Volume 45, pp. 183-200, 2016.

Operational Müntz-Galerkin approximation for Abel-Hammerstein integral equations of the second kind

P. Mokhtary

Abstract

Since solutions of Abel integral equations exhibit singularities, existing spectral methods for these equations suffer from instability and low accuracy. Moreover, for nonlinear problems, solving the resulting complex nonlinear algebraic systems numerically requires high computational costs. To overcome these drawbacks, in this paper we propose an operational Galerkin strategy for solving Abel-Hammerstein integral equations of the second kind which applies Müntz-Legendre polynomials as natural basis functions to discretize the problem and to obtain a sparse nonlinear system with upper-triangular structure that can be solved directly. It is shown that our approach yields a well-posed and easy-to-implement approximation technique with a high order of accuracy regardless of the singularities of the exact solution. The numerical results confirm the superiority and effectiveness of the proposed scheme.

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Key words

Abel-Hammerstein integral equations, Galerkin method, Müntz-Legendre polynomials, well-posedness

AMS subject classifications

45E10, 41A25

Links to the cited ETNA articles

[26]Vol. 44 (2015), pp. 462-471 Payam Mokhtary: High-order modified Tau method for non-smooth solutions of Abel integral equations
[27]Vol. 41 (2014), pp. 289-305 P. Mokhtary and F. Ghoreishi: Convergence analysis of the operational Tau method for Abel-type Volterra integral equations

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