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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 Fredholm-Nyström 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 Hammerstein-type 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 semi-linear elliptic partial differential equation. Here we provide the parallel development for its integral equation counterpart.

 

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