LEGENDRE FUNCTIONS AS SOLUTIONS TO

THE INHOMOGENEOUS HEAT EQUATION

by

GLEN A. BEAR, B.A.

A THESIS

IN

MATHEMATICS

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

Approved

August, 1995

•V-:' Are • ^^V'7

; i ' ^ ^ ACKNOWLEDGEMENTS

C • ^ I woukj like to express my sincere appreciation to my advisor Professor

Wayne T. Ford for his guidance and encouragement throughout the preparation

of this thesis.

I also wish to thank the State of Texas for support under the Texas Higher

Education Coordinating Board - Advanced Technical Program (Project No.

003644-162) and for additional support under the Minority Supplement Support

Program (Project No. 003644-162).

II

CONTENTS

ACKNOWLEDGEMENTS.

ABSTRACT iv

I. INTRODUCTION.

II. THE WEIGHTED HEAT EQUATION 4

III.REDUCTIONOFORDER 10

IV. ADDITIONAL SOLUTIONS 17

V. CONCLUSION 20

REFERENCES 21

III

ABSTRACT

This thesis is a study of the classical and weighted heat equations, with a

detailed examination of solutions for a particular case of the weighted equation.

This particular case Involves the substitution of a weighting factor into the

equation which reduces it to Legendre's differential equation. Several methods

of finding solutions for this particular case are given, along with solutions

computed using the given techniques.

IV

CHAPTER I

INTRODUCTION

Consider the classical heat equation in the form

d (duix,t)\ _ du dx\ dx J dt

X69t / > 0 (1.1)

and the weighted heat equation in the form

dx in Bt

(1.2)

where the w in w(x) stands for a weight. Although it might seem that (1.2) is quite

similar to (1.1), "Little attention has been paid to diffusion in nonhomogeneous

media in which the diffusion coefficients vary with distance measured in the

direction of diffusion" [6].

The outer partial in (1.2) with respect tox is indicated as an iterated partial to

emphasize that a major purpose of this paper is to insert a variable conductivity

between the outer and inner partials. This paper treats conductivity variable in

position in (1.2) rather than the nonlinear case, noted in Ford [10], as in

dx

du du

It (1.3)

which has been studied by Aronson [2] who gives a particular closed form

solution for the particular nonlinear version of the heat equation in

dx mu m-\ _i du

dx_ = —(u"') = ~

dx' dt m>\ (1.4)

Consider the heat equation itself in (1.1). So that the computations in later

discussions may be fully understood, let us review the method of solution of this

particular equation through separation of variables. To begin, assume the

solution is of the form

uix,t) = vix)zit) . (1.5)

1

Then

du dvix)

dx dx

d~u d'vix)

U) , (1 -6)

z(t) , (1.7) dx^ dx~

and - = v ( . ) — . (1.8)

Upon substituting (1.7), and (1.8) into (1.1), we find

d\ix) ^dzit) , Q. zit) = v(x)—— , (1.9)

dx^ dt

and dividing by v(x)z(/) gives

d\ix) dzit) dx"- _ dt v(x) zit)

(1-10)

As can be seen, the left side of (1.10) is dependent on x only and the right side

is dependent on t only, hence, this equation must be equal to a constant.

Set ^'v(x) dzit)

which implies

and

• ^ ^ = - ^ = - A ^ . ( 1 . 1 1 ) vix) zit)

d~vix)

- ^ = -A^ (1.12) vix)

dzit)

-^-X- . (1.13)

Upon multiplying (1.12) by v(x) and (1.13) by c(r)and rearranging both

equations, we have

^'v(x) + A'v(^) = 0 (1.14)

dx' and

^ + A z(r) = 0 . (1.15) dt

By (1.14), we find the solution to v(x) is

v(x)= AsinAx+BcosAt , (1-16)

where A and B are constants. Integrate (1.15) with respect to r to find

lnz(0 = -A / + C or

zit)=De-''' , (1.17)

where again, C and D are constants. Recalling uix,t) = vix)zit) and substituting

the above results gives

-kh M(x,r) = (AsinAx+BcosAx)D&"^' (1.18)

This is a well-known solution to the classical heat equation [20].

CHAPTER II

THE WEIGHTED HEAT EQUATION

Let us now return to the weighted heat equation

d_

dx

du du (2.1)

and apply methods similar to those used to find a solution to the unweighted

heat equation to attempt to find a solution to (2.1). If we again assume the

solution M(X,/) can be expressed in the form

M(X,/) = V(X)Z(0 (2.2)

then, as before,

and

du dvix)

dx dx

d^u _ d^vix)

dx' dx^

Zit)

Zit)

du dzit)

dt dt

Upon substituting (2.3) and (2.5) into (2.1), we find

^ ( 0 d[ ^ ^dvix) ^ 1 ^ -dt

(2.3)

(2.4)

(2.5)

(2.6)

or, equivalently

dwix)dvix)^^^ . , .,fi? v(x) ^ , ..^.dzit)

dx dx zit) + wix)-

dx' lit) = vix)-

dt

As before, divide by v(x)z(/) to get

dwix) dvix) ^ ^ ) / v ( x ) dzit)

dx dx vix)

+ dx' vix)

dt zit)

(2.7)

(2.8)

which again can be seen to be equal to a constant, due to the fact that the left

side is dependent on x only and the right side is dependent on r only.

Hence, set ^ ( x ) ^ x ) . .d'vix) dzit)

w(x) — dx dx j dx' _ dt _ ^ (2 9)

v(x) v(x) zit)

which implies

^ ( x ) ^ x ) , .d~vix) w(x) =—

dx dx . dx + and v(x) v(x)

dzit)

= -?: (2.10)

dt zit)

= -A= . (2.11)

Upon multiplying (2.10) by v(x) and (2.11) by z(r) and rearranging both

equations, we have

^ w ( x ) ^ x ) , ^d^x) dx dx ^ - ( x ) ^ ^ + A-v<x) = 0 (2.12)

and

^ ^ + A'::(/) = 0 . (2.13) dt

By (2.13), we arrive at the same solution for zHt) as before, namely

zit)=Ae-''' . (2.14)

As it is most likely that "The term wix) in (2.7) makes the medium

inhomogeneous, and the solution in closed form then ranges from difficult (for

special cases) to impossible" [21], the rest of this paper will be limited to

considering one special case of the weighted heat equation, namely when

wix) = 1 - x . Therefore, consider (2.12) again, but this time make the

substitution wix) = l-x' . Then we have

d /, 2\dvix) I ^sd\ix) .-, , , ^ — l - x ' ) - - 3 - ^ + ( l - x - ) — - ^ + A-v(^) = 0 2.15) dx dx dx

or, carrying through the derivation and rearranging gives

/, 2\d'vix) ^ dvix) <,. ^ ,/s-.,,v (1-x )-—^- 2x—^—^ + X-v{x) = 0 (2.16)

dx^ dx

Compare (2.16) with Legendre's differential equation

{l.,^-)£^.2x^^ + viy + mx)^0 . (2.17) ^ ^ dx^ dx

It is obvious these are the same equation with the identification

v(v + l) = A' . (2.18)

Carrying through the multiplication and rearranging we have

v' + v - A ' = 0 (2.19)

and upon solving for v using the quadratic formula, we find

-l±Vl+4A' V =

Make the following designations for clarity in future calculations:

and -1-V1+4A'

^ 2 =

(2.20)

-1+V1+4A' ,„^^, V, = z (2.21)

(2.22)

Solutions to Legendre's equation (2.17) are

r v(v +1) . . V(V+1XV-2XV-H3) 4 S = fln 1 X + X

1 2! 4! v(v-HXv-2Xv + 3Xv-4Xv + 5) , ^

and

T^a. iv-Wv + l) 3 (V-1XV-H2XV-3XV+4) 5 X — X + X

3! 5 (V-1XV + 2XV-3XV+4XV-5XV-H6) 7 1

7! J (2.24)

where a^ and a, are arbitrary constants [16]. For comparison with later

calculations, call the terms inside the brackets above 5; and T^, respectively.

As 5; and 7 are linearly independent, and each series is a solution of

Legendre's equation, any linear combination of S^^ and T;^ Is a solution. Hence,

the general solution is given by

v(x) = ao5;i+a,ri (2.25)

Given the computation of v, and v^ in (2.21) and (2.22), we are now in a

position to substitute Into (2.23) and (2.24) to determine two linearly

independent solutions to (2.16). Upon substituting v, into (2.23), we have

1 S,ix) = l - -

( 1 1 1 1 _i^iViT3F -^^-ivrr^ 2!LV 2 2 2 2

l_ 6!

. 2 2 A2 2 )]

SM) = • - x\^'V + [( 'XA^ -6)]x^ -1[(A^XA^ -6XA' -20)];c' +... 2! 4! 6!

5.(^) = 1 - ^ ^ ' ^ - ^ 120A' - 26A + t

6! x^+... (2.26)

Upon substituting Vj into (2.24), we have

5!LV 2 2 A2 2 A 2 2 A2 2 J.

7!LV 2 2 A 2 2 A 2 2 A2 2 J

2 2 2 2 X +...

T,(x) = x-^[X--2]x' +1[(A' _2XA' -12)]r

-:|;[(A' -2X A' -12X A' -30)]A:' +...

.-. 7i(x) = x -2-A' 3 24-14A'+A^ 5 720-444A' + 44A^-A' 7

3! 5! •X' +

7! x'+... .(2.27)

Hence, we have two solutions of (2.16). A calculation similar to that which led to

(2.26) and (2.27) will give additional solutions by substituting v. into (2.23) and

(2.24). However, these solutions will be the same as (2.26) and (2.27), as is

apparent by comparing the form of v,and v^.

Let us conclude this section by taking a brief look at a substitution which will

be considered in more detail in the next section, namely, let wix) = 14- x'. Upon

substitution into (2.12), and rearranging the equation, we see

d r dvix)-] dx

= -iMx) (2.28)

Then, make a change of variables, letting x = irf, hence dx = idr\. After

substituting into (2.28) we see

\d_ i dt]

1-ry dvjir]) i dr] _

= -A-v(/r]) (2.29)

Letting V(T]) = viij]) and simplifying, we are left with

_d_ dt]

1 - 7 ] ' dvir])

dr] 2 '^y

= A^ ^T)) (2.30)

8

As in (2.16) and (2.17), this is again simply Legendre's differential equation with the identification

i7(v+i)=-A- , (2.31)

where the "bar" designation on v distinguishes (2.31) from (2.18). Using the

quadratic formula as before, and solving for v, we find

- - l±Vl -4A' ,^^^, V = ^ . (2.32)

Make the following designations for clarity in future calculations:

_ -1+V1-4A' . = 2 (2.33)

and

' 2

Hence, by using the given solutions of Legendre's equation in (2.23) and (2.24),

we are now in a position to find additional solutions of (2.30). Namely, upon

substituting the above values of v, and v. into (2.23) and (2.24) and following

similar calculations which led us to (2.26) and (2.27) we find

- ^ ^ , A' 2 6A'+A' 4 120A'+26A'' + A' , ,^ ^.. S^ix) = 1 + —X + — X + X +... (2.35) • + X +

2! 4! 6!

and 2 + A' 3 24 + 14A'+A'' s

ri(x) = x-i- X +

+ •

3! 5

720-444A- + 44A'+A' 7!

x'+... (2.36)

where again the "bar" designation on the S and T distinguishes (2.35) and (2.36)

from (2.26) and (2.27).

CHAPTER ill

REDUCTION OF ORDER

We have been considering

dx wix)^\ = -?Cv{x)

dxj (3.1)

in the special case

_d^ (i+r) i.dvi^)

dx = - v ( ^

Let ir] = ^ to obtain

(3.2)

or

\d_ i dr)

dr)

1 _

1-77^ dvjr)) i dr)

il-r)')''^''^

-v(n)

dr) = V(77)

If we imagine vir)) to be transposed to the left side of (3.4), we obtain

Legendre's differential equation with the identification

v(v + l) = - l

which implies _

V = — ± / — 2 2

(3.3)

(3.4)

(3.5)

(3.6)

If V, is defined using the plus sign, and v is defined using the minus sign, it is

interesting to observe

and

v , + l =

V2 + 1 =

I .^^ . 1 s - - + 1-

2 2 J + 1 = - + / — = _v,

2 2

1 . ^ ^ , 1 .Ji 2 ^ 2

+1 = - - / - = - = - V , (3.7)

10

As (3.4) has a regular point at x = 0 , the power series method provides a

solution of (3.4) in a neighborhood of this point [22]. Hence, let

v(n) = X ;t * it=0

Then, suppress subscripts on v to obtain

^k+2 ~ m-^D-vi v-n)

ik + l)lik + 2) for 0<k<<^

(3.8)

(3.9)

It should be noted that If v is an integer, (3.8) reduces to a polynomial.

However, in the case wherev is not an integer, whether (3.8) is an acceptable

solution, and therefore of interest, depends upon its convergence properties.

The ratio test for convergence says that if in a series of positive numbers the

ratio of the (n+1)th term to the nth term approaches a limit L as n increases

without limit, and if L is less than one, the series converges. That is,

convergence requires

where L < 1

It is evident that in equations (2.23) and (2.24) the ratio is

N but from equation (3.9) it is readily seen ttiat

(3.10)

(3.11)

limlT^=lim k

A:(;t + l ) - v ( v + 1)

ik+m+2) = lim

k

V(V+1)

k + 2 ik+lXk+2) = 1 (3.12)

Hence, the condition that equations (2.23) and (2.24) converge is that x be less

than 1, and this is true only If |x| < 1 • Hence, for values of x in the range

-1 < X < 1, the series solutions are valid.

Alternatively, consider (3.1) In the special case

11

dx \^-^'')^V-X'yix) . (3.13) dx J

where the normalization replacing x by using = Ax has been dropped, and

wix) has a minus sign leading directly to Legendre's equation in the form

d__ dx

( l - ; c ^ ) ^ l + A^(x) = 0 , (3.14) dx J

Where we identify v(v + l) = v'+v = A' . (3.15)

Solve (3.15) for V in the form

v = ^ (3.16)

Overcome the tendency to imagine some restrictions on choices of A , and

consider a sequence of values of A defined by

l+4A'=(2v + l)^

4A'=4v-+4v ,

A = v + V and

A = ±Vv' + V (3.17)

where v takes the successive values in 5i = {o, l, 2, ... }corresponding to the

positive sign in (3.16). If the values in (3.14) were used with the negative signs

in (3.16) the successive values of v would lie in ^ = {-1,-2,-3 ... }. The sum of

each element in S^ and the corresponding element in 2 is - 1 . Then (3.14)

becomes

- ^ [ ( l - x ' ) ^ ^ 1 + «(/i + l)v(x) = 0 where n = v E:S, (3.18) dxL dx J

which has the Legendre polynomial given by Rodrigue's formula in

PnM-— \ „ ^ wheren = v e 5 , . (3.19)

as one of Its solutions.

12

If we let Vj =P„ ,a second solution of (3.18) can be obtained by reduction of order to obtain Vj „ = q„P^ using

-nin + l)q„ix)P^ix)

dq„ix) dx -'{ =-7-i (1 - xOI P , ( J : ) - ^ 5 ^ + q',(x)

dx dx J.

= [(i-.^)/',(.)^]

dx dx dxL ^ J (3.20)

where « = v 6 5*,. Set

^ ^ n ( ^ )

dx

and use /J in (3.18) to work with (3.20) as follows:

Thus,

0 = s n(n + l)P„(x) + ~ [ ( l - x ' ) ^ ^ 'q.ix)

^ £ r ( j _ , . ) ^ ( , ^ ^ L ( i _ , . ) ^ ^ dxL dx A dx dx

^±[i\^x')P„ix)z„ix)]+il-x'')^^^z„ix) dx' dx

= (1 - x ' ) P , ( x ) ^ ^ + [(1 -x')P^ix)]z^ix) dx dx

, dPix) +il-x')^^^zM) . dx

(3.21)

(3.22)

_(l_;,^)P,(;,)*5(i>={i-[(l_;,^)/>,(;c)]+(l-;c^)^k(;c) , (3.23) d[r l<xc dx )

and H ^ '' _

1 dzix) zjix) dx

1 -i[(l_^^)P^(;,)]^-i«^^W (l-x')P„(x)d!r^ " •• P„(x) dx

13

(3.24)

This may be integrated to obtain

ln[(l-x')P„(x)] + lnP,(x) + lnc„(x)

= ln[(l-x')P/(x)z„(x)] = 0 . (3.25) Hence, by (3.25) we see

.2\n2,

or l=(l-x^)P;(x):;(x)

1 ,ix) =

il-x')P^ix) (3.26)

Thus, we know by (3.21) that

and "<^ = l(l-i=(x) - P{ xf dx

• ' ( l -x')P;(x

(3.27)

(3.28)

where w = v € 5,. Perhaps the complexities in the above computations suggest checking the

solution in (3.28) against Legendre's equation. Compute

'^-'^'-'^'^.ixHP.M^'-dx dx

dPnix)

dx

1 dx '^''^''^•^(l-x^)P,(x)

(3.29)

and dxl

i\-x )• dx _

= - -^( l -x^) dx

4P„(X)^„(X)]^

dx

d_ dx

jn 2jdP„ix)^ dq„ix) -\ < ( 1 - X ), 1 qnix)^ ^_ Pnix)j 1~^

dxl 2,dPJ^

dx

dx

( l -x ' )^^^-^?.(x) j + —j^d-acO ^ . , dx " ""'J

-iK-'-^V'-^-'-'^^^^L^J 14

/ ^ i ^ D / . / . 1 dp^ix) ^ r 1 1 = -nin + l)PM)^ix) + i T + —I I

[P„ix)] dx dxUix)\ = -n(n + l)P,(x)^(x) = -n(/i + 1)V2 „(x) , (3.30)

and V2 (x) satisfies Legendre's equation.

To continue, use , from (3.27) to compute

. . _ f ax f ax •'(l-x=)/^^(x)"J(l-x>^

1 X

(3.31)

which has singularities for x € { - l , 0, 1} . However, use v,, from (3.28) to

compute

V2.,(jc) = /f(x)(?,(x) = x ,(x) = xlnJ-; I 1-x

- 1 (3.32)

which has singularities for x e {-1, 1} .

As an example, note that Legendre's polynomial for n = 1 compares with

(2.24) as shown in

/> = 7;(x) = x . (3.33)

To develop v^^ from (3.32) for comparison with (3.33) recall

I I 1 2 1 3 1 4 I n l + x =J<^—X + - X — X .+ ...

' 2 3 4 (3.34)

Although unnecessary, write

I i 1 2 1 3 1 4

ln|l-x| = - x - - x - - X - - X (3.35)

for Clarity in using (3.32) to compute

15

v.M+i = x l n J i i ^ =-ln — = i[ln|l+x|-ln|l-x|l (3.36) V l - x 2 1-x 2^ ' ' J

X

2 3 4 J

^r _ i 2 _ i 3_1 4_ 1 2 3 4 J

= X 1 3 1 2 1 4

X + - X +... = x +—X + 3 J 3

(3.37)

Thus, V2 i(x) = - 1 +x^ +-x'^ +... = -Sj^ix) (3.38)

16

CHAPTER IV

ADDITIONAL SOLUTIONS

Recall, from the variation of parameters method, a second solution to

Legendre's equation may be found as

^2.n = P.Mq.ix) (4.1)

where/i(x) are the associated Legendre polynomials and

"^^^'^-^IT^) (4.2)

Let us find a few more solutions to (3.15), given the following Legendre

polynomials [22]:

P,ix) = 1

P,ix)^x

P2(x) = 1(3x^-1) 2

P3(x)=i(5x^-3x) 2

P4(x) = -i(35x'^-30x'+3) o

1

(4.3)

From (4.2), we see

or

r dx '?o( ) = J^IT7^

^o(^)=jl° 1+X

1-X

(4.4)

(4.5)

Hence, by (4.1)

V2.o=Po( ) o( ) = ( l ( j l n l + x 1-x

Mm J 2

1-fx

1-x -"H (4.6)

17

We have already calculated a solution associated with /f(x). For comparative

purposes it will be restated here, that is

V2 ,(x) = /f (x)^,(x) = x^,(x) = X V l - x

- 1 •

To find V,. we have

dx f ax %M = } 7- ^

(1-JC')J-(3J:--1)J

or, by partial fractions

r i i f 1 2 2

^^' ~ -I I (1 - x) " (1 + x) '^JSx^ -1)' • (3X--1)

Hence,

1 6 3 ,,

+ : 'dtc

(4.7)

(4.8)

(4.9)

^2(-t) = ln ll + x

Vl-x 3x

(3x- -1)

and 1

v., =PJx)q2ix) = -i3x'--l) r

\\-x 3x 1

(3x^-1)

Hence,

V2.2 = P2(^)^2(^) = (3-^' - i^^^ylTlf 3x Y

Similar calculations will show that

%ix) = J dx

(l-x^-)[ix(5x^-3)] I2

(4.10)

(4.11)

(4.12)

(4.13)

or

qiix) Jl + x —

4 25x

9x ~ 9(5x- -3)

(4.14)

18

Again by (4.1)

V2.3 = P3( ) 3( ) = ^(5^' -3x)| li J 7 7 7 4 25r 1 9x 9(5x--3)J

(4.15)

Hence,

Vo 3 = -(Sx" -3x)ln J • 2 V l l - ^

5 o 2 —X" + -

2 3

(4.16)

Continued calculations in a manner similar to those above would generate an

infinite number of solutions to (3.14). It is interesting to note that each solution is

dependent on a combination of a logarithmic term times the nth degree

Legendre polynomial plus linear combinations of lower order polynomials.

Perhaps further computations would yield a simple algorithm for computing a

particular order solution without the necessary integrations.

19

CHAPTER V

CONCLUSION

In this thesis, we have studied the classical and weighted heat equations,

with an emphasis on the weighted equation

d^ dx

>v(x)--ox.

du

By assuming the solution is of the form

w(x,0 = v(x)z(0 (5.2)

and applying the method of separation of variables, we were able to reduce the

problem to one that resembled Legendre's differential equation as shown in

Chapter II. Solutions were then computed by substitution into well-known

solutions for Legendre's equation.

Furthermore, it was shown in Chapter III, that an infinite number of solutions

could be generated for (5.1) when wix) = 1 - x by using the method of reduction

of order. In this particular case, the additional solutions were

given by the formula

y2.n = P.M^n(x) (5.3)

where

'?»W = I(1_;,= )/. (;C) <^*)

and P (x) are the Legendre polynomials. Several solutions were calculated

using this formula, and It was noted that successive solutions were

combinations of preceding solutions. Perhaps further investigation would yield

a simple algorithm which would allow one to forego the prohibitive integrations

necessary to compute additional solutions. I feel the best step in this direction

could be made with careful considerations of inductive arguments.

20

REFERENCES

[1] M. Abramowltz and I. A. Stegun, Hanc^book of Mathematical Jmctions, NBS, 1964, Dover reprint, Mineola, NY, 1972.

[2] D. G. Aronson, "Regularity Properties of Flows Through Porous Media," SD/m 'JoiArml of/^piied Mathematics 17, pp. 461-467, 1969.

[3] Richard M. Barrer, Diffi^sion in ani^l Through Solic^s, Cambridge University Press, Cambridge, 1951.

[4] Jacob Bear, Dynamics ofjiuiids in Porous Media, Elsevier, 1972, Dover, Mineola, NY, 1988.

[5] William H. Beyer, Ci^e Standard Mathematical Tables. O^C Press, Boca Raton, FL, 1978.

[6] J. Crank, The Mathematics of Diffusion. 1st edition corrected, Oxford, Gary, NC, 1964.

[7] J. T. Gushing, Applied Analytical Mathematics for physical Sciences. Wiley, New York, 1975.

[8] Ross L. Finney and Donald R. Ostberg, Elementary Differential Equations with Linear Algebra. Addison-Wesley, Reading, MA, 1976.

[9] Wayne T. Ford, Elements of Simulation ofjluid Jlow in Porous Media. Texas Tech University Mathematical Series No. 8, 1983.

[10] Wayne T. Ford, Solutions of the Heat Equation in a inhomogeneous Rod. Texas Tech University, 1995.

[11 ] Wayne T. Ford, Porous Media Jlow: An introduction. Texas Tech University, 1994.

[12] Wayne T. Ford and Ronald M. Anderson, Mathematical Methodology for Evaluating Simulations of Jlow in Porous Media. Technical Information Center, U. S. Department of Energy (DE84004775), 1984.

[13] Robert A. Greenkorn, Jlow Phenomena in Porous Media. Marcel Dekker, Inc., Monticello. NY, 1983.

21

[14] Seize Ito, Diffusion Equations, Translations of Mathematical Monographs Series No. 114, American Mathematical Society, Providence, Rl, 1992.

[15] W. Jost, Diffusion in Solids. Liquids. Qases. Academic Press Inc., San Diego. CA, 1952.

[16] William Wayne Kitts, Legendre Junctions. Master's Thesis, Texas Tech University, 1949.

[17] ZhuU, The Discrete Observability of the Heat Equation. Master's Thesis, Texas Tech University, 1987.

[18] James A. Liggett and Philip L-F. Liu, The Boundary integral Equation Method for Porous Media Jlow. George Allen and Unwin, London, 1983.

[19] Jin Liu, A Numerical Method for inverse Heat Conduction Problems. Master's Thesis, Texas Tech University, 1989.

[20] Zaiman Rubenstein, A Course in Ordinary and Partial Differential Equations. Academic Press, San Diego, CA, 1969.

[21] Paul G. Shewmon, Diffusion in Solids. McGraw-Hill Book Company, New York, 1963.

[22] Walter A. Strauss, Partial Differential Equations. An introduction. John Wiley and Sons, New York, 1992.

[23] Morriss Tenenbaum and Harry Pollard, Ordinary Differential Equations. Harper and Row, New York, 1963.

[24] H. F. Weinberger, A Jirst Course in Partial Differential Equations. John Wiley and Sons, New York, 1965.

22

PERMISSION TO COPY

In presenting this thesis in partial fulfillment of the

requirements for a master's degree at Texas Tech University or

Texas Tech University Health Sciences Center, I agree that the Library

and my major department shall make it freely available for research

purposes. Permission to copy this thesis for scholarly purposes may

be granted by the Director of the Library or my major professor. It

is understood that any copying or publication of this thesis for

financial gain shall not be allowed without my further written

permission and that any user may be liable for copyright infringement.

Agree (Permission is granted.)

Student's Signature Date

Disagree (Permission is not granted.)

Student's Signature Date