Volume 44, pp. 462-471, 2015.

High-order modified Tau method for non-smooth solutions of Abel integral equations

Payam Mokhtary

Abstract

In this paper, the spectral Tau method and generalized Jacobi functions are fruitfully combined to approximate Abel integral equations with solutions that may have singularities (non-smooth solutions) at the origin. In an earlier work of P. Mokhtary and F. Ghoreishi [Electron. Trans. Numer. Anal., 41 (2014), pp. 289–305], a regularization process was used to handle the high-order Tau method based on classical Jacobi polynomials for the numerical solution of Abel integral equations. However, it was found that this scheme makes the resulting equation and its Tau approximation more complicated. In this work, we introduce and analyze a new modified Tau method for the numerical solution of Abel integral equations with non-smooth solutions. The main advantage of this method is that it gains a high order of accuracy without adopting any regularization process. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique.

Full Text (PDF) [164 KB]

Key words

modified Tau method, generalized Jacobi functions, Abel integral equations.

AMS subject classifications

45E10, 41A25.

Links to the cited ETNA articles

[7]Vol. 41 (2014), pp. 289-305 P. Mokhtary and F. Ghoreishi: Convergence analysis of the operational Tau method for Abel-type Volterra integral equations

ETNA articles which cite this article

Vol. 45 (2016), pp. 183-200 P. Mokhtary: Operational Müntz-Galerkin approximation for Abel-Hammerstein integral equations of the second kind

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