111

O . M x l O - l7

OA^xlOJ,' o.Mfxio

0.30H66 612558 5986x10

O.MfxlO -27 1

O.MfxlO""2 7

Succesive estimations confirm the accuracy of the order of the error

provided by the first e s t i m a t o r .

We will now consider Falkner Skan equation ( see G e b e c i and B r a d s h a w ^ ) ,

which is an example of a third order nonlinear differential e q u a t i o n d e -

fined over an infinite d o m a i n . T h i s equation is relevant to b o n d a r y laye

problems in fluid m e c h a n i c s . By u s i n g similarity variables F a l k n e r Skan

equation takes the familiar form

D y(x) := y'» (x) + y ( x ) y " ( x ) + {3 \ 1 - £ y* ( x ) l \ = 0

y ( o ) = y' (0) = 0 , y'(co ) = 1 , 0 ^ X 00 (13)

we wish to estimate the missing boundary condition yM

( 0 ) . We take in this

example a numerically i n t e r e s t i n g choice of j0 , namely/? = 2 , and use y1

(3

=1 instead of the infinity condition prescribed in (13)- Later on we

check the validity of this a s s u m p t i o n computing the derivative of our T a u

approximant of y(x) at x = 9 and reporting its difference with the value

For details on the used of the recursive formulation of the Tau method fo

nonlinear problems the reader is referred to Ortiz

- 7 -

For y"(0),/3 = 2 , N a1 1

reports h i s own value of 1.6995770, obtained by

using invariant imbedding, and the value 1,6872170 as a standard one

from the literature on this e q u a t i o n . The difference between Na's in -p

variant imbedding and the standard value is 1.236x10 . The differen-

ce between the standard value and the one obtained by using a Tau approxi-

mant of degree 20 defined over the interval 0 ^ x £ 9 is 9*9x10"^; the o

derivative value at x = 9 differs from 1 (value of y1

( oo) ) in 6.xl0 .

In Table 3 we give the error of T a u approximants of degrees n = 15,17,19

(with respect of the standard value y"(0) = 1.6872170) and also the values given by the Tau estimator ( e

n

( 0 ) ) . n m

TABLE_3 : Tau approximate values of y"(0),i3= 2 , and Tau error estimates

n error m Tau estimator

(e»(0)) n m

15 6.*fxlO- 5

20 6.5x10

17 l.oxlO"5

20 1 . 7 x l O ~5

19 3.2xlO"6

20 k.lxlO"6

In Table k we report values of y! ,

(0^ obtained with invariant imbedding

•or given as. standard values in the literature surveyed by Na"^, f or = 3 . 0 ,

1.6 and 2.0. We also give, for 1 . 0 , 1.6,1.8, and 2.0, the values obtained 3

by using the recursive formulation of the Tau method . When ^ 2 , the error

T A B L E J ^ i y"(0) estimated for various values of j3 .

0 Invariant imbedding

Standard values^!

Tau approximants of degree 20

1.0 1.2^-23^00 1.?325870 1.2325875

1.6 1.5327980 1.5215120 I.5215139

1.8 . 1.606^802

2.0 1.6995770 1.6872170 1.6872160

in y' (9) falls below 1 0 ~1 5

.

- 8 -

B E F E B S H C E S

1 T.H.Gawain and R . E . B a l i , Improved finite difference formulas for bou value problems, Int.J.num.Meth.Engng.12,1151-1160(1978)

2 R.E.Miller, On consistent finite difference formulae for ordinary di ferential equations, Int.J.num.Meth.Engng

114 ,1567-1573(1979)

3 E.L.Ortiz, The Tau m e t h o d , S I A M J.Numer.Anal.,6, *f80-**92( 1969)

E.L.Ortiz,Canonical p o l y n o m i a l s in Lanczos T a u method, In "Studies i Numerical Analysis'

1

, B . P . K . S c a i f e , E d . , Academic P r e s s , New York(l975

.5 P.«0numa/yi, P h . D . T h e s i s , I m p e r i a l College,(1981)

6 M.R.Crisci and E . L . O r t i z , Existence and convergence results for t h e numerical solution of d i f f e r e n t i a l equations with the Tau method^Ini'n rial College Res.Rep.l-l6 (1981)

7 C . L a n c z o s , T r i g o n o m e t r i c interpolation of empirical and a n a l y t i c a l f tions,J.Math.Phys.,17,123-199(1938)

8 P.Onumanyi, E.L.Ortiz and H.Samara,Software for a method of finite a proximations for the n u m e r i c a l solution of differential e q u a t i o n s , A

;;

M a t . Modelling, in the p r e s s .

9 T.Cebeci and P.Bradshaw,Momentum Transfer in Boundary Layers,McCiraw-New York.. (1977)

10 E.L.Ortiz, On the n u m e r i c a l solution of nonlinear and function-..! di:' tial equations with the Tau M e t h o d , in "Numerical Treatment of Diff tial Equations in A p p l i c a t i o n s " , R.Ansorge and W.Tornig, E d s . , ^ori:. Verlag, Berlin (1978)

11 T.Y.Na/'Computational M e t h o d s in Engineering Boundary Value Prcclem; Academic Press,Hew Y o r k

:( 1 9 7 9 )

E R R A T A PAGE

1 10 27

LINE 9

12 21

READS S o m m e r f i e l d

SHOULD R E A D S o m m e r f e l d i£N-Z , in v iew of the osc i l la tory-nature of the so lu t ion

34 15 e r i s e s a r i s e s 58 10 c o f f i c i e n t s c o e f f i c i e n t s 58 12 In t h e . . . Cons ider the. . . 62 1 i

64 2 s t e fan Stefan 68 20 0 ( h 4 ) f ini te d i f f e r e n c e s 0 ( h ) finite d i f f e r e n c e 78 20 d e f f e r e d d e f e r r e d 81 6 L o g a r i t h m Logar i thmic 94 6 A. . . There is a. . . 95 8 turned out produced by-n 96 21 >

96 26 s a t i s f y ach i eve 99 8 D e s c r i b e s . . . Th i s example ' d e s c r i b e s

104 11 go tend 120 19 d i f f er e t i a l d i f f erent ia l 127 6 Randwe r tanfgab ena l s Randwertaufgaben a l s 127 7 Randwe r tanfgab en Randwe r tauf g ab e n

f o r m u l a e

G e n e r a l Correc t ion: C h e b e y s h e v shou ld r e a d Chebyshev