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