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Spectral Petrov-Galerkin Methods for the Second Kind Volterra Type Integro-Differential Equations

Published online by Cambridge University Press:  28 May 2015

Xia Tao ,
Ziqing Xie  and
Xiaojun Zhou
Xia Tao*
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing, Ministry of Education of China, Hunan Normal University, Changsha, Hunan 410081, P.R. China
Ziqing Xie*
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing, Ministry of Education of China, Hunan Normal University, Changsha, Hunan 410081, P.R. China
Xiaojun Zhou*
Affiliation:
School of Mathematics and Computer Science, Guizhou Normal University, Guiyang, Guizhou 550001, P. R. China
*
Corresponding author.Email address:xtaohn@gmail.com
Corresponding author.Email address:ziqingxie@yahoo.com.cn
Corresponding author.Email address:zxj0702@126.com
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Abstract

This work is to provide general spectral and pseudo-spectral Jacobi-Petrov-Galerkin approaches for the second kind Volterra integro-differential equations. The Gauss-Legendre quadrature formula is used to approximate the integral operator and the inner product based on the Jacobi weight is implemented in the weak formulation in the numerical implementation. For some spectral and pseudo-spectral Jacobi-Petrov-Galerkin methods, a rigorous error analysis in both and L∞ norms is given provided that both the kernel function and the source function are sufficiently smooth. Numerical experiments validate the theoretical prediction.

Keywords

65M1078A48Volterra integro-differential equationspectral Jacobi-Petrov-Galerkinpseudo-spectral Jacobi-Petrov-Galerkinspectral convergence
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 2 , May 2011 , pp. 216 - 236
Copyright
Copyright © Global Science Press Limited 2011

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