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Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Equations with Smooth Solutions

Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Equations with Smooth Solutions
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Year:    2012

Author:    Yunxia Wei, Yanping Chen

Advances in Applied Mathematics and Mechanics, Vol. 4 (2012), Iss. 1 : pp. 1–20

Abstract

The theory of a class of spectral methods is extended to Volterra integro-differential equations which contain a weakly singular kernel $(t-s)^{-\mu}$ with $0<\mu<1$. In this work, we consider the case when the underlying solutions of weakly singular Volterra integro-differential equations are sufficiently smooth. We provide a rigorous error analysis for the spectral methods, which shows that both the errors of approximate solutions and the errors of approximate derivatives of the solutions decay exponentially in $L^\infty$-norm and weighted $L^2$-norm. The numerical examples are given to illustrate the theoretical results.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.10-m1055

Advances in Applied Mathematics and Mechanics, Vol. 4 (2012), Iss. 1 : pp. 1–20

Published online:    2012-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    20

Keywords:    Volterra integro-differential equations weakly singular kernels spectral methods convergence analysis.

Author Details

Yunxia Wei

Yanping Chen

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