Volume 6, Issue 2
A Spectral Method for Neutral Volterra Integro-Differential Equation with Weakly Singular Kernel

Numer. Math. Theor. Meth. Appl., 6 (2013), pp. 424-446.

Published online: 2013-06

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• Abstract

This paper is concerned with obtaining an approximate solution and an approximate derivative of the solution for neutral Volterra integro-differential equation with a weakly singular kernel. The solution of this equation, even for analytic data, is not smooth on the entire interval of integration. The Jacobi collocation discretization is proposed for the given equation. A rigorous analysis of error bound is also provided which theoretically justifies that both the error of approximate solution and the error of approximate derivative of the solution decay exponentially in $L^∞$ norm and weighted $L^2$ norm. Numerical results are presented to demonstrate the effectiveness of the spectral method.

• Keywords

Neutral Volterra integro-differential equation, weakly singular kernel, Jacobi collocation discretization, convergence analysis.

65R20, 45J05, 65N12

• BibTex
• RIS
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@Article{NMTMA-6-424, author = {}, title = {A Spectral Method for Neutral Volterra Integro-Differential Equation with Weakly Singular Kernel}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2013}, volume = {6}, number = {2}, pages = {424--446}, abstract = {

This paper is concerned with obtaining an approximate solution and an approximate derivative of the solution for neutral Volterra integro-differential equation with a weakly singular kernel. The solution of this equation, even for analytic data, is not smooth on the entire interval of integration. The Jacobi collocation discretization is proposed for the given equation. A rigorous analysis of error bound is also provided which theoretically justifies that both the error of approximate solution and the error of approximate derivative of the solution decay exponentially in $L^∞$ norm and weighted $L^2$ norm. Numerical results are presented to demonstrate the effectiveness of the spectral method.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2013.1125nm}, url = {http://global-sci.org/intro/article_detail/nmtma/5912.html} }
TY - JOUR T1 - A Spectral Method for Neutral Volterra Integro-Differential Equation with Weakly Singular Kernel JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 424 EP - 446 PY - 2013 DA - 2013/06 SN - 6 DO - http://doi.org/10.4208/nmtma.2013.1125nm UR - https://global-sci.org/intro/article_detail/nmtma/5912.html KW - Neutral Volterra integro-differential equation, weakly singular kernel, Jacobi collocation discretization, convergence analysis. AB -

This paper is concerned with obtaining an approximate solution and an approximate derivative of the solution for neutral Volterra integro-differential equation with a weakly singular kernel. The solution of this equation, even for analytic data, is not smooth on the entire interval of integration. The Jacobi collocation discretization is proposed for the given equation. A rigorous analysis of error bound is also provided which theoretically justifies that both the error of approximate solution and the error of approximate derivative of the solution decay exponentially in $L^∞$ norm and weighted $L^2$ norm. Numerical results are presented to demonstrate the effectiveness of the spectral method.

Yunxia Wei & Yanping Chen. (2020). A Spectral Method for Neutral Volterra Integro-Differential Equation with Weakly Singular Kernel. Numerical Mathematics: Theory, Methods and Applications. 6 (2). 424-446. doi:10.4208/nmtma.2013.1125nm
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