Volume 9, Number 2, Summer 2001
SEQUENTIAL EIGENFUNCTION
EXPANSION FOR CERTAIN
NONLINEAR HAMMERSTEIN EQUATIONS
K.K. TAM, ANDONOWATI AND M.T. KIANG
Introduction. In the numerical solution of nonlinear integral
equations [1], [3], [4], discretization schemes such as FredholmNyström
or projection schemes such as Galerkin lead to systems of N simultaneous
nonlinear algebraic or transcendental equations which have to be
solved by Newton’s method. In general it may be expected that the
accuracy of the approximation improves with increasing N, but the
difficulty in using Newton’s method for large N is well known. In this
note we consider a projection method in which the solution to the integral
equation is sought in terms of an expansion where the coefficients
of the projections are determined sequentially, one at a time through
iteration. Thus the need to solve a system of nonlinear equations is
obviated.
In Section 2, we illustrate the procedure for a Hammersteintype
equation for which the symmetric kernel has a complete orthonormal
set of eigenfunctions. We detail the development and give some
consideration to the convergence of the iteration scheme. In Section 3,
we consider the case where the kernel is degenerate. A few examples
are then considered in Section 4.
This idea of determining the coefficients of an eigenfunction expansion
for a nonlinear problem sequentially was developed by Tam, Andonowati
and Kiang [6] for a semilinear elliptic partial differential
equation. Here we provide the parallel development for its integral
equation counterpart.
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