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Spectral Collocation Methods for Second-Order Volterra Integro-Differential Equations with Weakly Singular Kernels

Spectral Collocation Methods for Second-Order Volterra Integro-Differential Equations with Weakly Singular Kernels
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Year:    2020

Author:    Xiulian Shi, Yanping Chen, Yunqing Huang, Fenglin Huang

Advances in Applied Mathematics and Mechanics, Vol. 12 (2020), Iss. 2 : pp. 480–502

Abstract

In this paper, a Jacobi spectral collocation approximation is proposed for the solution of second-order Volterra integro-differential equations with weakly singular kernels. The solution of such equations usually exhibits a singular behaviour at the origin. By using some suitable variable transformations, we obtain a new equation which is still weakly singular, but whose solution is as smooth as we like. Then the resulting equation is solved by standard spectral methods. We establish a rigorous error analysis for the proposed method, which shows that the numerical errors decay exponentially in $L^\infty$-norm and weighted $L^2$-norm. Finally, to perform the numerical simulation, a test example is considered with non-smooth solutions.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.OA-2019-0056

Advances in Applied Mathematics and Mechanics, Vol. 12 (2020), Iss. 2 : pp. 480–502

Published online:    2020-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    23

Keywords:    Volterra integro-differential equations weakly singular kernel spectral collocation methods.

Author Details

Xiulian Shi

Yanping Chen

Yunqing Huang

Fenglin Huang

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