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A new collocation-type method for Hammerstein integral equations

Authors: Sunil Kumar and Ian H. Sloan
Journal: Math. Comp. 48 (1987), 585-593
MSC: Primary 65R20; Secondary 45G10
MathSciNet review: 878692
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Abstract: We consider Hammerstein equations of the form

$\displaystyle y(t) = f(t) + \int_a^b {k(t,s)g(s,y(s))\,ds,\quad t \in [a,b],} $

and present a new method for solving them numerically. The method is a collocation method applied not to the equation in its original form, but rather to an equivalent equation for $ z(t): = g(t,y(t))$. The desired approximation to y is then obtained by use of the (exact) equation

$\displaystyle y(t) = f(t) + \int_a^b {k(t,s)z(s)\,ds,\quad t \in [a,b].} $

Advantages of this method, compared with the direct collocation approximation for y, are discussed. The main result in the paper is that, under suitable conditions, the resulting approximation to y converges to the exact solution at a rate at least equal to that of the best approximation to z from the space in which the collocation solution is sought.

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Article copyright: © Copyright 1987 American Mathematical Society