International Journal of Nonlinear Analysis and Applications

A numerical scheme for solving nonlinear parabolic partial differential equations with piecewise constant arguments

Document Type : Research Paper

Authors

1 Department of Applied Mathematics, Bandar Anzali Branch, Islamic Azad University, Bandar Anzali , Iran

2 Department of Applied Mathematics and Computer Science, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

Abstract

‎This article deals with the nonlinear parabolic equation with piecewise continuous arguments (EPCA)‎. ‎This study‎, ‎therefore‎, ‎with the aid of the $\theta$‎ ‎-methods,‎ ‎aims at presenting a numerical solution scheme for solving such types of equations which has applications in certain ecological studies‎. ‎Moreover‎, ‎the convergence and stability of our proposed numerical method are investigated‎. ‎Finally‎, ‎to support and confirm our theoretical results‎, ‎some numerical examples are also presented‎.

Keywords

[1] D. Agirseven, Approximate Solutions of delay parabolic equations with the Dirichlet condition, Abstr. Appl. Anal., Article Number 682752, 2012 (2012), https://doi.org/10.1155/2012/682752.
[2] J. Alavi, H. Aminikhah, Numerical Study of the Inverse Problem of Generalized Burgers–Fisher and Generalized Burgers–Huxley Equations, Adv. Math. Phys.,2021(2021), https://doi.org/10.1155/2021/6652108.
[3] A. Ashyralyev, D. Agirseven, Stability of parabolic equations with unbounded operators acting on delay terms, Electron. J. Differential Equations, 2014(160) (2014) 1–13.
[4] A. Ashyralyev, D. Agirseven, On source identification problem for a delay parabolic equation, Nonlinear Anal. Model. Control, 19 (3) (2014) 335–349.
[5] A. Ashyralyev, D. Agirseven, Stability of delay parabolic difference equations, Filomat 28(5)(2014) 995–1006.
[6] A. Ashyralyev, D. Agirseven, Well-posedness of delay parabolic equations with unbounded operators acting on delay terms, Bound. Value Probl., 2014(126) (2014), https://doi.org/10.1186/1687-2770-2014-126.
[7] A. Ashyralyev, D. Agirseven, Well-posedness of delay parabolic difference equations, Adv. Difference Equ., 2014(18) (2014), https://doi.org/10.1186/1687-1847-2014-18.
[8] A. Ashyralyev, D. Agirseven, On convergence of difference schemes for delay parabolic equations, Comput. Math. Appl., 66 (7) (2013), 1232–1244, https://doi.org/10.1016/j.camwa.2013.07.018.
[9] A. Ashyralyev, P. E. Sobolevskii, On the stability of the linear delay differential and difference equations, Abstr. Appl. Anal., 6(5) (2001), 267–297.
[10] A. Ashyralyev, D. Agirseven, Bounded solutions of nonlinear hyperbolic equations with time delay, Electron. J. Differential Equations, 2018(21) (2018) 1–15.
[11] H. Bereketoglu, M. Lafci Behavior of the solutions of a partial differential equation with a piecewise constant argument, Filomat, 31(19) (2017) 5931–5943.
[12] M. L. B¨uy¨ukkahraman, H. Bereketoglu, On a partial differential equation with piecewise constant mixed arguments, Iran J. Sci. Technol. Trans. Sci., 44 (2020), 1791–1801.
[13] R. H. Byrds, R. B. Schnabel, G. A. Shultz, A trust region algorithm for nonlinearly constrained optimization, SIAM J. Numer. Anal., 24(5) (1987) 1152–1170.
[14] M. Esmaeilzadeh, H. S. Najafi, H. Aminikhah, A numerical scheme for diffusion-convection equation with piecewise constant arguments, Comput. Methods Differ. Equ., 8(3) (2020) 573–584.
[15] M. Esmaeilzadeh, H. S. Najafi, H. Aminikhah, A numerical method for solving hyperbolic partial differential equations with piecewise constant arguments and variable coefficients, J. Difference Equ. Appl., 27(2) (2021) 172–194.
[16] N. A. Kudryashov, A.S. Zakharchenko, A note on solutions of the generalized Fisher equation, Appl. Math. Lett., 32 (2014) 53–56.
[17] H. Poorkarimi, J. Wiener, Bounded solutions of nonlinear parabolic equations with time delay, 15th Annu. Conf. Appl. Math., Univ. of Central Oklahoma, Electron. J. Differential Equations, Conference 02 (1999) 87–91.
[18] H. Poorkarimi, J. Wiener, Bounded solutions of non-linear hyperbolic equations with delay, Proceedings of the VII International Conference on Non-Linear Analysis, V. Lakshmikan-tham, Ed. 1 (1986) 471–478.
[19] S. M. Shah, H. Poorkarimi, J. Wiener, Bounded solutions of retarded nonlinear hyperbolic equations, Bull. Allahabad Math. Soc., 1 (1986) 1–14.
[20] H. Poorkarimi, J. Wiener, Almost periodic solutions of nonlinear hyperbolic equations with time delay, Conf. 07 (2001), 99–102.
[21] S. M. Shah, J. Wiener, Advanced differential equations with piecewise constant argument deviations, Internat. J. Math. Sci. 6 (1983), 671–703.
[22] M. H. Song, Z. W. Yang, M. Z. Liu, Stability of θ-methods for advanced differential equations with piecewise continuous arguments, Comput. Math. Appl. 49 (2005) 1295–1301.
[23] T. Veloz, M. Pinto, Existence, Computability and stability for solutions of the diffusion equation with general piecewise constant argument, J. Math. Anal. Appl. 426 (2015) 330–339.
[24] Q. Wang, Stability of numerical solution for partial differential equations with piecewise constant arguments, Adv. Diff. Eq. (2018), doi.org/10.1186/s13662-018-1514-1.
[25] Q. Wang, J. Wen, Analytical and numerical stability of partial differential equations with piecewise constant arguments, Num. Meth. Partial Diff. Eq., 30 (2014) 1–16.
[26] Q. Wang, Stability analysis of parabolic partial differential equations with piecewise continuous arguments, Inc. Numer Meth. Partial Differential Eq., 33 (2017) 531–545.
[27] J. Wiener, Generalized Solutions of Functional Differential Equations, World Scientific, Singapore, 1993.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 1
March 2022
Pages 783-789
  • Receive Date: 25 June 2021
  • Revise Date: 06 September 2021
  • Accept Date: 23 September 2021