Convergence and stability of Galerkin finite element method for hyperbolic partial differential equation with piecewise continuous arguments of advanced type
Abstract
This paper deals with the convergence and stability of Galerkin finite element method for a hyperbolic partial differential equations with piecewise continuous arguments of advanced type. First of all, we obtain the expression of analytic solution by the method of separation variable, then the sufficient conditions for stability are obtained. Semidiscrete and fully discrete schemes are derived by Galerkin finite element method, and their convergence are both analyzed in L2-norm. Moreover, the stability of the two schemes are investigated. The semidiscrete scheme can achieve unconditionally stability. The sufficient conditions of stability for fully discrete scheme are derived under which the analytic solution is asymptotically stable. Finally, some numerical experiments are presented to illustrate the theoretical results.
How to Cite
Chen, Y., & Wang, Q. (2023). Convergence and stability of Galerkin finite element method for hyperbolic partial differential equation with piecewise continuous arguments of advanced type. Mathematical Modelling and Analysis, 28(3), 434–458. https://doi.org/10.3846/mma.2023.16677
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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Q. Wang and X.M. Wang. Runge-Kutta methods for systems of differential equation with piecewise continuous arguments: convergence and stability. Numer. Func. Anal. Opt., 39(7):784–799, 2018. https://doi.org/10.1080/01630563.2017.1421554
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J. Wiener. Generalized Solutions of Functional Differential Equations. World Scientific, Singapore, 1993.
J. Wiener and L. Debnath. A wave equation with discontinuous time delay. Int. J. Math. Math. Sci., 15(4):781–788, 1992. https://doi.org/10.1155/S0161171292001017
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H.Z. Yang, M.H. Song and M.Z. Liu. Strong convergence and exponential stability of stochastic differential equations with piecewise continuous arguments for non-globally Lipschitz continuous coefficients. Appl. Math. Comput., 341:111–127, 2019. https://doi.org/10.1016/j.amc.2018.08.037
C.J. Zhang, B.C. Liu W.S. Wang and T.T. Qin. A multi-domain Legendre spectral collocation method for nonlinear neutral equations with piecewise continuous argument. Int. J. Comput. Math., 95(12):2419–2432, 2018. https://doi.org/10.1080/00207160.2017.1398321
C.J. Zhang and X.Q. Yan. Convergence and stability of extended BBVMs for nonlinear delay-differential-algebraic equations with piecewise continuous arguments. Numer. Algor., 87:921–937, 2021. https://doi.org/10.1007/s11075-020-00993-8
G.L. Zhang. Oscillation of Runge-Kutta methods for advanced impulsive differential equations with piecewise constant arguments. Adv. Differ. Equ., 2017(1):13–31, 2017. https://doi.org/10.1186/s13662-016-1067-0
F. Cavalli and A. Naimzada. A multiscale time model with piecewise constant argument for a boundedly rational monopolist. J. Differ. Equ. Appl., 22(10):1480– 1489, 2016. https://doi.org/10.1080/10236198.2016.1202940
C.J. Chen, X.Y. Zhang, G.D. Zhang and Y.Y. Zhang. A twogrid finite element method for nonlinear parabolic integro-differential equations. Int. J. Comput. Math., 96(10):2010–2023, 2019. https://doi.org/10.1080/00207160.2018.1548699
K.S. Chiu and T.X. Li. Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments. Math. Nachr., 292(10):2153–2164, 2019. https://doi.org/10.1002/mana.201800053
K.L. Cooke and J. Wiener. Retarded differential equations with piecewise constant delays. J. Math. Anal. Appl., 99(1):265–297, 1984. https://doi.org/10.1016/0022-247X(84)90248-8
Z.H. Feng, Y. Wang and X. Ma. Asymptotically almost periodic solutions for certain differential equations with piecewise constant arguments. Adv. Differ. Equ., 2020(1):1–22, 2020. https://doi.org/10.1186/s13662-020-02699-6
S. Ganesan and S. Lingeshwaran. Galerkin finite element method for cancer invasion mathematical model. Comput. Math. Appl., 73(12):2603–2617, 2017. https://doi.org/10.1016/j.camwa.2017.04.006
J.F. Gao. Numerical oscillation and non-oscillation for differential equation with piecewise continuous arguments of mixed type. Appl. Math. Comput., 299:16–27, 2017. https://doi.org/10.1016/j.amc.2016.11.031
J.W. Hu and H.M. Tang. Numerical Method of Differential Equations. Science Press, 2011. (in Chinese)
Y. Jang and S. Shaw. A priori error analysis for a finite element approximation of dynamic viscoelasticity problems involving a fractional order integro-differential constitutive law. Adv. Comput. Math., 47(3):1–30, 2021. https://doi.org/10.1007/s10444-021-09857-8
F. Karakoc. Asymptotic behaviour of a population model with piecewise constant argument. Appl. Math. Lett., 70:7–16, 2017. https://doi.org/10.1016/j.aml.2017.02.014
S. Kartal and F. Gurcan. Stability and bifurcations analysis of a competition model with piecewise constant arguments. Math. Meth. Appl. Sci., 38(9):1855– 1866, 2015. https://doi.org/10.1002/mma.3196
M. Li, C.M. Huang and P.D. Wang. Galerkin finite element method for nonlinear fractional Schrödinger equations. Numer. Algor., 74(2):499–525, 2017. https://doi.org/10.1007/s11075-016-0160-5
H. Liang, M.Z. Liu and W.J. Lv. Stability of θ-schemes in the numerical solution of a partial differential equation with piecewise continuous arguments. Appl. Math. Lett., 23(2):198–206, 2010. https://doi.org/10.1016/j.aml.2009.09.012
H. Liang, D.Y. Shi and W.J. Lv. Convergence and asymptotic stability of Galerkin methods for a partial differential equation with piecewise constant argument. Appl. Math. Comput., 217(2):854–860, 2010. https://doi.org/10.1016/j.amc.2010.06.028
X. Liu and Y.M. Zeng. Linear multistep methods for impulsive delay differential equations. Appl. Math. Comput., 321:555–563, 2017. https://doi.org/10.1016/j.amc.2017.11.014
Y. Liu, Y.W. Du, H. Li, S. He and W. Gao. Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction-diffusion problem. Comput. Math. Appl., 70(4):573–591, 2015. https://doi.org/10.1016/j.camwa.2015.05.015
M. Milošević. The Euler-Maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments. J. Comput. Appl. Math., 298:1–12, 2016. https://doi.org/10.1016/j.cam.2015.11.019
V. Niño-Celis, D.A. Rueda-Gómez and É.J. Villamizar-Roa. Convergence and positivity of finite element methods for a haptotaxis model of tumoral invasion. Comput. Math. Appl., 89:20–33, 2021. https://doi.org/10.1016/j.camwa.2021.02.007
S.M. Shah and J. Wiener. Advanced differential equations with piecewise constant argument deviations. Int. J. Math. Math. Sci., 6(4):671–703, 1983. https://doi.org/10.1155/S0161171283000599
V. Thomée. Galerkin Finite Element Methods for Parabolic Problems. SpringerVerlag, New York, 1986.
Q. Wang. Stability analysis of parabolic partial differential equations with piecewise continuous arguments. Numer. Meth. Part. D. E., 33(2):531–545, 2017. https://doi.org/10.1002/num.22113
Q. Wang. Stability of numerical solution for partial differential equations with piecewise constant arguments. Adv. Differ. Equ., 2018(1):1–13, 2018. https://doi.org/10.1186/s13662-018-1514-1
Q. Wang and X.M. Wang. Runge-Kutta methods for systems of differential equation with piecewise continuous arguments: convergence and stability. Numer. Func. Anal. Opt., 39(7):784–799, 2018. https://doi.org/10.1080/01630563.2017.1421554
Q. Wang and X.M. Wang. Stability of θ-schemes for partial differential equations with piecewise constant arguments of alternately retarded and advanced type. Int. J. Comput. Math., 96(12):2352–2370, 2019. https://doi.org/10.1080/00207160.2018.1562059
Q. Wang, Q.Y. Zhu and M.Z. Liu. Stability and oscillations of numerical solutions for differential equations with piecewise continuous arguments of alternately advanced and retarded type. J. Comput. Appl. Math., 235(5):1542–1552, 2011. https://doi.org/10.1016/j.cam.2010.08.041
A. Westerkamp and M. Torrilhon. Finite element methods for the linear regularized 13-moment equations describing slow rarefied gas flows. J. Comput. Phys., 389:1–21, 2019. https://doi.org/10.1016/j.jcp.2019.03.022
J. Wiener. Generalized Solutions of Functional Differential Equations. World Scientific, Singapore, 1993.
J. Wiener and L. Debnath. A wave equation with discontinuous time delay. Int. J. Math. Math. Sci., 15(4):781–788, 1992. https://doi.org/10.1155/S0161171292001017
J. Wiener and L. Debnath. Boundary value problems for the diffusion equation with piecewise continuous time delay. Int. J. Math. Math. Sci., 20(1):187–195, 1997. https://doi.org/10.1155/S0161171297000239
J. Wiener and W. Heller. Oscillatory and periodic solutions to a diffusion equation of neutral type. Int. J. Math. Math. Sci., 22(2):313–348, 1999. https://doi.org/10.1155/S0161171299223137
H.Z. Yang, M.H. Song and M.Z. Liu. Strong convergence and exponential stability of stochastic differential equations with piecewise continuous arguments for non-globally Lipschitz continuous coefficients. Appl. Math. Comput., 341:111–127, 2019. https://doi.org/10.1016/j.amc.2018.08.037
C.J. Zhang, B.C. Liu W.S. Wang and T.T. Qin. A multi-domain Legendre spectral collocation method for nonlinear neutral equations with piecewise continuous argument. Int. J. Comput. Math., 95(12):2419–2432, 2018. https://doi.org/10.1080/00207160.2017.1398321
C.J. Zhang and X.Q. Yan. Convergence and stability of extended BBVMs for nonlinear delay-differential-algebraic equations with piecewise continuous arguments. Numer. Algor., 87:921–937, 2021. https://doi.org/10.1007/s11075-020-00993-8
G.L. Zhang. Oscillation of Runge-Kutta methods for advanced impulsive differential equations with piecewise constant arguments. Adv. Differ. Equ., 2017(1):13–31, 2017. https://doi.org/10.1186/s13662-016-1067-0