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Superconvergence of a collocation-type method for simple turning points of Hammerstein equations

Author: Sunil Kumar
Journal: Math. Comp. 50 (1988), 385-398
MSC: Primary 65R20; Secondary 45G10
MathSciNet review: 929543
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Abstract: In this paper a simple turning point ($ y = {y^c}$, $ \lambda = {\lambda ^c}$) of the parameter-dependent Hammerstein equation

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

is approximated numerically in the following way. A simple turning point ($ z = {z^c}$, $ \lambda = {\lambda ^c}$) of an equivalent equation for $ z(t):=\lambda g(t,y(t))$ is computed first. This is done by solving a discretized version of a certain system of equations which has ($ {z^c}$, $ {\lambda ^c}$) as part of an isolated solution. The particular discretization used here is standard piecewise polynomial collocation. Finally, an approximation to $ {y^c}$ is obtained by use of the (exact) equation

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

The main result of the paper is that, under suitable conditions, the approximations to $ {y^c}$ and $ {\lambda ^c}$ are both superconvergent, that is, they both converge to their respective exact values at a faster rate than the collocation approximation (of $ {z^c}$) does to $ {z^c}$.

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Article copyright: © Copyright 1988 American Mathematical Society

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