Article Contents
Article Contents

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

• 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.

Mathematics Subject Classification: Primary: 26A33, 35R11, 65M06; Secondary: 65H20.

 Citation:

• Figure 1.  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)

Figure 2.  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)

Figure 3.  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)

Figure 4.  2D Numerical and exact travelling wave solution and absolute error of the Eq. (1) for Finite Difference and Laplace Perturbation Methods

Table 1.  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}$

Table 2.  $L_2$ 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
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