The finite-section approximation for integral equations ...

J. Austral. Math. Soc. Ser. B 28 (1987), 415-434

THE FINITE-SECTION APPROXIMATION FOR INTEGRAL EQUATIONS ON THE HALF-LINE FRANK DE HOOG1 AND IAN H. SLOAN2

(Received 20 September 1985; revised 21 February 1986)

Abstract Integral equations on the half line are commonly approximated by the finite-section approximation, in which the infinite upper limit is replaced by a positive number /?. A novel technique is used here to rederive a number of classical results on the existence and uniqueness of the solution of the Wiener-Hopf and related equations, and is then extended to obtain existence, uniqueness and convergence results for the corresponding finite-section equations. Unlike the methods used in the recent work of Anselone and Sloan, the present methods are constructive, and result in explicit asymptotic bounds for the error introduced by the finite-section approximation.

1. Introduction.

In this paper, we consider integral equations of the form y{t)~r

[w(t-s) + h(t,s)]y(s)ds=f(t), t>0, (1.1) A) where w e L 1 ( R ) , h satisfies the conditions indicated below, a n d f , y ^ L * : = LOO(U+). Our main concern is the approximation of this equation by the corresponding 'finite-section' equation [»>('-s)+h(t,s)]yf)(s)

ds=f(t),

Otit^P,

(1.2)

in which the upper limit of integration is replaced by a positive number /?.

1

DMS, C.S.I.R.O., P.O. Box 1965, Canberra, A.C.T. 2601, Australia Department of Applied Mathematics, University of New South Wales, Sydney, N.S.W. 2033, Australia © Copyright Australian Mathematical Society 1987, Serial-fee code 0334-2700/87 2

415

416

Frank de Hoog and Ian H. Sloan

[ 21

The conditions to be satisfied by h are snV r

J 0 t>0'O

Urn r

\h(t',s)lim r

\h(t,s)\ds
ao •'0

An important special case is h(t,s) = 0, in which case (1.1) is a Wiener-Hopf equation. We may conveniently write (1.1) as

(/+_ w+-

H+)y=f,

where I+ is the identity operator for functions defined on U +, and W+ and H+ are integral operators on R + with kernels w(t — s) and h(t,s) respectively. The Wiener-Hopf operator W+ is a bounded operator on L*, and the operator H+ is a compact operator on L* (see for example Atkinson [2]). In a recent paper by Anselone and Sloan [1], the convergence of y^(t) to y(t), uniformly for f on an arbitrary finite interval, was established under suitable conditions. The proofs used in that work were non-constructive, being based on a repeated use of the Arzela-Ascoli theorem, and so provide no foundation for the estimation of errors. In this paper we rederive many of the results of Anselone and Sloan [1] by constructive arguments, and then go on to study the nature of the approximation of y(t) by yp(t) (see Theorem 5.2). A key result obtained in [1] is that the solution operator for (1.2) (or more precisely, for that equation amended by allowing / to run over U+, and then viewed in the space X+ of bounded, continuous functions on U+) exists and is uniformly bounded for all /? sufficiently large. Results of this kind are useful not only for establishing the convergence of y^(t) to y{t), but also, as in a recent paper by Sloan and Spence [6], for establishing convergence and error bounds for numerical methods. In the present work the existence and uniform boundedness of the solution operator for the equation (1.2) is established in an appropriate space (namely LX(Q, /})), with the important difference that an explicit bound on the norm of the solution operator is now obtained (see (4.8) and (4.9)).

2. The Wiener-Hopf equation Before we can attempt to answer questions about the existence and uniqueness of solutions of the finite-section equation (1.2), we require similar results for the Wiener-Hopf equation. In this section we derive these and other related results in a somewhat novel way. It should be said, however, that none of the results in this

13 ]

Integral equations on the half-line

417

section are new. Moreover, not all of the known results for the Wiener-Hopf equation can be obtained in this manner (see Krein [5] for a full treatment of the Wiener-Hopf equation). Nevertheless, the techniques used are of interest in their own right and, more importantly, can be extended to analyse the finite-section equation. The Wiener-Hopf equation, that is (1.1) with h(t, s) set equal to zero, is written here as (),

t>0,

(2.1)

•'o or (I+- W+)y+=f+. (2.1') It turns out to be convenient to study also the corresponding equation on the negative real axis, y-(t)-f°

w(t-s)y-(s)ds=f-(t),

t0,

Part (i) of the theorem therefore follows from Theorem 2.4 part (i). In a similar way, the pair of equations

(l+-W+)y+=f+,

y+,f+^E+,

and (r-W-)y-=0,

y-eE-,

(I-W+A)y=f,

y.feE,

is equivalent to (2.16)

[9 ]

Integral equations on the half-line

423

provided w

/o,

It now follows from Theorem 2.4 part (ii) that equation (2.16) has a solution y e E (and hence (2.14) has a solution y+e E+)ii and only if

x-(-s)ds = 0

(2.17)

for every solution x~ of (2.15). Thus (ii) holds. If (2.14) holds for a/// + e E+ then (2.17) must hold for all / + e £ + , from which it follows that x~= 0, so that (2.15) has only the trivial solution. Conversely, if (2.15 ) has only the trivial solution, then, by (ii), (2.14) has a solution for each / + e E+, so proving (iii). We can now deduce a number of equivalent conditions for ensuring that the Wiener-Hopf equation (2.1) and the associated equation on the real line (2.8) have unique solutions for arbitrarily chosen right-hand sides. 2.6. Let condition (2.7) hold. Then the following are equivalent. (a) / - W + A has a bounded inverse on Lp for allp satisfying 1 < p < oo. (b) I — W + A has a bounded inverse on Lp for some p satisfying 1 < p < oo. (c) / — W + A has a bounded inverse on Z. THEOREM

(d) The equation (/ - W + A)x = 0 has no nontrivial solution x e Z. (e)/ + — W+ and I~— W~ have bounded inverses on L* and L~ respectively for allp satisfying 1 < p < oo. ( f * ) / * — W± has a bounded inverse on Lp for some p satisfying 1 «£ p < oo. (g±) I±— W± has a bounded inverse on Z*. (h) Neither ( / + - W+)x+= 0 nor ( / " - W~)x~= 0 has a nontrivial solution ± x ^Z±. (Note that (f + ) and (f~), and similarly (g + ) and (g~), are to be considered as separate equivalent conditions in the theorem.) PROOF. The implications (a) => (b) =» (d) and (c) =» (d) are trivial, and the implications (d) =» (a) and (d) =» (c) follow from Theorem 2.4. Thus (a), (b), (c), (d) are equivalent.

424

Frank de Hoog and Ian H. Sloan

[ 10 ]

The implications (e) => (f *) are trivial, and the implications (f *) => (h) and ( g * ) =» (h) follow from Theorem 2.5, as also do (h) => (e) and (h) => (g*). Thus (e), (f ± ) , (g *), (h) are equivalent. x\a±. Finally, (d) and (h) are equivalent under the correspondence x±=

3. Modified Wiener-Hopf equation. In this section we return to the full equation on the half-line, equation (1.1), which we now write as

( / + - W+- H+)y+ = f+. (3.1) + Since H is a compact operator on the space L+ (see section 1), it will be convenient to consider this equation in the space L+. As in Section 2, we shall also consider an equation on the negative real axis, (3.2) (/-- W-)y'=f-. Equations (3.1) and (3.2) can then be viewed as a single equation on the whole real line, namely (l-W + A-H)y=f, (3.3) where W and A are as in section 2, and H is an integral operator with kernel h{t,s) for t, s > 0, and zero otherwise. THEOREM

3.1. Let condition (2.7) hold, and suppose that (I - W + A - H)x = 0, x^Lx,

(3.4)

has no nontrivial solution x. Then I — W + A — H has a bounded inverse on Lx, and I — W + A has a bounded inverse on Lp, 1 < /> < oo.

Since H+ is compact on L+, it follows easily that H is compact on L^. Thus /— W + A — # as an operator on Lx is a Fredholm operator with index zero. Since the homogeneous equation (3.4) has only the trivial solution, it follows from the Fredholm alternative that / — W + A — H has a bounded inverse on PROOF.

It then follows from (3.3), on restricting attention to the negative real axis, that (3.2) has a unique solution y~e L~ for each / ~ e L~, and hence that / " - W has a bounded inverse on L~. The remaining result, that I - W + A has a bounded inverse on Lp for 1 < p < oo, now follows from Theorem 2.6 (a) and (f-). We may also state an equivalent result for operators directly relevant to the half-line. The result is similar to one obtained by Anselone and Sloan [1, Theorem 9.3].

[ill THEOREM

Integral equations on the half-line

425

3.2. Let condition (2.7) hold, and suppose that neither

nor

(r-w-)x-=o, +

x-eL-,

+

+

has a nontrivial solution. Then I - W — H has a bounded inverse on L+, and I±— W± have bounded inverses on L*, 1 < p < oo.

4. The finite-section equation In this section we show that a unique solution of the finite-section equation (1.2) exists when /? is sufficiently large. Our approach to the finite-section equation is simliar to that in the previous sections. We begin by incorporating (1.2) into an equation on the whole line. Specifically, we consider the three equations w(t-s)y(s)ds=f(t),

t>/3,

y(t) - [fiw(t - s)y(s)ds- \P h(t,s)y(s) ds = /(/), •'o •'o w(t-s)y(s)ds=f(t),

0 < t < 0, } (4.1)

t0,

s)y(s)ds,

and

f° w(t - s)y(s) ds - r h(t,s)y(s)ds, •'-oo

•'O

- ^ ° ° h(t,s)y(s)ds,

0 0 , ds,

/ < 0,

f°° \v(s)\ds,

t > 0 ,

J'

t < 0 .

\v(s)\ds,

It is easy to verify that z0 - z = - (/ - W)-lAzQ = - (I + V)Az0.

The result now follows from the definitions of A and V, after application of some elementary inequalities. We shall now combine the results of the previous section with Lemma 5.1, to obtain some insight into the behavior of the finite-section approximation when /? is large. If we denote by o(l) any terms in an expression that vanish in norm as /? -» 00, then (4.4) may be written as / - W + A - H + Ap- bp = (I- W + A- H)(I - W)'\l - W + Ap) + o(l). From this it follows, for /} sufficiently large, that ( / - W+A-H

+ A/,- A^)"1

= ( / - W + A$)~\l-

W)(I - W + A - H)'1 + o(l)

= ( / - W + A - H)'1 + ( / - W + Ap)'1 - ( / - W)'1 + (l - W + Ap)'lAp(I

- W)'\A - H){I -W + A- H)'1 + o(l).

It follows from an argument similar to that used in the appendix to prove (4.5) that "AJl-W)'\A-H)\\-*0

as )3-oo,

[is)

Integral equations on the half-line

429

in consequence of which the fourth term of the above may be absorbed into the o(l) term. Thus for /? sufficiently large we have

(I-W

+ A-H + Ap- A^)"1 = ( / - W + A - H)'1 + ( / - W + A ^ - i l - W)-l + o{\). (5.3)

Now, let y, y^, Zp e Lx be the (unique) solutions of

( / - W + A-H)y=f, ( / - W+A-H + Ap-^)yp=f, and (I-W+Ap)zfi=f respectively, where / e Lx. Then from (5.3) we have yn = y + *p-* + o{\), where z is the solution of (5.2). Equation (5.4) has a simple interpretation, if we rewrite it as yp-y = zp-z + o(l),

(5.4)

(5.5)

and restrict attention to the interval (0,/?). On this interval y is the solution of (1.1), the full equation on the half-line, while yp is the solution of the finite-section equation (1.2). On the other hand z is the solution of the equation

z{t)-rw{t-s)z{s)ds = -oo

whereas, because t < /$, Zp is the solution of

That is, z and z^ satisfy the same equations as y and y^ respectively, except that the kernel h(t,s) is absent, and secondly, the lower limit of integration is now -oo instead of 0. To the extent to which the approximation y$-y~Zp-z is valid (and it is valid uniformly for /? sufficiently large), the error yp — y is unaffected by the kernel h(t,s), and shows no effects from the lower limit of integration. Rather, the error can be expected to be small over the entire interval (0, /?), except in the vicinity of the right-hand end. To obtain a more precise result, we observe that an obvious extension of Lemma 5.1 yields

430

Frank de Hoog and Ian H. Sloan

[ 16 ]

where, with the aid of (4.7),

Then from (5.5), together with Theorems 3.1 and 4.2, we obtain the error bound in the following theorem. The constant cx in the theorem may be taken to be ||(7 - W + A)'^]. THEOREM

5.2 Assume that condition (2.7) holds, and that neither ( 7 + - W+- H+)x+=0,

x+eL+

(5.6)

nor

( / " - W-)x~= 0,

x - e L~

(5.7)

has a nontrivial solution. Then for /? sufficiently large there exists c1 independent of /} such that

0 < t < P, where y and y^ are the solutions of (1.1) and (1.2), and e^ -* 0 aS /J -> oo.

6. Example A convenient illustration of the error bound in Theorem 5.2 is furnished by the integral equation e~^'~siy(s) ds = 1,

y(t) ~ T I

t 3* 0,

(6-1)

for which both the exact solution y and the finite-section approximation y^ can be obtained analytically. For this example h(t,s) = 0 and W(A

= -L-KI. A

Thus the Fourier transform w is

-( \ W(")

2 =

\

_J_ l

+

v2>

and therefore the condition (2.7) is satisfied provided X £ [0,2]. And in fact for X £ [0,2] a solution y e L^ of (6.1) exists and is unique (see [2]), and is given explicitly by

[ 17 ]

Integral equations on the half-line

431

where

with the positive branch of the square root to be taken for X e U \ [0,2]. It can easily be verified that for this example the integral operator V defined in Lemma 2.1, with kernel v(t — s), has v given explicitly by „(,) = ^-e-"l'". Then one easily computes, using (2.6),

^4

^ x ^

and in a similar way the quantities y and TJ defined in Lemma 5.1,

One also notes that for X £ [0,2] equation (5.7) has no nontrivial solution, since otherwise (because w is even, and H+= 0) there exists a nontrivial solution of (5.6), obtained by reflection in the origin. Thus for this example Theorem 5.2 implies, for X £ [0,2] and /? sufficiently large, the following bound on the error in y^.

-y(t)\
0

J

\h(t,s)\ds.

B/3

Since v e Llt each of the above bounds converges to zero as /? -» 00. Thus - H)(I - WylAp\\ -» 0 as 0 -» 00. References [1] P. M. Anselone and I. H. Sloan, "Integral equations on the half-line", / . Integral Equations 9 (suppl.) (1985), 3-23. [2] K. E. Atkinson, "The numerical solution of integral equations on the half-line," SI AM J. Numer. Anal. 6 (1969), 375-397. [3] K. Jorgens, Linear integral operators (Pitman, Boston, 1982).. [4] M. A. Krasnosel'skii, "On a theorem of M. Riesz", Soviet Math. 1(1960), 229-231. [5] M. G. Krein, "Integral equations on a half-line with kernel depending on the difference of the arguments", Amer. Math. Soc. Transl. (2) 22(1963), 163-288. [6] I. H. Sloan and A. Spence, "Projection methods for integral equations on the half-line", IMA J. Numer. Anal., (to appear). [7] N. Wiener, The Fourier integral and certain of its applications (University Press, Cambridge, 1933).