ASYMPTOTICAL STABILITY OF NEUTRAL REACTION-DIFFUSION EQUATIONS WITH PCAS AND THEIR FINITE ELEMENT METHODS∗

doi:10.1007/s10473-023-0424-9

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (4): 1865-1880.doi: 10.1007/s10473-023-0424-9

ASYMPTOTICAL STABILITY OF NEUTRAL REACTION-DIFFUSION EQUATIONS WITH PCAS AND THEIR FINITE ELEMENT METHODS^∗

Hao HAN^1,2, Chengjian ZHANG^1,2,†

1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China;
2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China

收稿日期:2022-02-21 修回日期:2022-07-25 发布日期:2023-08-08
作者简介:Hao HAN, E-mail: D201880002@hust.edu.cn
基金资助:
*NSFC (11971010).

ASYMPTOTICAL STABILITY OF NEUTRAL REACTION-DIFFUSION EQUATIONS WITH PCAS AND THEIR FINITE ELEMENT METHODS^∗

Hao HAN^1,2, Chengjian ZHANG^1,2,†

1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China;
2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China

Received:2022-02-21 Revised:2022-07-25 Published:2023-08-08
About author:Hao HAN, E-mail: D201880002@hust.edu.cn
Supported by:
*NSFC (11971010).

摘要/Abstract

摘要： This paper focuses on the analytical and numerical asymptotical stability of neutral reaction-diffusion equations with piecewise continuous arguments. First, for the analytical solutions of the equations, we derive their expressions and asymptotical stability criteria. Second, for the semi-discrete and one-parameter fully-discrete finite element methods solving the above equations, we work out the sufficient conditions for assuring that the finite element solutions are asymptotically stable. Finally, with a typical example with numerical experiments, we illustrate the applicability of the obtained theoretical results.

关键词: neutral reaction-diffusion equations, piecewise continuous arguments, asymptotical stability, finite element methods, numerical experiment

Abstract: This paper focuses on the analytical and numerical asymptotical stability of neutral reaction-diffusion equations with piecewise continuous arguments. First, for the analytical solutions of the equations, we derive their expressions and asymptotical stability criteria. Second, for the semi-discrete and one-parameter fully-discrete finite element methods solving the above equations, we work out the sufficient conditions for assuring that the finite element solutions are asymptotically stable. Finally, with a typical example with numerical experiments, we illustrate the applicability of the obtained theoretical results.

Key words: neutral reaction-diffusion equations, piecewise continuous arguments, asymptotical stability, finite element methods, numerical experiment

Hao HAN, Chengjian ZHANG. ASYMPTOTICAL STABILITY OF NEUTRAL REACTION-DIFFUSION EQUATIONS WITH PCAS AND THEIR FINITE ELEMENT METHODS^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1865-1880.

Hao HAN, Chengjian ZHANG. ASYMPTOTICAL STABILITY OF NEUTRAL REACTION-DIFFUSION EQUATIONS WITH PCAS AND THEIR FINITE ELEMENT METHODS^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1865-1880.

参考文献

[1] Bereketoglu H, Lafci M. Behavior of the solutions of a partial differential equation with a piecewise constant argument. Filomat, 2017, 31: 5931-5943
[2] Chen S, Zhao J. Estimations of the constants in inverse inequality for finite element functions. J Comput Math, 2013, 31: 522-531
[3] Esmailzadeh M, Najafi H S, Aminikhah H. A numerical scheme for diffusion-convection equation with piecewise constant arguments. Comput Meth Differ Equ, 2020, 8: 573-584
[4] Esmailzadeh M, Najafi H S, Aminikhah H. A numerical method for solving hyperbolic partial differential equations with piecewise constant arguments and variable coefficients. J Differ Equ Appl, 2021, 27: 172-194
[5] Han H, Zhang C. One-parameter Galerkin finite element methods for neutral reaction-diffusion equations with piecewise continuous arguments. J Sci Comput, 2022, 90: Art 91
[6] Kolmanovskii V, Myshkis A.Introduction to the Theory and Applications of Functional Differential Equations. Dordrecht: Kluwer Academic Publishers, 1999
[7] Kuang J, Xiang J, Tian H. The asymptotic stability of one-parameter methods for neutral differential equations. BIT, 1994, 34: 400-408
[8] Kuang Y.Delay Differential Equations with Applications in Population Dynamics. New York: Academic Press, 1993
[9] Li C, Zhang C. Block boundary value methods applied to functional differential equations with piecewise continuous arguments. Appl Numer Math, 2017, 115: 214-224
[10] Liang H, Liu M, Lv W. Stability of $\theta$-schemes in the numerical solution of a partial differential equation with piecewise continuous arguments. Appl Math Lett, 2010, 23: 198-206
[11] Liang H, Shi D, Lv W. Convergence and asymptotic stability of Galerkin methods for a partial differential equation with piecewise constant argument. Appl Math Comput, 2010, 217: 854-860
[12] Liu M, Spijker M N. The stability of $\theta$-methods in the numerical solution of delay differential equations. IMA J Numer Anal, 1990, 10: 31-48
[13] Liu Y. Stability analysis of $\theta$-methods for neutral functional-differential equations. Numer Math, 1995, 70: 473-485
[14] Thomée V. Galerkin Finite Element Methods for Parabolic Problems. Berlin: Springer-Verlag, 2006
[15] Veloz T, Pinto M. Existence, computability and stability for solutions of the diffusion equation with general piecewise constant argument. J Math Anal Appl, 2015, 426: 330-339
[16] Wang Q, Wen J. Analytical and numerical stability of partial differential equations with piecewise constant arguments. Numer Meth Part Differ Equ, 2014, 30: 1-16
[17] Wang Q. Stability analysis of parabolic partial differential equations with piecewise continuous arguments. Numer Meth Part Differ Equ, 2017, 33: 531-545
[18] Wang Q. Stability of numerical solution for partial differential equations with piecewise constant arguments. Adv Differ Equ, 2018, 2018: Art 71
[19] Wang W, Li S. Stability analysis of $\theta$-methods for nonlinear neutral functional differential equations. SIAM J Sci Comput, 2008, 30: 2181-2205
[20] Wiener J, Debnath L. A parabolic differential equation with unbounded piecewise constant delay. Int J Math Math Sci, 1992, 15: 339-346
[21] Wiener J, Debnath L. A wave equation with discontinuous time delay. Int J Math Math Sci, 1992, 15: 781-788
[22] Wiener J, Debnath L. Boundary value problems for the diffusion equation with piecewise continuous time delay. Int J Math Math Sci, 1997, 20: 187-195
[23] Wiener J, Heller W. Oscillatory and periodic solutions to a diffusion equation of neutral type. Int J Math Math Sci, 1999, 22: 313-348
[24] Wiener J.Generalized Solutions of Functional Differential Equations. Singapore: World Scientific, 1993
[25] Wiener J, Debnath L. A survey of partial differential equations with piecewise continuous arguments. Int J Math Math Sci, 1995, 18: 209-228
[26] Weaver H J.Theory of Discrete and Continuous Fourier Analysis. London: John Wiley & Sons, 1989
[27] Wu J.Theory and Application of Functional Differential Equation. New York: Springer, 1996
[28] Zhang C, Li C, Jiang J. Extended block boundary value methods for neutral equations with piecewise constant argument. Appl Numer Math, 2020, 150: 182-193
[29] Zhang C, Sun G. The discrete dynamics of nonlinear infinite-delay-differential equations. Appl Math Lett, 2002, 15: 521-526

ASYMPTOTICAL STABILITY OF NEUTRAL REACTION-DIFFUSION EQUATIONS WITH PCAS AND THEIR FINITE ELEMENT METHODS^∗

ASYMPTOTICAL STABILITY OF NEUTRAL REACTION-DIFFUSION EQUATIONS WITH PCAS AND THEIR FINITE ELEMENT METHODS^∗

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 6

Metrics

本文评价

推荐阅读 10

[1]	牛原玲, 张诚坚, 段金桥. A DELAY-DEPENDENT STABILITY CRITERION FOR NONLINEAR STOCHASTIC DELAY-INTEGRO-DIFFERENTIAL EQUATIONS[J]. 数学物理学报(英文版), 2011, 31(5): 1813-1822.
[2]	王绍利, 冯新龙, 何银年. GLOBAL ASYMPTOTICAL PROPERTIES FOR A DIFFUSED HBV INFECTION MODEL WITH CTL IMMUNE RESPONSE AND#br# NONLINEAR INCIDENCE[J]. 数学物理学报(英文版), 2011, 31(5): 1959-1967.
[3]	李明霞, 陈竑焘, 毛士鹏. ACCURACY ENHANCEMENT FOR THE SIGNORINI PROBLEM WITH FINITE ELEMENT METHOD[J]. 数学物理学报(英文版), 2011, 31(3): 897-908.
[4]	乔梅红, 齐欢, 陈迎春. QUALITATIVE ANALYSIS OF HEPATITIS B VIRUS INFECTION MODEL WITH IMPULSIVE VACCINATION AND TIME DELAY[J]. 数学物理学报(英文版), 2011, 31(3): 1020-1034.
[5]	孙清滢. GLOBAL CONVERGENCE RESULTS OF A THREE TERM MEMORY GRADIENT METHOD WITH A NON-MONOTONE LINE SEARCH TECHNIQU[J]. 数学物理学报(英文版), 2005, 25(1): 170-178.
[6]	陈蔚. IMPLICIT-EXPLICIT MULTISTEP FINITE ELEMENT-MIXED FINITE ELEMENT METHODS FOR THE TRANSIENT BEHAVIOR OF A SEMICONDUCTOR DEVICE[J]. 数学物理学报(英文版), 2003, 23(3): 386-.