The numerical solution of Hammerstein equations by a method based on polynomial collocation
Sunil Kumar
(Received 22 January 1988; revised 15 March 1989)
Abstract
In recent papers we have considered the numerical solution of the Hammerstein equation
ó 1 y(t) = f(t) + ô k(t, s) g(s, y(s)) ds,
t Î [-1, 1],
õ-1
by a method which first applies the standard collocation procedure to an equivalent equation for z(t) : =
g(t, y(t)), and then obtains an approximation to y by use of the equation
ó 1 y(t) = f(t) + ô k(t, s) z(s) ds,
t Î [-1, 1].
õ-1
In this paper we approximate z by a polynomial zn of degree £ n - 1,
with coefficients determined by collocation at the zeros of the nth degree Chebyshev polynomial of the first kind.
We then define the approximation to y to be
ó 1 yn(t) : = f(t) + ô k(t, s) zn(s) ds,
t Î [-1, 1],
õ-1
and establish that, under suitable conditions, limn®¥ yn(t) =
y(t), uniformly in t.
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Author
Sunil Kumar
School of Mathematics, The University of New South Wales, Sydney, NSW 2033, Australia.
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