Some recent numerical methods for solving nonlinear Hammerstein integral equations

Mathl. Comput. Modelling Vol. 18, No. 9, pp. 55-62, Printed in Great Britain. All rights reserved

1993 Copyright@

08957177/93 $6.00 + 0.00 1993 Pergamon Press Ltd

Some Recent Numerical Methods for Solving Nonlinear Hammerstein Integral Equations B.SOME Facultb

des Sciences

et Techniques (FA. S. T.) DBpartement de Mathhmatiques et Informatique 03 B.P. 7021 Ouagadougou 03, Universit6 de Ouagadougou, Burkina Faso

(Received

April

1993;

accepted

May 1993)

survey is given of some recent numerical methods for fixed points of nonlinear Hammerstein integral operators. These methods include projection methods (Galerkin and collocation) and Adomian’s decomposition method. Proof of convergence of Adomian’s method applied to Hammerstein nonlinear integral equation is given. A discussion is given of some recent works on iteration methods for solving these nonlinear equations.

Abstract-A

1. INTRODUCTION In the following survey, we consider applied to a wide variety of nonlinear be of the second kind,

numerical methods integral equations.

of a general The integral

nature, those that can be equations are restricted to

2 = H(z),

(1.1)

where H is a nonlinear integral operator. An important equation is considered here as the following:

special

case, the Hammerstein

integral

R z(t) = ?m

A well-known

example

+

s

with R > 0.

_R K(4s)f(s,4s))ds,

is the Chandrasekhar

(1.2)

H-equation

’ t H(t)

H(t) = ; /

H(s) ds t-k.9

0

*

(1.3)

It can be rewritten in the form (1.2) by letting z(t) = l/H(t) and rearranging the equation. For an exact solution and some recent works on (1.3) and related equations, see [l]. Another rich source of Hammerstein integral equations is the reformulation of boundary value problems for both ordinary and partial differential equations. The numerical solution of nonlinear equations has two major aspects. First, the equation z = H(z) is discretized, generally by replacing it with a sequence of finite-dimensional approximating problems (projection methods) z,, = H,(z,), with n --) +oo, and n is some discretization parameter. Second (Adomian’s method), the solution z of the equation z = H(z) is replaced by a series x = E z,,, and H(z) = E A,,, where the A, are polynomials of ~0,. . . , zn (Adomian’s n=O n=O polynomials). Projection methods are discussed in Section 2, Adomian’s method in Section 3, and a discussion follows in Section 4.

55

B.

56

SOME

2. PROJECTION

METHODS

General Description We give only the most important features of the analysis of projection methods and then discuss some applications. General theoretical frameworks for projection methods have been given by a number of researchers, among these are Atkinson [2,3] and Potra [3], Golberg [4], Kumar [5], and Saramen [6]. Let E be a Banach space, usually C”( [-R, R]) or L’([-R,

RI); and let E,,, n 2 1 be a sequence

of finite-dimensional subspaces being used to approximate z* (solution of (1.1)). Let P, : E z E, be a bounded projection, n 2 1 and, for simplicity, let E, have dimension n. It is usually assumed that P*x-+x,

CS?l++CXl,

x E E,

(2.1)

for all x in some dense subspace of E containing the range of H. In abstract form, the projection method amounts to solve 2, = P, H(xn). (2.2) For motivation about connecting this with a more concrete integral equation, see the discussion in [7, pp. 54-711 for a linear integral equation. Assume P, can be written

pnx= ~4(XMj,x

E E,

(2.3)

j=l

with (4j)l
a set of bounded linear functionals that are inde-

det[lj(4j)l

# 6.

(2.4)

To reduce (2.2) to a finite nonlinear system, let (2.5)

Xn=kOZj4j j=l

be solved for {ej},

j = 1,. . . , n from the nonlinear system

(2.6) The choice of (41,. . . , &} and (11, . . . , In} determine the particular method. framework for the error analysis of projection methods was given in [3].

A very general

THEOREM. Let H : R c E + E be completely continuous, with E, Banach and R open. Assume

that the sequence of bounded projections {P,,}

on E satisfies

SUP IIU - Pn) Wx)I/ + 0,

ZEB

asn--,+m,

(2.7)

for all bounded sets B c R. Then, for any bounded open set B, with B c R, there is an N such that I - H and I - P,, H have the same rotation on the boundary of B, for n 1 N. In the particular case that x* is an isolated fixed point of nonzero index of H, there is a neighborhood & = {x I lb - x*(1 5 E}, with P,, H h aving fixed points in B, that are convergent to x*. (The index of x* is defined as the rotation of I - H over the surface S, of any Be which contains only the one fixed point x*. In case [I - H’(x)] - 1 exists on E into E, the index of x* is fl.) For the proof of the above theorem, see [3]. Generally, the projections are assumed to be pointwise convergent on E, as in (2.1) and in that case (2.7) follows in straightforward way. However, only equation (2.7) is actually needed.

Nonlinear Hammerstein Integral Equations

Iterated Projection

57

Methods

Given the projection

method solution x,,, define

(2.8)

P, = H(x& then using (2.2) we have

(2.9) and f,

satisfies f,

with H differentiable

(2.10)

= H(P,&&),

in a neightborhood of x*,

- CalI I CllX811x*

xnll,

n L N,

(2.11)

c > 0.

Thus, Z,, ---) x* at least as rapidly ss x,, + x*. For many methods, especially Galerkin methods, the convergence & + Z* can be shown to be more rapid than that of x,, + x*. For the nonlinear iterated projection method, see [3]. Galerkin’s Method Let E be C”((-R, R]) with the uniform norm (( - lloo or L2([-R, RI), and let En be a finite dimensional subspace of E. Define P, x to be the orthogonal projection of x onto E,,, based on using the inner product of L2((-R,R])

and regarding E,, as a subspace of L2([-R,

RI); thus (2.12)

y E En, (PnX>Y> = (XVY)? a.11 with (a, .) the inner product in L2([-R,R]). products are also used in some applications. Let xn = ~~=‘=, cq &, with (&)lsjgn,

Other Hilbert spaces (e.g., H’([-R,R])

a basis of En. Solve for {&},

and inner

using

(2.13)

Generally, the integrals must be evaluated numerically, thus introducting this is done, it is called the discrete Gale&in method. In general, for any projection

(lx* -&II

new errors.

When

method,

I c, IIV - pn)~*Il~

c, = c max{]](l

- Pn) x8]],

with

llH’(~*)(I- Pn)llI-

(2.14)

If E is a Hilbert space, then c, converges to zero, because ]]H’(z*) (I - Pn)ll ---, 0, as n + 00. This uses the fact that H/(x*) and H’(x*)* are compact linear operators. Thus, &., converges to Z’ more rapidly than xn does. In any Hilbert space, 5, + x* more rapidly than z,, + x’ does. Similar results can be shown for Galerkin’s method in C’([-R, R]) with the uniform norm, see [8]. For numerical examples of application of the Galerkin’s method to nonlinear integral equations, see [2].

B. SOME

58 Collocation

Method RI), let ti, . . . , t, E [-R, R] be such that

Let E = C”([-R,

deQj(ti)]

(2.15)

= O. x at the nodes ti, . . . , t,.

Define P,, x to be the element in En that interpolates of En, solve the nonlinear system

( )

&Wd=H &j4j

j=l

(t&

i=l,...,

To find x, element

(2.16)

n;

j=l

the integrals must usually be evaluated numerically, introducing a new error, this is called the discrete collocation method. In using the general error result [2], ]]x* - P, x* ]loo is simply an interpolation

error.

For the iterated

collocation

method,

with 11x* - xnll I cn11x* -G&II, c, = c max{]]x* - hll, ll(I - pJ*LlI, en)

(2.17) and (2.18)

To show superconvegence

of f,

to x*, one must show lim e, = 0. n+cG

Studies of this and various tion are given in [3]. Case of Hammerstein Consider

types of collocation

(2.19)

methods

applied

to the Urysohn

integral

equa-

Equation

using the projection

method

to solve

R x(t) = Y(t)+

s -R

with R > 0.

Wt,s)f(s,x(s))ds,

(2.20)

For nonlinear integral equations, the Galerkin and collocation methods can be quite expensive to implement. But, for this equation, there is an alternative formulation which can lead to a less expensive projection method. We consider the problem when using a collocation method. Let xn(t) = Cy=“=, aj 4j(t) and solve for {oj}, using

2

aj $j(ti) =

Y(h)

j=l

+

/” K(t, S)f -R

(sy$oj#j(&))

ds,

fori=I,...,n.

(2.21)

In the iterative solution of this system, many integrals will need to be computed, which usually becomes quite expensive. In particular, the integral on the right side will need to be re-evaluated with each new iterate. Kumar [5] and Kumar and Sloan [9] recommend the following variant approach. Define R 6 Y(t)+

z(t)= f

and obtain

W&s) s -R

(

z(t) from x(t) = y(t) + s_“, K(t, s) z(s) ds; the collocation

Z(t) =

k@j 4j(t) = f

j=l

h,

Y(h) +

(2.22)

f(s,x(s))ds , ) method

k/Jj1-1 K(ti, s> bj(s) ds

for (2.22) is

*

(2.23)

j=l

The integrals on the right hand side need to be evaluated only once, since they are dependent only on the basis, not on the unknowns {aj}. Many fewer integrals need to be calculated to solve this system. For the results on Hammerstein’s integral equations, see [6].

Nonlinear Hammerstein

Integral Equations

3. ADOMIAN’S

59

METHOD

General Description Consider,

for example,

the general

functional

equation

Ax=y, where

A is an operator

from an Hilbert

for x E V satisfying

are looking

(3.3).

(3-I)

V into V, y is a given

space

We assume

that

function

(3.1) has a unique

in V, and we

solution

for z E V.

If the operator A has linear and nonlinear terms, the linear term is decomposed into L + R, where L is easily invertible and R is the remainder of the linear operator. Then, the operator A is decomposed as following: A = L + R + N, where N represents the nonlinear term. With these considerations,

the equation

(3.1) becomes Lx+Rx+Nx=y.

Further,

if L-’

is the inverse of the linear operator

(3.2) L, then the solution

x of (3.1) or (3.2) verifies

z=L-‘~-L-‘Rx-L-~Nx. The initial

Adomian’s

method

[lO,ll]

consists

x=

The nonlinear

operator

is decomposed

(3.3)

of representing

x as a series,

-&k.

(3.4)

as

Nx

= FAn,

(3.5)

where the A, are polynomials z = C:“=, Xi xi, N(z) having

that of xc, . . ,x, (Adomian plynomials), = CT=“=, X, A,, with X being a parameter introduced

,!A,=&=

[N(~Yxi)l,_,;

n=0,1,2

we obtain

,....

It allows to determine the An’s by formula (3.6). Generally it is possible to obtain functions of xc, xi, . . . , x, from the nonlinearity N. With these decompositions, can be written 5

n=L-'y-L-lR

km 18:9-E

the other terms

of the above series

Rx0 - L-l Ao,

x2 = -L-l

Rx1 - L-’ Al,

x, = -L-=

Rx+~

=

(3.6) exactly A, as formula (3.3)

-L-1(zA+

-L-l

x1

and

F (nEoxq

n=O

Taking x0 = L-l y, we can identify algorithm

by writing

for convenience,

- L-l A,_1.

E x, by the following n=O

B. SOME

60

Adomian’s

Method

We consider

Applied to Hammerstein

a Hammerstein

integral

equation

Integral Equation

of the form

R

44 = Y(t) + From (3.7), we obtain

directly

Ax=y,

J KG, s>f(% x(s)) -R

Adornian’s

fundamental

where Ax(t)

= x(t) -

with R > 0.

6

functional

J

equation

(3.7) by writing

R

WC s) f(s, 4s))

ds.

(3.8)

-R

Here, the operator

A can be decomposed

as: A = L + N; where Lx = x is the linear

is no remainder linear term) and Nz = - J_“, K(t, s) f(s, z(s)) ds is the nonlinear nonlinear. Then (3.8) can be written Lx+Nx=y

or

According to Adomian’s technique the solution are calculated by the following algorithm:

term (there term,

with f

x+Nx=y.

(3.9)

x of (3.9) verifies x = 5 z,, n=O

where the terms x,,

x0 =Y, 21 =

Ao,

22

-41,

=

x, = A,_1. We obtain

the Adomian’s

polynomials

N @A%)

= -[IK(r,s)f

where X is a parameter introduced From (3.10) we obtain

Then

A, from the nonlinear

(+%(a))

term by writing

da = ~,‘X,

(3.10)

for convenience.

we have A, = ;

(3.11)

REMARK. Formula (3.11) allows us to calculate z,, and then the solution x of the Hammerstein integral equation. Numerically (3.11) is not expensive to implement [12]. Convergence of Adomian’s

Method

Applied to Hammerstein

Let us come back to our nonlinear integral equation technique, equation (3.9) can be replaced by

(3.9) x + N x = y. According

to Adomian’s

(3.12)

&+gAn=y,

n=O

Integral Equations

n=O

Nonlinear

but the convergence What

hypothesis

problem

will ensure

Hammerstein

remains

Integral

in suspense.

the convergence

61

Equations

We have to answer to the following zn and En”=-, A,?

of the series C,“=,

question: In a recent

paper [13], Y. Cherruault has given an answer to the above question of convergence. In a general case, he has given a proof of the convergence of Adomian’s method by using the fixed point theorem. proving

He introduced that

a new formulation

S, is a solution

of the method

N (x0 + S) = In the

following,

series substituted G. Saccomandi We consider

we will give a new convergence in another

S, = xi + x2 +. . . + x, and

by setting

of the fixed point equation

s.

theorem

using

the properties

series, and recent results of convergence

obtained

of the entire

by Y. Cherruault,

and B. Some [12]. a Hammerstein

nonlinear

integral

equation

of the form

R

x(t) = Y(t) + calling

J wt, s>f(s, x(s))

with R > 0;

ds,

(3.13)

-R

Na: = - J_“, K(t, s) f(s, X(S)) ds, equation

(3.13) becomes

x+Nx=y.

If x can be developed as an infinite parameterized by introducing X

The series (3.15) is absolutely

series, we can write

convergent

(3.14)

x =

E x,. n=O

This expression

00 xx = cx,x,.

can be

(3.15)

for JX] 5 1, and for such values of X the series

(3.16)

where m is the upper limit of the series’ elements and 0 < p < 1, is a majorant On the other hand, if f is an analytical function with respect to x in the interval

series for (3.15). [-R, R] we have

Nx=&,x,.

(3.17)

n=O

Then,

subtituting

(3.15) into (3.17) we obtain

the following

array

~o+a~~o+a2s~+~~~+.~x~+'.~+u~x~X+202x~x~X+...+71,~,~;;L-~~~~ +... +alx2X2

+ a2(x? +2x0x2)+....

(3.18)

The i-row of this array converges to the Ai defined as in (3.11) (for details, see [12]), when we set X = 1 because Nx can be developed in a Taylor series. Our problem is now to prove the convergence of the double serie in the array (3.18) for X = 1. It is necessary to have X = 1 inside the radius of convergence: otherwise, we are not able to recover the solution (3.14). In our situation, the classical theorems of elementary calculus on double series cannot ensure that 1 will be inside the radius of convergence of (3.18). The following theorem states a sufficient condition to ensure the convergence of (3.18) for X = 1. Obviously, we must require some stronger hypothesis than usual but generally the equations coming from physics satisfy such hypotheses.

B. SOME

62 THEOREM.

If f is an analytical

function

series x = Cr=, x,, the parameterization and the series x can be majored by

of x in (-R,

R), x can be decomposed

xx = 5 x, X, is absolutely n=O

convergent

as an infinite for X E [-1, 11,

(3.19) where m’ 2 m (m the upper

converges

This theorem recent

limit of the x,),

E > m/R,

and p I 1, then the double

series (3.18)

way as the proof of our theorem

given in the

for X = 1.

can be demonstrated

in a similar

work [12, pp. 88-891

4. DISCUSSION In practice, Adornian’s method is a numerical elegant method that can solve wide classes of nonlinear equations (algebraic differential, intergro-differential, partial differential equations, etc.) It avoids the cumbersome integrations of the projection methods and can solve physical problems which cannot be done by other numerical methods (iterative collocation methods, etc.) Adomian’s method is especially well-adapted to Hammerstein integral equations because of the form of the nonlinear functional equation chosen by Adomian. Convergence hypotheses are easy to verify and we obtain rapidly an approximation of the solution. The solution is given by a function, and not only at some grid points as is in the collocation method [5]. This method is powerful, more efficient, and faster than projection methods (Galerkin’s method, collocation method, etc.) [2]. The only difficulties, could arise from the calculations of Adomian’s polynomials applied to nonlinear integral equations. But, in most pratical cases, the nonlinearities to simple A,‘s, which can be easily determined.

A,

when give rise

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Dans. Theory Stat. Physics 18, 185-204 C.T. Kelley, A fast two-grid method for matrix H-equations, (1989). K.E. Atkinson, A survey of numerical methods for solving nonlinear integral equations, J. of Integral Equations 4 (l), 15-46 (1992). K. Atkinson and F. Potra, Projection and iterated methods for nonlinear integral equations, SIAM. J. Num. Anal. 24, 1352-1373 (1989). M. Golberg, Editor, Perturbed projection methods for various classes of operators and integral equations in numerical solution of integral equations, Plenum Press, New York, (1990). S. Kumar, A discrete collocation type method for Hammerstein equations, SIAM J. Num. Anal. 25, 328-341 (1988). J. Saranen, Projection methods for a class of Hammerstein equations, SIAM J. Num. Anal. (to appear). K. Atkinson, A survey of numerical methods for Fredholm integral equations of the second kind, SIAM J. Num. Anal. 24, 248-262 (1976). K. Atkinson and G. Chandler, BIE methods for solving Laplace’s equation with nonlinear boundary conditions: The smooth boundary case, Math. Comp. 55, 451-472 (1990). S. Kumar and I. Sloan, A new collocation type method for Hammerstein integral equations, Math. Comp. 48, 585-593 (1987). G. Adomian, Convergent series solution of nonlinear equations, J. of Comp. and App. Maths. 11, 225-230 (1984). G. Adomian, A review of the decomposition method and some recent results for nonlinear equations, Mathl. Comput. Modelling 13 (7), 17-43 (1990). Y. Cherruault, G. Saccomandi and B. Some, News results for convergence of Adomian’s method applied to integral equations, Mathl. Comput. Modelling 16 (2), 85-93 (1992). Y. Cherruault, Convergence of Adomian’s method, Kybernetes 18 (2), 31-38 (1989).