奇异摄动 Volterra 积分微分方程参数一致的数值方法

数学物理学报 ›› 2025, Vol. 45 ›› Issue (2): 576-583.

奇异摄动 Volterra 积分微分方程参数一致的数值方法

刘利斌^*(),廖仪戈(),隆广庆()

南宁师范大学应用数学中心南宁 530100

收稿日期:2022-10-10 修回日期:2024-10-06 出版日期:2025-04-26 发布日期:2025-04-09
通讯作者: 刘利斌 E-mail:liulibin969@163.com;lyg199600@163.com;longgq@amss.ac.cn
作者简介:廖仪戈，E-mail: lyg199600@163.com;|隆广庆，E-mail: longgq@amss.ac.cn
基金资助:
国家自然科学基金(12361087);国家自然科学基金(12261062)

A Parameter-Uniform Numerical Method for a Singularly Perturbed Volterra Integro-Differential Equation

Liu Libin^*(),Liao Yige(),Long Guangqing()

Center for Applied Mathematics of Guangxi, Nanning Normal University, Nanning 530100

Received:2022-10-10 Revised:2024-10-06 Online:2025-04-26 Published:2025-04-09
Contact: Libin Liu E-mail:liulibin969@163.com;lyg199600@163.com;longgq@amss.ac.cn
Supported by:
NSFC(12361087);NSFC(12261062)

摘要/Abstract

摘要：

针对一类奇异摄动 Volterra 积分微分方程, 在 Vulanović-Bakhvalov 网格上构造了一个一阶参数一致收敛的有限差分格式. 进一步, 基于 Richardson 外推技术, 将数值格式的收敛阶从 $O(N^{-1})$ 提高到 $O(N^{-2})$, 其中 $N$ 是网格剖分数. 最后, 数值实验证明了数值方法的有效性.

关键词: 奇异摄动, Volterra 积分微分方程, Richardson 外推, Vulanovi?-Bakhvalov 网格

Abstract:

A singularly perturbed Volterra integro-differential equation is considered. The problem is discretized by using a simple first-order finite difference scheme on a Vulanović-Bakhvalov mesh, the accuracy of which is first-order uniformly convergent with respect to the perturbation parameter $\varepsilon$. Furthermore, based on the Richardson extrapolation technique, the $\varepsilon$-uniform accuracy of the presented approximation scheme can be improved from $O(N^{-1})$ to $O(N^{-2})$, where $N$ is the number of mesh intervals. Finally, the theoretical finds are illustrated by two numerical experiments.

Key words: singularly perturbed, Volterra integro-differential equation, Richardson extrapolation, Vulanovi?-Bakhvalov mesh

中图分类号:

O241.8

刘利斌,廖仪戈,隆广庆. 奇异摄动 Volterra 积分微分方程参数一致的数值方法[J]. 数学物理学报, 2025, 45(2): 576-583.

Liu Libin,Liao Yige,Long Guangqing. A Parameter-Uniform Numerical Method for a Singularly Perturbed Volterra Integro-Differential Equation[J]. Acta mathematica scientia,Series A, 2025, 45(2): 576-583.

参考文献

[1]	Iragi B C, Munyakazi J B. A uniformly convergent numerical method for a singularly perturbed Volterra integro-differential equation. Int J Comput Math, 2020, 97(4): 759-771
[2]	Hoppensteadt F C. An algorithm for approximate solutions to weakly filtered synchronous control systems and nonlinear renewal processes. SIAM J Appl Math, 1983, 43(4): 834-843
[3]	Lodge A S, McLeod J B, Nohel J A. A nonlinear singularly perturbed Volterra integrodifferential equation occurring in polymer rheology. Proc R Soc Edinburgh Sect A, 1978, 80(1/2): 99-137
[4]	Jordan G S. A nonlinear singularly perturbed Volterra integrodifferential equation of nonconvolution type. Proc R Soc Edinburgh Sect A, 1978, 80(3/4): 235-247
[5]	Yapman ?, Amiraliyev G M. A novel second-order fitted computational method for a singularly perturbed Volterra integro-differential equation. Int J Comput Math, 2020, 97(6): 1293-1302
[6]	?evgin S. Numerical solution of a singularly perturbed Volterra integro-differential equation. Adv Difference Equ, 2014, 171: 1-15
[7]	Huang J, Cen Z, Xu A, Liu L B. A posteriori error estimation for a singularly perturbed Volterra integro-differential equation. Numer Algorithms, 2020, 83(2): 549-563
[8]	Sumit S, Kumar J, Vigo-Aguiar. Analysis of a nonlinear singularly perturbed Volterra integro-differential equation. J Comput Appl Math, 2021, Article 113410
[9]	Nhan T A, Vulanovi? R. Analysis of the truncation error and barrier-function technique for a Bakhvalov-type mesh. Electron Trans Numer Anal, 2019, 51: 315-330
[10]	Long G, Liu L B, Huang Z. Richardson extrapolation method on an adaptive grid for singularly perturbed Volterra integro-differential equations. Numer Funct Anal Optim, 2021, 42: 739-757
[11]	Lin? T. Error expansion for a first-order upwind difference scheme applied to a model convection-diffusion problem. IMA J Numer Anal, 2004, 24: 239-253
[12]	Bakhvalov N S. The optimization of methods of solving boundary value problems with a boundary layer. Comp Math Math Phys, 1969, 9(4): 139-166
[13]	Boglaev I P. Approximate solution of a nonlinear boundary value problem with a small parameter at the highest-order derivative. USSR Comput Math Math Phys, 1984, 24(6): 30-35
[14]	Andreev V B, Kopteva N V. On the convergence, uniform with respect to a small parameter of monotone three-point finite difference approximations. Differ Equations, 1998, 34(7): 921-929
[15]	Kopteva N V. On the uniform with respect to a small parameter convergence of the central difference scheme on condensing meshes. Comp Math Math Phys, 1999, 39(10): 1594-1610
[16]	Kopteva N. Uniform pointwise convergence of difference schemes for convection-diffusion problems on layer-adapted meshes. Computing, 2001, 66: 179-197
[17]	Vulanovi? R. On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh. Univ u Novom Sadu Zb Rad Prir Mat Fak Ser Mat, 1983, 13: 187-201
[18]	Lin? T. Sufficient conditions for uniform convergence on layer-adapted grids. Appl Numer Math, 2001, 37: 241-255
[19]	Roos H G, Lin? T. Sufficient conditions for uniform convergence on layer-adapted grids. Computing, 1999, 63: 27-45
[20]	Kudu M, Amirali I, Amiraliyev G M. A finite-difference method for a singularly perturbed delay integro-differential equation. J Comput Appl Math, 2016, 308: 379-390
[21]	Amiraliyev G M, ?evgin S. Uniform difference method for singularly perturbed Volterra integro-differential equations. Applied Mathematics and Computation, 2006, 179(2): 731-741