An Efficient Approximation to Numerical Solutions for the Kawahara Equation Via Modified Cubic B-Spline Differential Quadrature Method

The main purpose of this work is to obtain the numerical solutions for the Kawahara equation via the Crank–Nicolson–Differential Quadrature Method based on modified cubic B-splines (MCBC-DQM). First, the Kawahara equation has been discretized using Crank–Nicolson scheme. Then, Rubin and Graves linearization technique has been utilized and differential quadrature method has been applied to obtain algebraic equation system. Four different test problems, namely single solitary wave, interaction of two solitary waves, interaction of three solitary waves, and wave generation, have been solved. Next, to be able to test the efficiency and accuracy of the newly applied method, the error norms L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ L _{2}$$\end{document} and L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ L _{\infty }$$\end{document} as well as the three lowest invariants I1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ I _{1}$$\end{document}, I2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ I _{2}$$\end{document}, and I3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ I _{3}$$\end{document} have been computed. Besides those, the relative changes of invariants have been reported. Finally, the newly obtained numerical results have been compared with some of those available in the literature for similar parameters. The comparison of present results with earlier works showed that the newly method may be provide significant benefit in case of numerical solutions of the other nonlinear differential equations.