# American Institute of Mathematical Sciences

## 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 Faculty of Science, Department of Mathematics-Computer, Necmettin Erbakan University, Konya, 42090, Turkey

* Corresponding author: mehmetyavuz@erbakan.edu.tr

Received  June 2019 Revised  July 2019 Published  January 2020

In this paper, we have investigated some analytical, numerical and approximate-analytical methods by considering the time-fractional nonlinear fractional Burger–Fisher equation (FBFE). (1/G$'$)-expansion method, finite difference method and Laplace perturbation method have been considered to solve the FBFE. Firstly, we have obtained the analytical solution of the mentioned problem via (1/G$'$)-expansion method. Also, we have compared the numerical method solutions and have obtained that which method is more effective and accurate. 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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020258
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##### References:
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. (37)
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. (38)
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. (39)
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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