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

DOI:
https://doi.org/10.1090/S0025-5718-1987-0878692-4

MathSciNet review:
878692

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Abstract: We consider Hammerstein equations of the form \[ y(t) = f(t) + \int _a^b {k(t,s)g(s,y(s)) ds,\quad t \in [a,b],} \] 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 \[ y(t) = f(t) + \int _a^b {k(t,s)z(s) ds,\quad t \in [a,b].} \] 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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© Copyright 1987
American Mathematical Society