Volume 40, Issue 2
Numerical Analysis of a Nonlinear Singularly Perturbed Delay Volterra Integro-Differential Equation on an Adaptive Grid

J. Comp. Math., 40 (2022), pp. 258-274.

Published online: 2022-01

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• Abstract

In this paper, we study a nonlinear first-order singularly perturbed Volterra integro-differential equation with delay. This equation is discretized by the backward Euler for differential part and the composite numerical quadrature formula for integral part for which both an a priori and an a posteriori error analysis in the maximum norm are derived. Based on the a priori error bound and mesh equidistribution principle, we prove that there exists a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter. The a posteriori error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Furthermore, we extend our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations. Numerical results are provided to demonstrate the effectiveness of our presented monitor function. Meanwhile, it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay differential equations with a turning point.

65L05, 65L20, 65L50

liulibin969@163.com (Libin Liu)

yanpingchen@scnu.edu.cn (Yanping Chen)

l_ying321@163.com (Ying Liang)

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@Article{JCM-40-258, author = {Liu , LibinChen , Yanping and Liang , Ying}, title = {Numerical Analysis of a Nonlinear Singularly Perturbed Delay Volterra Integro-Differential Equation on an Adaptive Grid}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {2}, pages = {258--274}, abstract = {

In this paper, we study a nonlinear first-order singularly perturbed Volterra integro-differential equation with delay. This equation is discretized by the backward Euler for differential part and the composite numerical quadrature formula for integral part for which both an a priori and an a posteriori error analysis in the maximum norm are derived. Based on the a priori error bound and mesh equidistribution principle, we prove that there exists a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter. The a posteriori error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Furthermore, we extend our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations. Numerical results are provided to demonstrate the effectiveness of our presented monitor function. Meanwhile, it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay differential equations with a turning point.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2008-m2020-0063}, url = {http://global-sci.org/intro/article_detail/jcm/20186.html} }
TY - JOUR T1 - Numerical Analysis of a Nonlinear Singularly Perturbed Delay Volterra Integro-Differential Equation on an Adaptive Grid AU - Liu , Libin AU - Chen , Yanping AU - Liang , Ying JO - Journal of Computational Mathematics VL - 2 SP - 258 EP - 274 PY - 2022 DA - 2022/01 SN - 40 DO - http://doi.org/10.4208/jcm.2008-m2020-0063 UR - https://global-sci.org/intro/article_detail/jcm/20186.html KW - Delay Volterra integro-differential equation, Singularly perturbed, Error analysis, Monitor function. AB -

In this paper, we study a nonlinear first-order singularly perturbed Volterra integro-differential equation with delay. This equation is discretized by the backward Euler for differential part and the composite numerical quadrature formula for integral part for which both an a priori and an a posteriori error analysis in the maximum norm are derived. Based on the a priori error bound and mesh equidistribution principle, we prove that there exists a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter. The a posteriori error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Furthermore, we extend our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations. Numerical results are provided to demonstrate the effectiveness of our presented monitor function. Meanwhile, it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay differential equations with a turning point.

Libin Liu, Yanping Chen & Ying Liang. (2022). Numerical Analysis of a Nonlinear Singularly Perturbed Delay Volterra Integro-Differential Equation on an Adaptive Grid. Journal of Computational Mathematics. 40 (2). 258-274. doi:10.4208/jcm.2008-m2020-0063
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