Revisited Fisher’s equation and logistic system model: a new fractional approach and some modifications

In the present paper, a new technique Laplace residual power series (L-RPS) method is used for presenting approximate solutions (ASs) of the nonlinear fractional Fisher partial differential equation (NLFF-PDE), and then, we make a comparison between our results and that obtained via residual power series (RPS) method in the literature [1]. Moreover, we show that the 1st and 2nd coefficients of RPS-ASs obtained in [1] are identical to Eq. (2.12) and (2.16), respectively. In addition, we have corrected and improved the following error results seen in [1]: Firstly, we find that the 3rd, 4th and 5th coefficients of RPS-ASs are not agreement with Eqs. (2.20)–(2.22) that obtained in [1, Page 85]. Secondly, the authors in [1] misused used Caputo fractional derivative (C-FD) D2α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^{2\alpha }$$\end{document} in Eq. (2.18) on page 84 which leads the terms in Eqs. (2.19) and f3(x),f4(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_3 (x), \ f_4 (x)$$\end{document} and f5(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_5 (x)$$\end{document} in Eqs. (2.20)-(2.22) are incorrect. Finally, we present the numerical and graphical solutions of fractional Fisher logistic model (FFLM) based on our new approach. To check the robustness, accuracy and efficiency of our proposed technique, we compute the absolute error for the ASs of L-RPS, sinc-collocation-finite difference (SCFD) and RPS methods.