Cubic B-spline finite element method for generalized reaction-diffusion equation with delay

Yıl 2023, Cilt: 41 Sayı: 2, 256 - 265, 30.04.2023

Gemeda Tolessa Lubo , Gemechis File Duressa

Öz

In this paper, a cubic B -spline finite element method is constructed based on redefined cubic B-spline basis functions for solving the generalized reaction-diffusion equations with delay. The time discretization process is based on Crank-Nicolson method. Examples are worked out to validate the theoretical convergence analysis. The numerical results given in graphs and tables demonstrate that the present method approximates the exact solution very well. The accurateness of the numerical scheme is confirmed by computing L2 and L∞ error norms.

Anahtar Kelimeler

Generalized Reaction-Diffusion Equation, Finite Element, Crank-Nicolson Method, Convergence Analysis

Kaynakça

• REFERENCES
• [1] Li D, Zhang J, Zhang Z. Unconditionally optimal error estimates of a linearized Galerkin method for nonlinear time fractional reaction sub-diffusion equations. J Sci Comput 2018;76:848-866. [CrossRef]
• [2] Li D, Zhang J. Efficient implementation to numer-ically solve the nonlinear time fractional parabolic problems on unbounded spatial domain. J Comput Phys 2016;322:415-428. [CrossRef]
• [3] Li D, Wang J. Unconditionally optimal error anal-ysis of Crank-Nicolson Galerkin FEMs for a strongly nonlinear parabolic system. J Sci Comput 2017;72:892-915. [CrossRef]
• [4] Kongson J, Amornsamankul S. A model of the sig-nal transduction process under a delay. East Asian J Appl Math 2017;7:741-751. [CrossRef]
• [5] Culshaw RV, Ruan S, Webb G. A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay. J Math Biol 2003;46:425-444. [CrossRef]
• [6] Kumar R, Sharma AK, Agnihotri K. Dynamics of an innovation diffusion model with time delay. East Asian J Appl Math 2017;7:455-481. [CrossRef
• [7] Mei M, Ou C, Zhao XQ. Global stability of mono stable traveling waves for nonlocal time delayed reaction-diffusion equations. SIAM J Math Anal 2010;42:2762-2790. [CrossRef]
• [8] Adam AM, Bashier EB, Hashim MH, Patidar KC. Fitted Galerkin spectral method to solve delay par-tial differential equations. Math Methods Appl Sci 2016;39:3102-3115. [CrossRef]
• [9] Reyes E, Rodrguez F, Martn JA. Analytic-numerical solutions of diffusion mathematical models with delays. Comput Math Appl 2008;56:743-753.[CrossRef]
• [10] Wu F, Li D, Wen J, Duan J. Stability and conver-gence of compact finite difference method for par-abolic problems with delay. Appl Math Comput 2018;322:129-139. [CrossRef]
• [11] Zhang Q, Chen M, Xu Y, Xu D. Compact -method for the generalized delay diffusion equation. Appl Math Comput 2018;316:357-369. [CrossRef]
• [12] Martn JA, Rodrguez F, Company R. Analytic solu-tion of mixed problems for the generalized dif-fusion equation with delay. Math Comput Model 2004;40:361- 369. [CrossRef]
• [13] Garcia P, Castro MA, Martn JA, Sirvent A. Numerical solutions of diffusion mathematical models with delay. Math Comput Model 2009;50:860-868.[CrossRef]
• [14] Huang C, Vandewalle S .Unconditionally stable dif-ference methods for delay partial differential equa-tions. Numer Math 2012;122:579-601. [CrossRef]
• [15] Li D, Zhang C, Wen J. A note on compact finite difference method for reaction-diffusion equations with dela. Appl Math Model 2015;39:1749-1754.[CrossRef]
• [16] Li D, Zhang C. On the long time simulation of reac-tion-diffusion equations with delay. Sci World J 2014;186802. [CrossRef]
• [17] Blanco-Cocom L, Vila-Vales E. Convergence and stability analysis of the -method for delayed dif-fusion mathematical models. Appl Math Comput 2014;23:16-25. [CrossRef]
• [18] Wu F, Wang Q, Cheng X, Chen X. Linear θ -Method and Compact Method for Generalised Reaction-Diffusion Equation with Delay. Intern J Differ Eq 2018;2018:6402576.
• [19] Tian H. Asymptotic stability analysis of the linear θ -method for linear parabolic differential equa-tions with delay. J Differ Equ Appl 2009;15:473-487.[CrossRef]
• [20] Sun ZZ, Zhang ZB. A linearized compact difference scheme for a class of nonlinear delay partial differ-ential equations. Appl Math Model 2013;37:742- 752. [CrossRef]
• [21] Zhang Q, Zhang C. A compact difference scheme combined with extrapolation techniques for solving a class of neutral delay parabolic differential equa- tions. Appl Math Lett 2013;26:306-312. [CrossRef]
• [22] Li D, Zhang C, Qin H. LDG method for reac-tion-diffusion dynamical systems with time delay. Appl Math Comput 2011;217:9173-9181. [CrossRef]
• [23] Bhrawy AH, Abdelkawy MA, Mallawi F. An accu-rate Chebyshev pseudospectral scheme for multi-di-mensional parabolic problems with time delays. Bound Value Probl 2015;1:1-20. [CrossRef]
• [24] Aziz I, Amin R. Numerical solution of a class of delay differential and delay partial differential equations via Haar wavelet. Appl Math Model 2016;40:10286- 10299. [CrossRef]
• [25] Chen X, Wang L. The variational iteration method for solving a neutral functional-differential equa-tion with proportional delays. Comput Math Appl 2010;59:2696-2702. [CrossRef]
• [26] Jackiewicz Z, Zubik-Kowal B. Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. Appl Numer Math 2006;56:433-443. [CrossRef]
• [27] El-Dib YO. Periodic solution and stability behavior for nonlinear oscillator having a cubic nonlinearity time-delayed. Int Ann Sci 2018;5:12-25. [CrossRef]
• [28] Kutluay SE, Esen AL, Dag I. Numerical solutions of the Burgers equation by the least-squares quadratic B-spline finite element method. J Comput Appl Math 2004;167:21-33. [CrossRef]
• [29] Tasbozan O, Esen A. Quadratic B-spline Galerkin method for numerical solutions of fractional tele-graph equations. Bull Math Sci Appl 2017;18:23-39. [CrossRef]
• [30] Kutluay S, Ucar Y. Numerical solutions of the cou-pled Burgers equation by the Galerkin quadratic Bspline finite element method. Math Methods Appl Sci 2013;36:2403-2415. [CrossRef]
• [31] Karako SBG, Zeybek H. A cubic B-spline Galerkin approach for the numerical simulation of the GEW equation. Stat Optim Inf Comput 2016;4:30-41.[CrossRef] [32] Soliman AA. A Galerkin solution for Burgers' equa-tion using cubic B-spline finite elements. Abstr Appl Anal 2012;2012:527467. [CrossRef]
• [33] Goh J, Majid AA, Ismail AIM. Numerical method using cubic B-spline for the heat and wave equation. Comput Math Appl 2011;62:4492-4498. [CrossRef]
• [34] Aksan EN. An application of cubic B-Spline finite element method for the Burgers equation. Therm Sci 2018;22:195-202. [CrossRef]
• [35] Mirzaee F, Alipour S. Bi-cubic B-spline functions to solve linear two-dimensional weakly singular sto-chastic integral equation. Iran J Sci Technol Trans A Sci 2021;45:965-972. [CrossRef]
• [36] Mirzaee F, Alipour S. Cubic B-spline approxi-mation for linear stochastic integro-differential equation of fractional order. J Comput Appl Math 2020;366:112440. [CrossRef]
• [37] Mirzaee F, Alipour S. An efficient cubic B-spline and bi-cubic B-spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations. Math Methods Appl Sci 2020;43:384-397. [CrossRef]
• [38] Korkmaz A, Akso AM, Dag I. Quartic B-spline dif-ferential quadrature method. Int J Nonlinear Sci 2011;11:403-411.
• [39] Mirzaee F, Alipour S. Quintic B-spline colloca-tion method to solve n-dimensional stochastic Volterra integral equations. J Comput Appl Math 2021;384:113-153. [CrossRef]
• [40] Kutluay, S, Yagmurlu NM. The modified Bi-quintic B- spline base functions, an application to diffusion equation. Int J 2017;5:26-32.
• [41] Liang H. Convergence and asymptotic stability of Galerkin methods for linear parabolic equations with delays. Appl Math Comput 2015;264:160-178.[CrossRef]
• [42] Brunner H. Collocation methods for Volterra inte-gral and related functional differential equations. Annotated ed. Cambridge: Cambridge University Press; 2004. [CrossRef]