Volume 4, Issue 1
Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Equations with Smooth Solutions

Yunxia Wei & Yanping Chen

Adv. Appl. Math. Mech., 4 (2012), pp. 1-20.

Published online: 2012-04

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  • 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.

  • Keywords

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

  • AMS Subject Headings

45J05 65R20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-4-1, author = {Yunxia Wei and Yanping Chen}, title = {Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Equations with Smooth Solutions}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2012}, volume = {4}, number = {1}, pages = {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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.10-m1055}, url = {http://global-sci.org/intro/article_detail/aamm/103.html} }
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