### NUMERICAL ANALYSIS OF A NONLINEAR SINGULARLY PERTURBED DELAY VOLTERRA INTEGRO-DIFFERENTIAL EQUATION ON AN ADAPTIVE GRID

Libin Liu1, Yanping Chen2, Ying Liang3

1. 1. School of Mathematics and Statistics, Nanning Normal University, Nanning 530023, China;
2. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
3. School of Mathematics and Statistics, Nanning Normal University, Nanning 530023, China
• Received:2020-03-16 Revised:2020-08-18 Online:2022-03-15 Published:2022-03-29
• Contact: Yanping Chen,Email: yanpingchen@scnu.edu.cn
• Supported by:
This work is supported by the State Key Program of National Natural Science Foundation of China (11931003) and National Science Foundation of China (41974133,11761015,11971410),the Natural Science Foundation of Guangxi (2020GXNSFAA159010)

Libin Liu, Yanping Chen, Ying Liang. NUMERICAL ANALYSIS OF A NONLINEAR SINGULARLY PERTURBED DELAY VOLTERRA INTEGRO-DIFFERENTIAL EQUATION ON AN ADAPTIVE GRID[J]. Journal of Computational Mathematics, 2022, 40(2): 258-274.

In this paper, we study a nonlinear first-order singularly perturbed Volterra integrodifferential 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.

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