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

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.

45J05, 65R20

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@Article{AAMM-4-1, author = {Yunxia Wei , and Chen , Yanping}, 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} }
TY - JOUR T1 - Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Equations with Smooth Solutions AU - Yunxia Wei , AU - Chen , Yanping JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 1 EP - 20 PY - 2012 DA - 2012/04 SN - 4 DO - http://doi.org/10.4208/aamm.10-m1055 UR - https://global-sci.org/intro/article_detail/aamm/103.html KW - Volterra integro-differential equations, weakly singular kernels, spectral methods, convergence analysis. AB -

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.

Yunxia Wei & Yanping Chen. (1970). Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Equations with Smooth Solutions. Advances in Applied Mathematics and Mechanics. 4 (1). 1-20. doi:10.4208/aamm.10-m1055
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