Radial Basis Functions Collocation Method for Numerical Solution of Coupled Burgers’ and Korteweg-de Vries Equations of Fractional Order

The fractional nonlinear coupled viscous Burgers and Korteweg-de Vries (KdV) evolutionary equations model various interesting phenomena in engineering and applied sciences. Therefore, their accurate numerical modeling and solution behavior are very important. In this article, radial basis functions (RBFs) approach is proposed and analyzed for the numerical solutions of time-fractional coupled Burgers’ and KdV equations. RBFs together with collocation method are employed in space approximation. A simple quadrature formula combined with finite difference of O(Δt2-α),(0<α≤1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {O}{(\Delta t^{2-\alpha })},\,(0<\alpha \le 1)$$\end{document} is used for temporal discretization. For the proposed method, eigenvalue stability analysis is carried out theoretically and confirmed via numerical examples for RBFs shape parameter β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}. The proposed method is meshfree thus reduces the computational cost of mesh generation. Various test problems are considered for the method validation. Simulated results show good agreement with exact solutions and earlier works presented in graphical and tabulated forms. Accuracy and efficiency of the proposed method are assessed using discrete e2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${e}_{2}$$\end{document}, e∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${e}_{\infty }$$\end{document} and erms\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${e}_{\text {rms}}$$\end{document} error norms.