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.

中文翻译：

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分数 Burger-Fisher 方程数值方法和解析方法的新比较

在本文中，我们通过考虑时间分数非线性 Burger-Fisher 方程 (FBFE) 来研究一些解析、数值和近似解析方法。(1/G$'$)-展开法、有限差分法(FDM)和拉普拉斯微扰法(LPM)被考虑求解FBFE。首先，我们通过(1/G$'$)-展开法得到上述问题的解析解。此外，我们比较了数值方法的解决方案，并指出哪种方法更有效和准确。我们研究了FDM的截断误差、收敛性、冯诺依曼稳定性原理和线性稳定性分析。此外，我们调查了 FDM 的 $ L_{2} $ 和 $ L_\infty $ 范数误差。根据这项研究的结果，可以得出结论，有限差分法的误差水平低于拉普拉斯微扰法。尽管如此，这两种方法都完全解决了获得分数阶微分方程的有效结果。