computational error analysis of periodic solutions for singular semilinear parabolic problems

Computational Error Analysis of Periodic S Sing@w Semi&near Parabolic Problems

C. Y. Ghan’ and Benedict ha. Wong

Department of Mathematics University of Southwestern Louisiana Lafayette, Louisiana 70504-1010

ABSTRACT

By using the monotone method, a theoretical and computational method is given to find, to the degree of accuracy desired, approximate solutions of a class of singular semilinear parabolic problems. So that the error between the actual solution and its approximation is within a given error tolerance, the number of iterations is determined. Since each iterate is in terms of an infinite series, the number of terms to be retained in each iterate is determined so that its error from the exact iterate is within a given error tolerance. An improved rate of convergence is then given to show that it is possible to reduce the number of terms retained in each iterate. An algorithm is also described to obtain numerical solutions. For illustration of the computational methods developed, a numerical example is given.

1. INTRODUCTION

Let

b L,u = u,, + ;ux + c( t)u - u,,

where the constant b < 1, c < 0 and c is Lipschitz continuous on [O, 1’1. Also, let

fim = (0,a) X (--M),m)

*The work of this a uthor was partially supported hy the Board of Regents of the State of

Louisiana under Grant LEQSF(?36-89)-RD-A-11.

APPLlED MATHEMATZCS AND CQMPWATZON S&279-306 (1992)

0 Elsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010

279

0096-3003/92/$5.00

280 C. Y. CHAN AND BENEDICT M. WONG

and 0, be its closure. In [3], we establish existence and uniqueness of a time-periodic solution u of period T for the following singular semilinear parabolic problem:

i&u = -f( x, t,u) in Cl,, (14

U(W) = gl(t), u(a, t) = g2(t), --oc < t < w (1.2)

The main purpose here is to provide a theoretical as well as a computational method to find an approximate solution of u to the degree of accuracy desired. The ideas introduced here can be applied to many other nonlinear problems solved by iterative methods.

For ease of reference, we use the same notations of [3] unless otherwise specified. Throughout this paper, we also assume that the conditions given in [3, Corollary 153 f or existence and uniqueness of the solution hold. The monotone method (used in [3, pp. 426-4271) gives a sequence { uj( r, t)} of upper bounds as well as a sequence {E~( x, t)) of lower bounds, and each of these nonverges monotonically to the solution u. Since we require their explicit ‘kns, which are not stated there, let us list them here (by using Corollary rf of 131). For further reference on the monotone method, we refer to [41.

Let

c = /‘C(s) i&/T, 0

(131 ’ I

where n is a positive integer, v = (1 - b)/2, Jy(x) is the Bessel function of the first kind of order v, and A, (arranged as A, < A, < A,. . . ) satisfy I,(h%a> = 0. Also let

c(s) ds - C(t - mT)

281 Computational Error Analysis

for any integer m such that mT < t < (m + l)T,

G&&&r)= i i?n(t-~;C)~b$n(~)&(x) forO<t-T<<, ?I= 1

(14) .

G&,t;&,r) =i&(T+t-r;C)tb4n(f)4n(~) for-TT<t-r<O, fl= 1

(15) .

a( x, t) = a-2v[(a2v - x2v)g1( t) i- xZYg2( t)] ( 1.6)

[a, p. 4181, and Gc_P,(x, t; 5, r) be G,(x, t; 8, T) with C replaced by C - J,Q, where p., is a constant described in [3, p. 4271. For simplicity, we introduce the following notations:

&(h)(t, z) = /‘xbh( x, t, z)&(x) dx, 0

I,(h)(t) = /“Xbh(x,t)&(X)& 0

I,,(h) = jaxbh( x)&(x) dx, 0

12(h)(t, z) = /“xbh2( x, t, z) dx, 0

12(h)(t) = ,(‘xbh2( x, t) dx, . 0

12(h) = ,faxbh2( x) dx, 0

and for every non-negative integer j,

i!$( X, t) =f(X, t, iiij) + /l+lii;ii,

282 C. Y. CHAN AND BENEDICT ha. WONG

Since Bj and zj are T-periodic, it suffices to consider t in 10, Tl. It follows that

M,(u) = [HT(t)]+ ~T~aC,(x,t;5.~)HT(~) 0 0

x[~(O) +L,cu(~,r)]d~d~+(~(x,t)

= [r-r,(t)]-‘(.r; [&,(u+ L,a)(&(t - cC)HT(~) dT

X&(x) + i jTZn( c + L,a)(&(T + t - 7;C) n=l t

x&(r) dr&( x) 1 -+ a( x,t), (l-7)

where a(x, t) is a (given) function described in 13, pp. 422 and 4261,

+L c-& 6, T)] de d7 + a( x, t)

= [&(t)]+ (i /‘&(3 + L~,_,,+)&,(t - Cc - IQ) n=l O

XH&) de”(x) + ii lTI,(Fj+L,-,,+) n=l t

xE,(T + t - r;C - pJ&(7) d?+,,(x) 1

+ +,t), W)

Computational Error Analysis 283

+L c-p,a( 6, r)] dS Jr + a( x, t)

= [f&(t)]-‘{ $ [I”(4 +&p)(+f,,(t - cc - ~1) n=

x~?~(T + t - CC - lcJHT(+Wn(x) 1 + a(W), (l-9)

where n-+,(x, t) = y(x, t), which is 421-4221.

a Igiven) function described in [S, pp.

Although the monotone method shows that as j + 00, Rj< x, t) J, u( x, t)

AJ zj_Ci; t 1 t u( x, t), in practice we can never reach u( x, t). In Section 2, we determine the number j + 2 of iterations required such that llu - Rj+ 1 11

or Ilu - _m.+,ll I

is within a given error tolerance; here 11 l II denotes the sup norm of t e function on the closure a- of fi = (0, a) X (0, T). Since

G,(x, t; &?) and G,_,, (x, t; 5,~) are in terms of infinite series [cf. (I.4 and (L.5?1, we cannot compute their exact values numerically. In practice, we need to truncate the series. By establishing error estimates due to truncation, we can determine the number of terms to be retained in the series in each iterate SO that the approximate solution $j+ 1(:x, t 1 (Fj+ It X, t)) is such that - 1lM j-t-1 - lii,+le llEj+l - Fj+llJ) is within a given error tolerance. This means that given a positive c > 0, it is possible to determine the number

j -t- 2 of iter_ations and the number of terms (in the series) in each iterate so that 11~ - Mj+ Jl < E (11~ - ej+ ,I1 < E). In Section 3, we give an improved rate of convergence so that in computing ij or F~, it is possible that fewer terms are needed in each iteration. In Section 4, we describe an algorithm to obtain the numerical values of G (x, t) and ~9_$ x, t). For illustration of the computational methods develope d , a numerical example is given in Section 5. We remark that because of the singular term bu,/x and the fact that u,(R t) is not necessarily zero, difficulties arise in obtaining the Z’-petioik solution u numerically, for exmaple, by the finite difference method.

C. Y. GHAN AND BENEDICT M. WONG

2. ERROR ESTIMATES

The following result follows from [7, pp. 490-4941 and [I, p. 3701.

b’ulMA 1. If -l/2 < v< l/2, thenforn = 1,2,3 ,...,

( n - l/4 + v/Q?r< h!,i2a < (n - l/8 + v/4)~;

if v > a and h!,j2a 2 (2v + 1x2 v + 3)/n for some n 2 Nl where Nl is a positive integer, then for n >, N,,

( n - l/2 + v/2)Ir< hii2a < (n - l/4 + v/2)W.

By Bessel’s inequality (cf. [8, p 7311,

I# -S)(T)1 < [12(4 -q)(T)]1’2.

If b >, 0, then

< (Q2 + ~lab/2)2all~ - sjI12,

where Q2 [3, p. 4271 is a Lipschitz constant such that for UC x9 t! aud V( x, t) lying between m,-,(x, t) and-H&, t),

(“b’2[f(W,U) -f(x,t,v)]I G Q21+,t) - u(x,t)l. (2.1)

If b < 0, then

sup O,<T<T

[12(&--$)(r)] < (amb/ZQ, + p1)2al( Cbi2(y - Ej)1)2


Therefore if b > 0, then

SUP ( ln[Fj -S)(T) 1 \< (Qz + ~~ab’2)a”211ii?j - ~jll; (2.2) O<TdT

if b < 0, then

In [3, p. 4261, we let

p2 = max [ HT(t)] -l: t E [0, T]} , 1 (2.4)

& = max(H,(t): t E [(AT]). (W

Let

p = p2 &( Q2 + plabi2 )a1/2,

Sp”Jr) = sup 1 i rb’21+n(X)l/r ’

OdXGO n=p 1

Then for b 3 0, it follows from (1.8), (l.Q), and (2.2) that

Ilii?,+l - Ej+l II G Psl,a( rl,>llaj - Ejll

< [ ~sl.,(r~,)]j+llldM, - i!?_%$ (2 6) l

Similarly for b < 0, it follows from (1.81, (l.Q), and (2.3) that

II ( .+2 a j+l


Since

Ilir?,,, - E~+JI G a-6/2l( x”‘~(M~+~ - ~~+~)ll for b < 0,

it follows that

IIii?i, 1 - n_zj+ JI < [ d’/pS;,,( q,)] j+ 1a-b’211 X~‘~( i&, - _m,) 11. (2.7)

Because t&e solution u of the problem (l.l)-(1.2) is unique, each of the sequences {Mi) and (~j} converges to u as j + 00. The following result is useful for detemking the number (j + 2) of iterations so that llu - M7,+ 1 11

orllu - gj+ Jl is within the given error tokrance E.

THEOREM 2. Suppose

ps,_,( Q”) < 1 forb 3 0,

a -b’2ps;,m( rln) < 1 for b < 0.

Then given any E > 0,

provided

j + 1 > (ln( 8/IIM, - !?!oll)/ln[ PS,,,(r,“)]) forb 2 09 (2.8)

j + 1 3 {lnf 8/(a-b/2JI Pi2( a0 - _m,) II)] /ln[ a-b/2PSt,,(r,,)]}

forb < 0. (2 9) .

PROOF. lows that

Since ~j+ 1 i\< u < il?i+ 1 for all non-negative integers j, it fol-

lb - ii;i,,,ll < llBj+, -~j+lll, Il” - Zj+1ll G Ilitij+l - m -j+ 111*


For the theorem to hold, it sufkes that for any given c^ > G, the right-hand sides of (2.6) and (2.7) are each less than 8. A direct computation from these inequalities leads to (2.8) and (2.9), respectively.

Now, we want to determine the number of terms to be retained in the infinite series so that the “truncation errors” can be made as small as we wish. Although the number N of terms retained in each iteration can be different, we assume (for convenience of discussion) that it is the same for all iterates. Let n;i,C*, t) and ~j( x, t) denote the approximation of f;?i(x, t)

and q( x, t ), respectively; C&T, t; 6, 7) denote the Nth partial sum

of G&x, t; & 7); GC_&, (x, t; 5, T) denote CC,< x, t; 6,~) with C replaced by

c - PI*

From (1.7) and (LB),

x[u(&T) +L,a(&r)] d&h+ a(o), (2.10)

x[Qw + L,-p, a( 6, T)] dedr + a( x, t). (2.11)

By Bessel’s inequality,

By iz.l),

(2.12)

(2.13)

288 C. Y. CHAN AND BENEDICT M. WONC

Again by Bessel’s inequality,

I ( I,, Fj + L,-,I+)~2 < I” 4 + L_,+). ( (2.15)

From Corollary 15 of 13, p. 4271, Y < ~j < il?j < Go for all positive integers j. By assumptions, 7(x, t) and x ‘/‘y( x, t) are bounded. Also, a&, t) and x~/~~,-,(x, t) are bounded (cf. the proof of x”/‘M,(x, t) in [3, p. 4241). Let

l/2 K, = sup sup [12(4 + LA,_,, :j=O,1,2 ,...

O<tfT

K, = sup I{

sup [I”($+ L,_p,a)(t)]}1’2:j =0,1,2 ,...). OGtdT

Since xb/‘f(x, t, z,) is in C(& X (-00,~)) [3, p. 4191, and b > - 1 when g,(t) or g2(t) is not equal to zero [3, p. 4181, it follows that K, and K, exist and are finite. Therefore for b 2 0, it follows from (1.81, (2.11)-(2.13), (2.15), and (2.16) that

Similarly from (1.71 and (2.101,

II&) - &II G P2 P&&l+I.co(r;l)r


where

K, = ( sup [12(o.+L,u)(t)]}1’2. O<t,<T

(2.17)

Therefore,

k=o

(2.18)

By using an argument as for the case b 3 0, it follows from (1.8), (2.111, (2.14)-(2.16), (1.7), (2.101, and (2.17) that

X f: [a- “/“psf&-,,,)]k for h<O. k=O

Since

llii;ij+, - tij+ 111 < (1-b/2l) Xbi2 (Mj+ 1 - n;i,, 1)11 for b < 0,

it follows that

[a-b/2/?Sf,N(rln)]k k=O

for b < 0.


By Lemma 1, O(A,) = 0(n2). From Lemma l(i) and (ii) of 13, I). 4~1 YUUJ,

ikwl G hx -U2 for x E (0, a],

I+“( X) 1 G k2Q4 for x E [(A a]

for some constants k, and k,. We have

0 ( sup [Ix O<x<n

b/2+“(x) l/Q”]) = o(n-%

0 ( SUP [Ix O<XftZ

b@4”(X)l/r,]) = 0(n-“)y

Hence for each fixed j, the right-hand sides of (2.18) and (2.19) tend to zero as N tends to infinity. We thus have the following result.

THEOREM 3. Let E be any given positive constant. For each fi:xed j, if N satisfies

P2P3 [ PS,,N(rln)]j+1K2SN+I,~(r”) + KOSN+1,&-ln) i [ psi

k=O * .<%>]”

< E for b 2 0, aVb/2P2 p3 [ a-b/2PS; N( ~,,)]‘+‘&s~+ l,=hrl) .

+~os~+l,m(rl,b i [a-b/W; .h~l" 1 < E for b c 0, . k=o

then Ilii?i+, - ~j+111 < E.

By choosing ~~~ x, t) = _mo( x, t) = y( x, t ), an argument as for Theorem 3 gives the following result.

Cornputatiod EWW d%UlySiS 291

TWOREM 4. kt E be my given positive constant. For each fixed j, if N satisfies

P2wGsN+l c-h-1”) i [PSl zv<f-dl" < 8 forb 2 0, . k=O ’

a-b’p2B,KlS~+l,~(rln) i [a-b/2PSf,~(rh)]k < 6 forb < 0, k-o

then llq, 1 - rllf+Jl < 8. Theorems 2, 3, and 4 imply that we can choose the number j + 2 of

iterations and the number N of terms in each iterate so that Ilu - iUj+ 1 II and

Ilu - ??j+1 II can be made as small as we wish.

3. IMPROVED RATE OF CONVERGENCE

In this section, we show tha.t it is possible to decrease the number of terms needed by imposing additional conditions on fh, t, d, ah, t), Y(X, t) ad ah, t).

From [6, pp. 266-2671,

2

[ 1 l/2

Hy(l)( 2) = pz exp(i[ x - n( 42 + WI)

m-1

X c i*A,( v)/z” + q,),,l( z, v) ,

n=O 1 2

Hi2'(z) = 722 [ 1 l/2

exp(-i[z - r( v/2 + WI1

m~l(-i)nAn(v),W’ + 7?“,.2(Z9 4 l 1 n=O Here, H,")(z) and Hi2)( z) are Hankel functions of the first kind and the second kind, respectively;

An(v) = (4v2 - 12)(4v2 - 32). . . [4v2 - (2n - l)2]/(n!8n);

qm, &z, v) and q,Jz, v) are the error terms satisfying

292

Laet

C. Y. GHAN AND BENEDICT M. WONG

K,(z, m) =

i 4 i (-1)” A;y’ +I n= -1 II1 I1 0 = - cos[ z - 442 -t l/4)]

i (- 1)’ Ay2:,I(IY) 1 1

N 1 sin[ z - 7r( v/2 + l/4)]

+ [exp(i[ z - mc v/2 + l/4)1 blm, 1( =, v)

+exp( -i[ z - =( 42 + l/4)1 )%.2( =9 VI]/2

if m - 1 is even, and

m - 2

K&m) = i t-1)" A%,(V)

i 1 n=O z 2,, OS&Z - e/2 + l/4)1

+ [exp(i[ z - =( v/2 + l/4)3 )%,. s Z¶ v)

+exp( -i[ x - ev2 + WJl)rl,, 2(2, vJ1/2 3 .

if m - 1 is odd. Since JV( x) = [ H,“)(z) + Hj2)( z )]/2, it follows from (3.1) and (3.2) that

K,,(G m) = (7~~/2)~‘~5,(=) for m = 1,2,3,. . . .

Y+l(A!i2ffj = [ ~]1’2Kv+l(A~i2n. mjh,‘?


We note that 1 K,, I( #!“a, m)l > 0 for dl m and n since jJv+ @f,/‘a)i > 0 for all n. Thus, we have the following result.

LEMMA 5. ]lV+ ,(Af,/2a)]-’ = [7ra/2]““h!/]&+ I(Af,/“a, m)]-‘.

Let

BV,(fr) = (h(x,t): sup TV(h)(t) < -) o<t<r

where TV&X0 denotes the total variation of h(x, t) with respect to x on [O, a] for each t. Also, let 0 = x0 < x 1 < l *a < x,, = a be any partition of [O, a]. Since J&z) is positive and increasing for 0 < z < v (cf. 15, p. 29]), it follows that for 0 < xk < V/A!{‘,

From Formula 25 of [S, p. 1911,

By the mean value theorem and the fact that IJy(z)] < 1 (cf. [l, p. 36211, we have

< 21’2 {

v”~J~( v) A, 1’4 + [( 1 + 2v) A;!4/(2~ ‘i2) + ( A,, a) “‘1 a}

Hence,

294 C. Y. U-IAN AND BENEDICT M. WBNG

< [ HT(t)] -I lV]“( v) A, “4

+ [(l + 2~)hf,/~/(2v’9 + ( A,,#‘]a}

X ‘Z,,(a+Lca)(&(t-r;C)HT(~)d~ I

+izl X~“Ly( Xi, i = 1

Let

t) - XF_!fa( Xi-12 t)i* (3 3) .

K, = sup I( =y J( MI V z s ,

O<I<Z 0

V(h) = SUP (wx ““h)(t) + I( x”‘“h)(o, t) + ( ~“/%)(a, t) I]. 0<t<T

If xb/%(x t) x”/%(x t) and x”/“L ar( x t) are in follows from [i, 1;. 5951, [9,‘p. 571 and Leima ; that

BV,(Cl-) , then it

II,(a+ L+)(r)) 4 2d/“K,V(a + L,.Lu)h,~1/P/[a1/21K,+I(A~~~a, n*)l].

(3 4) .

Then from (3.3) and Lemma 5, we have

G 2lrp2 P3Q -‘K,V(o + L,a)

1 Ku+ I( h!,i2a, m) I-“[ v~/~],,( v)/c Aa2rn) n= 1

+( I + 2v)a/[(2v)““r,,] + A!/4n3i2/r,,]

+ sup Tv( xb’2a)( t). 0,<trT

(3 5) .

Computational Ermr Analysis 295

From Lemma 1, the foregoing series is convergent, and hence the right-hand side of (3.3) is bounded. Because this bound is independent of the choice of partition, it follows that x h/2RO(x t) is in W,(W). Similarly, we can

’ show I&(x, t) is in BVt(W). Suppose x”/~~(x, t, h(x, t)) E I3v,(n-) for every h(x, 0 E W,W).

Then analogous to (3.31, we have

< [H,.(t)]-’ 2 21’2 ( {

vq#( v) A, 1’4 #l = 1

+ ((1 + 2v)h!!4/(2v1’“) + (h,,a)‘/“]a)

X [I/ ( fl,, ~,+L,lr,(Y)(T)~,,(t-?;C-lll)H1.(7)~7 0 I

+ I/ ( t =I,, ~~~L,_,la)(T)~,,(T+t-~;C-Cll)HT(7)d~ II

+ i Ixy2a( xi, t) - xy:a( Xi-l, t)l* i = 1

Because ji&( X, t) E BV’( a-1, it follows from the hypothesis on f that &Zf(x, t, M,) E Wt(Q-). Since x b/2ii&(~, t), xbi2a(x, t) and xbi2~ c&r, t) are in BV,(fi-), it follows from [7, p. 5951, 19, p 571, and LemLa 5 that

< 2~1/~~,v(i7-, + L,_,,a)A; 1/2/[G1’21 K,, 1( A!,12a, m)l] l (3.6)


Analogous to (33, we have

Y

cl *“‘“M,( Xi ) t ) - X~~~~,( Xi_ 1) t ) 1 i

i= 1

+ sup Tv( x%)( t). OdC<T

Followmg the previous argument for s”%&, t), we have s”/“@,(s, t) E BV,(fi-). Similarly, H,( x, t) E BV,( W ). By using mathematical induction, we have the following lemma.

LEMMA 6. Suppose x”/‘~(;i, i), x’%(x, t) and x”/‘L&x, t) are in BV,(W). Furthermore, suppose x”/:‘f(x, t, !dx, t)) E BV,(ln-Lfor evey h(~, t) E By(n-1. T:Len for j = 0,1,2,. . . , x”‘“iii;i(“, t) and Mj(x, t) are in BV$K).

By a similar argument, we obtain the following result.

LEMMA 7. Suppose x”/‘y(x, t), x”/%(x, t) and x”/‘L,a(x, t) are in BI$n-). K.dienmre, suppose x “/‘ftx t, h(x, t)) E BVt(fl-) for every h(x s f! E B&(0-). Then for j = 0,1,2,. .‘. , x”‘“~j(x’ t) and ~j(x, t) ore in BV&W).

Analogous to (3.61, we have for j = 0, 1,2, . . . ,

Let

Sn = r h1/2)KY+I(A!!2a, m)l, n n

sin = flnA~‘21 K,+,(hk’2a, *)I.


When b a 0, it Co11 ows from (1.7), (2.10, (2.131, and (3.7) that

xV(5 + k.-,,+N+ I.&J

< [ &N( ?-J]j+‘ll&) - n;i,,ll

+2& &7C”K,a-““S~+ 1.J %,)

x i [ /3S,,,(r,,,)]“V(Fj_k + L,._,,a)- k=O

From (1.7), (2.10), and (3.4),

IliU.. - A&J < 2& &7r1bQ7( u + L,.c)a-‘/“SN+,.z( s,,). (3.8)

Hence when b 2 0,

< 2& &~WC$7( C+ Lca)a-““[ pS,,,( rl,,)ji+lS~+~ &,) .

+w, P3~1’2K3a-1’2Slv+ I&n)

W)

When b < 0, we use (1.7), (2.11), Q2.141, (3.71, and (3.8); analogous to (3*%

we have

< Z/3, &T~/~KJ( (T + Lca)a++ ‘Ii2

C. V. CHAN AND BENEDKI’ RI. WONG

+2p, P3?11’2K311-(h+1)‘4S~+l,x(sln)

.

x i [ PS;,~(+Jkv(iTj-k + Lc-p,Q)* (3*1Q) k-0

By an argument as in Section 2, we have

sup [l~b~24,,(~~l/~~,,]) = OW”h Ogr<tl

0 ( sup [IXh’2 OCrgs

A(x) I/%]) = O( n-3)*

O( \

sup [l4,~(X)l/S~,,]) = o(n-“/2), O<XStl

O( sup [I&(x) iA]) = O( n-5’2)- OGX<td

Hence by (3.9) and (3.101, we have the following result.

THEOREM 8. Suppose the conditions in Lemma 6 hold. For each fixed j and for any given 6 > 0, if N satisfies

w, P3w li2K3V( u + L,cY)a-1/2 [ PSI. N(rd]~+ %v+1 4%) .

+2P, f13rr’/2K3a-*/2S,+1 ,x(%) k 1 PW+ln)lkV k=o

x (3-k + LcBlrl(y) < E for b 2 0,

w, 83m 1’2K3V( a + LCLy)a-(b+l)‘2[ ps; . N( 1-J” ls;+J S”)

+2p, P3rr”2K3a-(h+l)‘2Sl;r+l.r( Sin)

x i [ ~s~,drdlkv(iTi_k + Lc-,,a) k=O

< E for b < 0,

Colnpu tational Error Anal@ 299

then

By a similar argument (with %o = _m, = 71, we have the following result.

THEOREM 9. Suppose the conditions in Lemma 7 hold. For each jked j and for any given E > 0, if N satisfies

.

2& &r’/“K,a-1~2S,+ 1 .r(S,,J i [ PS,.dr,,,)l”V(~-k + L/L,4 k=o

<s forbao,

2p, &~1/2K,a-(b+‘)/2Si;+, .rh,,) i [ i%hJlkq~-k + L-p,4 k=o

<E firb<O,

then

4. ALGORITHM

The number of iterations and the number of terms in each iterate a? determined by the theory deve!oped in Sections 2 and 3. To compute M (j > O), we use the representation formulas [(2.10) for G,( x, t ) and (2. llj for Mi(;r, t)] and (double precision) subroutines from the IMSL MATH/LIBRARY (Version 1.1, January, 1989; MALB-USM-PERFCT- EN8901-1.1) and IMSL SFUN/LIBRARY (Version 2.0, April 1987; SFLB- USM-UNBND-2.0).

Our algorithm for computing ~j(x, t) involves the following steps:

1. With the help of Lemma 1, subroutines DZREAL (to find the real zeros of a real function using MuIIer’s method) and DBSJS (to compute Jy( s)) are used to evaluate the first N zeros ha2a of ly( s). Then, DBSJS is used again to compute &( x).


2. We subdivide the x-interval into m subintervals and the t-interval into k subintervals with end-points x,. and t, satisfying 0 = xg < x1 < l =* < x,,, =a, 0 = t, < t, < l -• < tk = T. For each point ( x,, t,), we use the subroutine DTWODQ (to compute a two-dimensional iterated integral) to evaluate A&< x r, t,) by using (2.10).

3. To compute M,( x, t), (a) The subroutine DBS2IN (to compute a two-dimensional tensor-

product spline interpolant, retuping the tensor-product B-spline coeffi- cients) is used to interpolate A&(x, t) with data points M,,( x,, t,), r =

0, I, 2, - ..,m, s = o, 1,2,. . . , k. (b) The subroutine DBS2VL (to evaluate a two-dimensional tensor-

product spline, given its tensor-product B-sp!ine representation) is used to ev.aluate M, at any point (x, t). Thus, f( x, t, M,,( x, t N can also be computed.

(c) For each fixed point ( x, , t,),

compute ti&, t,) by using (2.11). the subroutine DTWODQ is used to

(d) The subroutine DBS2IN is used to interpolate G,( x, t ) with data points &x,, t,), I = 0, 1,2, . . . , m, and s = 0, 1,2, . . . , k.

(e) the subroutine DBS2VL is used to evaluate J& at any point (x, t).

4. By repeating the procedures (c)-(e) of Step 3, Mj< s, t ) is computed. Similarly, ej( x, t) can be computed.

5. AN EXAIulPLE

Let us consider non-negative time-periodic solutions of period 1 for the problem (l&(1.2) with n = 1, b = 0 = c(t),

f( x, t, u) = -(u” - l)sin”27rt/lO,

and gJt) = 1 = g,(t). For illustration, we use the error estimates estab- lished in Sections 2 and 3 to determine the number j + 2 of iterations and the number of terms N such that ]]u - ~j+ Jl < 0.001 since the procedures for determining those for I]u - ~j+ 1 11 < 0.001 is similar. Without loss of generality in illustrating the procedures, let us determine j and N by using

lb - Mi+lll < 0.0005 and IIMi+ I - Mi+ 111 < 0.0005. Since c(t) = 0 and g,(t) = 1 = g,(t), it follows from (1.3) and (1.6) that C = 0 and Q! = 1. We choose y(x, t) = 0 and a( x, t) = l/10. Since v = (1 - b)/2, it follows from Formulae 44 and 45 of [S, p. 1921 that for any positive integer n, pa ii = nrr and 4,,(x) = 2’/*sin(n7rx). From (1.7),

Conzputational Error Analysis 301

M,,h 9

= 1+ 2 /‘/’ 2’4in( n~~)/lO c@ E,,( t - 7; 0) cl7 V”sin( n7rx) n = 1 0 0

+ ii /‘/I 21’“sin(n7rt)/10 cI6 E,,( 1 + t - 7; 0) d7 2’/“sin(n7rx). II = 1 t 0

This gives

C ns3. ff - 1

By using Ec=,nB3 = 1.2020569 (to 7 decimal points) (cf. [2, p. 5411, we obtain

We note that

IIK(~All < 1.0155073. (5-l)

af I I du = u sin” (2nt)/5.

It follows from (2.1) that

Qz < 1.0155073/S = 0.2031015. (5 2) .

Here, we choose pl = 0.2031015 (cf. [3, p. 4271). Since c(t) = 0, it follows from (2.4) and (2.5) that & = 1 = p3. From (5.2) and the fact that

X

E

n-2 = 1.6449341 (to 7 decimal places)

n= 1

(cf. [2, p. 54]), we obtain

pS,,,( A,, + E.c,) G 2(0.2031015) i 21i2( nfl)-2 n= 1

< 0.0957430, (5 3) .


which is less than 1 (as required in Theorem 2). We note that m,, = 7 = 0. From (5.1) and (5.3),

in(0.0005/11M01()/1n[ PS,.,( A,, + ccl)] f 3.2463793.

It follows from Theorem 2 that

IlU - x&11 < 0.0005. (5=Q

This means that we can fm the number of iterations to be 5. For any non-negative integer j,

I4(5 + L_l")(t)

1 _, /[ ( 0 - T- l)sin’(2rrt)/lO + ~,(~j - l)]‘dt

= l’[ - (ii;ii + l)sin”(27rt)/lO + p,]@ - lr de.

It follows from 0 = ~(1 G zj < ii?i < M,, and (5.1) that Ilil?i - 111 < I, and

I- (Gj + l)sin”(elrt)/lO + PI)

4 ma{llM, + PI/lo, PI>

-c max(2.0155073/10,0.2031015)

= 0.2031015.

He; _‘.C

K,, < 0.2031015.

We note that K, = l/10, and

(5 5) .

X

c n=N+l

n-2<~~+l(x-1)2m=N-*,


pmtided N 2 1. From (5.3), (5.5), and the fact that & = 3 = &,

+K s () N+l.r(A,, + Y,) c [~Sl..(A,~ + k)lk

\< 0.0321822/N. (5 6) .

So that 0.0321822/N < 0.0005, we need N > 64.3644000. Therefore, it follows from (5.6) and Theorem 3 that if we retain 65 terms in the series of each iterate,

II@ - A&II < 0.0005.

This together with (5.4) gives

IlU - nri,ll < 0.001.

To decrease the number of terms N, we use the error estimates estab- lished in Section 3. It follows from Formulae 44 and 45 of [5, p. 1921 that

z211+-1/gsin( z) & < 23/Pqr-l/2, (5.7)

IJ3,2wl-’ l

= 2- l/Znl/;??r

By Lemma 5,

We note that

l&,2(“% 41 = 1.

V(u) = l//5.

From

(5 8) .

(5 9) .

+5-l g 11~‘sin(n?rf)d5~,,(1+t-*;0)d7n~COS(n?TX), n=l t 0

C. Y. CHAN AND BENEDICT M. WONG

we obtain

II 11 aM,, 2 zs 11 I- dx 11 G y-g ,FI n-’ G 0.0666667.

Since

it follows from (5.1) and (5.10) that

(5.10)

(Fj.11)

From (5.10) and (5.1 l), we obtain, respectively,

sup (ZV( M,,)( t)) < 0.0666667, O,<f<T

SUP {W[f( (“igt<T

x, t, Q](t)} G 0.0135401.

Hence with L+Pia = -I_c,,

< 0.0270802.

From (3.63. (5.71, and (5.8),

(5.12)

I ( 4, 41 - /.+)I < 0.1531887(~+‘.

It follows that

Thus,

sup (w( ii&)(t)) < 0.0361069. O<t<T

Since M,, > IT, 3 0, it follows from (5.1) that

30.5

Hence,

sup {Tv[f(x, t, n/I,)](t)} G 0.0073334. O,<t<T

Therefore,

V(F) +L,_,,cu) < sup {~[f(G q]w} +cc1 sup pwd(t)l O<t<T O<t<T

< 0.0146667. (5.13)

By a similar argument, we have

v(F2 + L,._,,a) < 0.0079435,

v(b + L/L, ) a < 0.0043022.

From (5.3), (5.71, (5.9), and (5.12)-615),

(5.14)

(5.15)

2q ps,, N( A,, + Cc,)]4KJ(a + L,cu)S,+ Lx(V)

+2n’12K s (Ay2 3 N+l,= + ju,q2) ; [ PS,, N(4, + k)lk

k=O

XV(&_k + L,_/p) G 0.0013514 I? n-IV+ 1

C. Y. CHAN AND BENEDICT M. WONG

provided N 2 1. So that

0.0013514/(2 N ‘) < 0.0005,

we need

N > 1.1624973.

Therefore, it follows from Theorem 8 that instead of 65 terms, we only require to retain 2 terms in the series such that

IlM, - n;i,ll < 0.0005.

We note that u( x, t) = 1 is the solution of the preceding example. For n_umericaI verification, we implement our algorithm in Section 4 to compute M,, by using 65 terms in each iterate. Again, & is computed by using 2 terms in each iterate. In each case, the numerical results obtained verify the fact that 111 - &II < 10w3.

REFERENCES

M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathenuttical Functions with Formulas, Gpiphs and Mathematical Tables, National Bureau of Standards, Applied Muthenzatics Series 55, Washington, 1972, pp. 362, 370. W. H. Beyer, Ed., CRC Standard Mathenzatical Tables, 27th ed., CRC Press, Inc., Boca Raton, Florida, 1984, p. 54. C. Y. Chan and B. M. Wong, Periodic solutions of singular linear and semilinear parabolic problems, Quart. Appl. Math. 47:405-428 (1989). G. S. bdde, V. Lakshmikantham, and A. S. Vatsala, Monotone Zteratioe Tech- niques for Nonlinear Diffetwatial Equations, Pitman Advanced Publishing Pro- gram, Boston, Massachusetts, 1985. N. W. McLachlan, Bessel Functions for Engineers, 2nd ed., Oxford at the Clarendon Press, London, 1955, pp. 29, 191-192.

F. W. J. Olver, Asymptotics and Special Functions, Academic Press, Inc., New York, 1974, pp. 266-267.

G. N. Watson, A Treatise on the Theo y of Bessel Functions, 2nd ed., Macmillan, New York, 1944, pp. 499-494,595. H. F. Weinberger, A First Course in Partial Differential Equations, Xerox College Publishing, Lexington, Massachusetts, 1965, p. 73. E. T. Whittaker and G. N. Watson, A Course of No&m Analysis, 4th ed., Cambridge at the University Press, London, 1940, p. 57.

computational error analysis of periodic solutions for singular semilinear parabolic problems

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