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Volume 4, Issue 2
Spectral Petrov-Galerkin Methods for the Second Kind Volterra Type Integro-Differential Equations

Xia Tao, Ziqing Xie & Xiaojun Zhou

Numer. Math. Theor. Meth. Appl., 4 (2011), pp. 216-236.

Published online: 2011-04

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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 $L_{\omega^{\alpha,\beta}}^2$ and $L^\infty$ norms is given provided that both the kernel function and the source function are sufficiently smooth. Numerical experiments validate the theoretical prediction.

  • AMS Subject Headings

65M10, 78A48

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-4-216, author = {}, title = {Spectral Petrov-Galerkin Methods for the Second Kind Volterra Type Integro-Differential Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2011}, volume = {4}, number = {2}, pages = {216--236}, 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 $L_{\omega^{\alpha,\beta}}^2$ and $L^\infty$ norms is given provided that both the kernel function and the source function are sufficiently smooth. Numerical experiments validate the theoretical prediction.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2011.42s.6}, url = {http://global-sci.org/intro/article_detail/nmtma/5966.html} }
TY - JOUR T1 - Spectral Petrov-Galerkin Methods for the Second Kind Volterra Type Integro-Differential Equations JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 216 EP - 236 PY - 2011 DA - 2011/04 SN - 4 DO - http://doi.org/10.4208/nmtma.2011.42s.6 UR - https://global-sci.org/intro/article_detail/nmtma/5966.html KW - Volterra integro-differential equation, spectral Jacobi-Petrov-Galerkin, pseudo-spectral Jacobi-Petrov-Galerkin, spectral convergence. AB -

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 $L_{\omega^{\alpha,\beta}}^2$ and $L^\infty$ norms is given provided that both the kernel function and the source function are sufficiently smooth. Numerical experiments validate the theoretical prediction.

Xia Tao, Ziqing Xie & Xiaojun Zhou. (2020). Spectral Petrov-Galerkin Methods for the Second Kind Volterra Type Integro-Differential Equations. Numerical Mathematics: Theory, Methods and Applications. 4 (2). 216-236. doi:10.4208/nmtma.2011.42s.6
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