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

Yunxia Wei and Yanping Chen

10.4208/aamm.10-m1055

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

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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.

  • History

Published online: 2012-04

  • AMS Subject Headings

45J05, 65R20

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