Volume 14, Issue 6
Piecewise Spectral Collocation Method for Second Order Volterra Integro-Differential Equations with Nonvanishing Delay

Adv. Appl. Math. Mech., 14 (2022), pp. 1333-1356.

Published online: 2022-08

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• Abstract

In this paper, the piecewise spectral-collocation method is used to solve the second-order Volterra integral differential equation with nonvanishing delay. In this collocation method, the main discontinuity point of the solution of the equation is used to divide the partitions to overcome the disturbance of the numerical error convergence caused by the main discontinuity of the solution of the equation. Derivative approximation in the sense of integral is constructed in numerical format, and the convergence of the spectral collocation method in the sense of the $L^∞$ and $L^2$ norm is proved by the Dirichlet formula. At the same time, the error convergence also meets the effect of spectral accuracy convergence. The numerical experimental results are given at the end also verify the correctness of the theoretically proven results.

• Keywords

Second-order Volterra type integro-differential equation, delay function, piecewise spectral-collocation method.

65M10, 78A48

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@Article{AAMM-14-1333, author = {Zhenrong and Chen and and 24229 and and Zhenrong Chen and Yanping and Chen and and 24230 and and Yanping Chen and Yunqing and Huang and and 24231 and and Yunqing Huang}, title = {Piecewise Spectral Collocation Method for Second Order Volterra Integro-Differential Equations with Nonvanishing Delay}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2022}, volume = {14}, number = {6}, pages = {1333--1356}, abstract = {

In this paper, the piecewise spectral-collocation method is used to solve the second-order Volterra integral differential equation with nonvanishing delay. In this collocation method, the main discontinuity point of the solution of the equation is used to divide the partitions to overcome the disturbance of the numerical error convergence caused by the main discontinuity of the solution of the equation. Derivative approximation in the sense of integral is constructed in numerical format, and the convergence of the spectral collocation method in the sense of the $L^∞$ and $L^2$ norm is proved by the Dirichlet formula. At the same time, the error convergence also meets the effect of spectral accuracy convergence. The numerical experimental results are given at the end also verify the correctness of the theoretically proven results.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2021-0334}, url = {http://global-sci.org/intro/article_detail/aamm/20850.html} }
TY - JOUR T1 - Piecewise Spectral Collocation Method for Second Order Volterra Integro-Differential Equations with Nonvanishing Delay AU - Chen , Zhenrong AU - Chen , Yanping AU - Huang , Yunqing JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1333 EP - 1356 PY - 2022 DA - 2022/08 SN - 14 DO - http://doi.org/10.4208/aamm.OA-2021-0334 UR - https://global-sci.org/intro/article_detail/aamm/20850.html KW - Second-order Volterra type integro-differential equation, delay function, piecewise spectral-collocation method. AB -

In this paper, the piecewise spectral-collocation method is used to solve the second-order Volterra integral differential equation with nonvanishing delay. In this collocation method, the main discontinuity point of the solution of the equation is used to divide the partitions to overcome the disturbance of the numerical error convergence caused by the main discontinuity of the solution of the equation. Derivative approximation in the sense of integral is constructed in numerical format, and the convergence of the spectral collocation method in the sense of the $L^∞$ and $L^2$ norm is proved by the Dirichlet formula. At the same time, the error convergence also meets the effect of spectral accuracy convergence. The numerical experimental results are given at the end also verify the correctness of the theoretically proven results.

Zhenrong Chen, Yanping Chen & Yunqing Huang. (2022). Piecewise Spectral Collocation Method for Second Order Volterra Integro-Differential Equations with Nonvanishing Delay. Advances in Applied Mathematics and Mechanics. 14 (6). 1333-1356. doi:10.4208/aamm.OA-2021-0334
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