# American Institute of Mathematical Sciences

July  2021, 14(7): 2591-2606. doi: 10.3934/dcdss.2020258

## Novel comparison of numerical and analytical methods for fractional Burger–Fisher equation

 1 Faculty of Science, Department of Actuary, Firat University, Elazig 23200, Turkey 2 Department of Mathematics and Computer Sciences, Necmettin Erbakan University, Konya 42090, Turkey

* Corresponding author: mehmetyavuz@erbakan.edu.tr

Received  June 2019 Revised  September 2020 Published  July 2021 Early access  May 2021

In this paper, we investigate some analytical, numerical and approximate analytical methods by considering time-fractional nonlinear Burger–Fisher equation (FBFE). (1/G$'$)-expansion method, finite difference method (FDM) and Laplace perturbation method (LPM) are considered to solve the FBFE. Firstly, we obtain the analytical solution of the mentioned problem via (1/G$'$)-expansion method. Also, we compare the numerical method solutions and point out which method is more effective and accurate. We study truncation error, convergence, Von Neumann's stability principle and analysis of linear stability of the FDM. Moreover, we investigate the $L_{2}$ and $L_\infty$ norm errors for the FDM. According to the results of this study, it can be concluded that the finite difference method has a lower error level than the Laplace perturbation method. Nonetheless, both of these methods are totally settlement in obtaining efficient results of fractional order differential equations.

Citation: Asif Yokus, Mehmet Yavuz. Novel comparison of numerical and analytical methods for fractional Burger–Fisher equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2591-2606. doi: 10.3934/dcdss.2020258
##### References:
 [1] K. A. Abro and J. Gómez-Aguilar, A comparison of heat and mass transfer on a waltersb fluid via Caputo-Fabrizio versus Atangana-Baleanu fractional derivatives using the fox-H function, The European Physical Journal Plus, 134 (2019), 101. [2] A. Allwright and A. Atangana, Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systems, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 443-466.  doi: 10.3934/dcdss.2020025. [3] J. Alzabut, T. Abdeljawad, F. Jarad and W. Sudsutad, A Gronwall inequality via the generalized proportional fractional derivative with applications, J. Inequal. Appl., 2019 (2019), Paper No. 101, 12 pp. doi: 10.1186/s13660-019-2052-4. [4] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular Kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A. [5] E. Balcı, İ. Öztürk and S. Kartal, Dynamical behaviour of fractional order tumor model with Caputo and conformable fractional derivative, Chaos Solitons Fractals, 123 (2019), 43-51.  doi: 10.1016/j.chaos.2019.03.032. [6] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, 2012. doi: 10.1142/9789814355216. [7] E. Bas, B. Acay and R. Ozarslan, Fractional models with singular and non-singular kernels for energy efficient buildings, Chaos, 29 (2019), 023110, 7 pp. doi: 10.1063/1.5082390. [8] E. Bonyah, A. Atangana and M. A. Khan, Modeling the spread of computer virus via Caputo fractional derivative and the beta-derivative, Asia Pacific Journal on Computational Engineering, 4 (2017), 1. doi: 10.1186/s40540-016-0019-1. [9] A. G. Bratsos and A. Q. M. Khaliq, An exponential time differencing method of lines for Burgers–Fisher and coupled Burgers equations, J. Comput. Appl. Math., 356 (2019), 182-197.  doi: 10.1016/j.cam.2019.01.028. [10] H. 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Comput., 152 (2004), 403-413.  doi: 10.1016/S0096-3003(03)00565-4. [26] D. Kaya, S. Gülbahar, A. Yokuş and M. Gülbahar, Solutions of the fractional combined KdV-mKdV equation with collocation method using radial basis function and their geometrical obstructions, Adv. Difference Equ., 2018 (2018), Paper No. 77, 16 pp. doi: 10.1186/s13662-018-1531-0. [27] D. Kaya, A. Yokus and U. Demiroglu, Comparison of exact and numerical solutions for the Sharma-Tasso-Olver equation, In Numerical Solutions of Realistic Nonlinear Phenomena, Springer, Cham, (2020), 53–65. [28] A. Keten, M. Yavuz and D. Baleanu, Nonlocal cauchy problem via a fractional operator involving power kernel in banach spaces, Fractal Fract., 3 (2019), 27. doi: 10.3390/fractalfract3020027. [29] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002. [30] D. Kumar, J. Singh, H. M. Baskonus and H. 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##### References:
 [1] K. A. Abro and J. Gómez-Aguilar, A comparison of heat and mass transfer on a waltersb fluid via Caputo-Fabrizio versus Atangana-Baleanu fractional derivatives using the fox-H function, The European Physical Journal Plus, 134 (2019), 101. [2] A. Allwright and A. Atangana, Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systems, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 443-466.  doi: 10.3934/dcdss.2020025. [3] J. Alzabut, T. Abdeljawad, F. Jarad and W. Sudsutad, A Gronwall inequality via the generalized proportional fractional derivative with applications, J. Inequal. Appl., 2019 (2019), Paper No. 101, 12 pp. doi: 10.1186/s13660-019-2052-4. [4] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular Kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A. [5] E. Balcı, İ. Öztürk and S. Kartal, Dynamical behaviour of fractional order tumor model with Caputo and conformable fractional derivative, Chaos Solitons Fractals, 123 (2019), 43-51.  doi: 10.1016/j.chaos.2019.03.032. [6] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, 2012. doi: 10.1142/9789814355216. [7] E. Bas, B. Acay and R. Ozarslan, Fractional models with singular and non-singular kernels for energy efficient buildings, Chaos, 29 (2019), 023110, 7 pp. doi: 10.1063/1.5082390. [8] E. Bonyah, A. Atangana and M. A. Khan, Modeling the spread of computer virus via Caputo fractional derivative and the beta-derivative, Asia Pacific Journal on Computational Engineering, 4 (2017), 1. doi: 10.1186/s40540-016-0019-1. [9] A. G. Bratsos and A. Q. M. Khaliq, An exponential time differencing method of lines for Burgers–Fisher and coupled Burgers equations, J. Comput. Appl. Math., 356 (2019), 182-197.  doi: 10.1016/j.cam.2019.01.028. [10] H. Bulut, D. Kumar, J. Singh, R. Swroop and H. M. Baskonus, Analytic study for a fractional model of HIV infection of CD4+ T lymphocyte cells, Math. Nat. Sci., 2 (2018), 33-43. [11] M. Caputo, Linear models of dissipation whose Q is almost frequency independentII, Geophysical Journal International, 13 (1967), 529-539. [12] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2015), 1-13. [13] A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer, 2014. [14] V. Chandraker, A. Awasthi and S. Jayaraj, Numerical treatment of Burger-Fisher equation, Procedia Technology, 25 (2016), 1217-1225. [15] W. Chen, L. Ye and H. Sun, Fractional diffusion equations by the Kansa method, Comput. Math. Appl., 59 (2010), 1614-1620.  doi: 10.1016/j.camwa.2009.08.004. [16] F. Evirgen and M. Yavuz, An alternative approach for nonlinear optimization problem with Caputo-Fabrizio derivative, ITM Web of Conferences: EDP Sciences, 22, (2018), 01009. doi: 10.1051/itmconf/20182201009. [17] A. Ghorbani, Beyond Adomian polynomials: He polynomials, Chaos Solitons Fractals, 39 (2009), 1486-1492.  doi: 10.1016/j.chaos.2007.06.034. [18] Z. Hammouch and T. Mekkaoui, Traveling-wave solutions of the generalized Zakharov equation with time-space fractional derivatives, Journal MESA, 5 (2014), 489-498. [19] Z. Hammouch and T. Mekkaoui, Approximate analytical and numerical solutions to fractional KPP-like equations, Gen. Math. Notes, 14 (2013), 1-9. [20] Z. Hammouch, T. Mekkaoui and F. B. Belgacem, Numerical simulations for a variable order fractional Schnakenberg model, AIP Conference Proceedings, 1637 (2014), 1450-1455.  doi: 10.1063/1.4907312. [21] J. Hristov, Space-fractional diffusion with a potential power-law coefficient: Transient approximate solution, Progress in Fractional Differentiation and Applications, 3 (2017), 19-39. [22] H. N. A. Ismail and A. A. A. Rabboh, A restrictive padé approximation for the solution of the generalized Fisher and Burger-Fisher equations, Appl. Math. Comput., 154 (2004), 203-210.  doi: 10.1016/S0096-3003(03)00703-3. [23] F. Jarad and T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 709-722.  doi: 10.3934/dcdss.2020039. [24] F. Jarad, T. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos Solitons Fractals, 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006. [25] D. Kaya and S. El-Sayed, A numerical simulation and explicit solutions of the generalized Burgers–Fisher equation, Appl. Math. Comput., 152 (2004), 403-413.  doi: 10.1016/S0096-3003(03)00565-4. [26] D. Kaya, S. Gülbahar, A. Yokuş and M. Gülbahar, Solutions of the fractional combined KdV-mKdV equation with collocation method using radial basis function and their geometrical obstructions, Adv. Difference Equ., 2018 (2018), Paper No. 77, 16 pp. doi: 10.1186/s13662-018-1531-0. [27] D. Kaya, A. Yokus and U. Demiroglu, Comparison of exact and numerical solutions for the Sharma-Tasso-Olver equation, In Numerical Solutions of Realistic Nonlinear Phenomena, Springer, Cham, (2020), 53–65. [28] A. Keten, M. Yavuz and D. Baleanu, Nonlocal cauchy problem via a fractional operator involving power kernel in banach spaces, Fractal Fract., 3 (2019), 27. doi: 10.3390/fractalfract3020027. [29] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002. [30] D. Kumar, J. Singh, H. M. Baskonus and H. Bulut, An effective computational approach for solving local fractional telegraph equations, Nonlinear Sci. Lett. A: Math. Phys. Mech, 8 (2017), 200-206. [31] V. F. Morales-Delgado, J. F. Gómez-Aguilar, H. Yépez-Martínez, D. Baleanu, R. F. Escobar-Jimenez and V. H. Olivares-Peregrino, Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular, Adv. Difference Equ., 2016 (2016), Paper No. 164, 17 pp. doi: 10.1186/s13662-016-0891-6. [32] P. A. Naik, M. Yavuz, S. Qureshi, J. Zu and S. Townley, Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan, The European Physical Journal Plus, 135 (2020), 1-42. [33] I. Podlubny, Fractional Differential Equation: An Introduction to Fractional Derivatives, Fractional Differential Equations, To Methods of their Solution and some of their Applications, Mathematics in Science and Engineering, 198. 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Traveling wave solution ${u_1}(x,t)$ of Eq. (1) by substituting the values $\mu = 1,\;\lambda = - 0.5,\;{c_1} = 1,\;\alpha = 0.8,\;\delta = - 2,$ in Eq. (41)
Traveling wave solution ${u_2}(x,t)$ of Eq. (1) by substituting the values $\mu = 1,\;\lambda = - 0.5,\;{c_1} = 1,\;\alpha = 0.8,\;\delta = - 2,$ in Eq. (42)
Traveling wave solution ${u_3}(x,t)$ of Eq. (1) by substituting the values $\mu = 1,\;\lambda = - 0.5,\;{c_1} = 1,\;\alpha = 0.8,\;v = 1,$ in Eq. (43)
2D Numerical and exact travelling wave solution and absolute error of the Eq. (1) for Finite Difference and Laplace Perturbation Methods
Exact solution, numerical results and absolute error of FDM and LPM for Eq. $(1)$ at $\Delta x = \Delta t = 0.01$
 FDM LPM FDM LPM $x_{i}$ $t_{j}$ Numerical Numerical Exact Errors Errors $0.00$ $0.01$ -0.182279 -0.182996 -0.182350 7.09713$\times$10$^{-5}$ 6.45583$\times$10$^{-4}$ $0.01$ $0.01$ -0.182296 -0.183012 -0.182367 7.09118$\times$10$^{-5}$ 6.45497$\times$10$^{-4}$ $0.02$ $0.01$ -0.182312 -0.183027 -0.182383 7.09118$\times$10$^{-5}$ 6.44425$\times$10$^{-4}$ $0.03$ $0.01$ -0.182328 -0.183043 -0.182399 7.07930$\times$10$^{-5}$ 6.44366$\times$10$^{-4}$ $0.04$ $0.01$ -0.182344 -0.183058 -0.182415 7.07337$\times$10$^{-5}$ 6.4332$\times$10$^{-4}$ $0.05$ $0.01$ -0.182360 -0.183074 -0.182431 7.06744$\times$10$^{-5}$ 6.4329$\times$10$^{-4}$
 FDM LPM FDM LPM $x_{i}$ $t_{j}$ Numerical Numerical Exact Errors Errors $0.00$ $0.01$ -0.182279 -0.182996 -0.182350 7.09713$\times$10$^{-5}$ 6.45583$\times$10$^{-4}$ $0.01$ $0.01$ -0.182296 -0.183012 -0.182367 7.09118$\times$10$^{-5}$ 6.45497$\times$10$^{-4}$ $0.02$ $0.01$ -0.182312 -0.183027 -0.182383 7.09118$\times$10$^{-5}$ 6.44425$\times$10$^{-4}$ $0.03$ $0.01$ -0.182328 -0.183043 -0.182399 7.07930$\times$10$^{-5}$ 6.44366$\times$10$^{-4}$ $0.04$ $0.01$ -0.182344 -0.183058 -0.182415 7.07337$\times$10$^{-5}$ 6.4332$\times$10$^{-4}$ $0.05$ $0.01$ -0.182360 -0.183074 -0.182431 7.06744$\times$10$^{-5}$ 6.4329$\times$10$^{-4}$
$L_2$ and $L_\infty$ error norm when $0\leq \Delta x = \Delta t\leq 1$
 $\Delta x=\Delta t$ $L_2$ $L_\infty$ $0.1$ 0.000444585 0.000216239 $0.05$ 0.000343115 0.000190017 $0.02$ 0.000148088 0.000114257 $0.01$ 0.000067741 0.000070912 $0.002$ 9.27231$\times$10$^{-6}$ 0.0000210066 $0.001$ 3.82314$\times$10$^{-6}$ 0.0000121997
 $\Delta x=\Delta t$ $L_2$ $L_\infty$ $0.1$ 0.000444585 0.000216239 $0.05$ 0.000343115 0.000190017 $0.02$ 0.000148088 0.000114257 $0.01$ 0.000067741 0.000070912 $0.002$ 9.27231$\times$10$^{-6}$ 0.0000210066 $0.001$ 3.82314$\times$10$^{-6}$ 0.0000121997
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