While the usual polynomial-based spectral collocation methods are capable of providing highly accurate simulation results for nonlinear integral equations with smooth solutions, its accuracy can be deleteriously degraded when the solutions exhibit locally singular behaviours. Recovering non-smooth solutions of nonlinear systems of weakly singular integral equations is a highly non- trivial task. Nonetheless, such problems arise in many applications of practical interest. The present letter develops high-order spectral collocation method for a coupled system of Abel integral equations of the second kind with non-smooth solutions on two dimensional domains. The proposed solver builds an approximation to the solution via a smoothing transformation for the integral equation. The principal advantage of the proposed method is that it is direct (as opposed to iterative) and efficiently regains high accuracy without excessively increasing the number of degrees of freedom used.
A. Abdelkawy, Mohamed and G. Ameen, Ibrahem
"A Spectral Collocation Method for Coupled System of Two Dimensional Abel Integral Equations of the Second Kind,"
Information Sciences Letters: Vol. 8
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Available at: https://digitalcommons.aaru.edu.jo/isl/vol8/iss3/2