Approximation of weakly singular integral equations by sinc projection methods

Abstract

In this paper, two numerical schemes for a nonlinear integral equation of Fredholm type with weakly singular kernel are studied. These numerical methods blend collocation, convolution, and approximations based on sinc basis functions with iterative schemes like the steepest descent and Newton's method, involving the solution of a nonlinear system of equations. Exponential rate of convergence for the convolution scheme is shown and collocation method is analyzed. Numerical experiments are presented to illustrate the sharpness of the theoretical estimates and the sensitivity of the solutions with respect to some parameters in the equations. The comparison between the schemes indicates that the sinc convolution method is more effective.

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Key words

Fredholm integral equation, Urysohn integral operator, weak singularity, convolution method, collocation method

AMS subject classifications

45B05, 45E99, 65J15, 65R60.

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