An Improvised Cubic B-spline Collocation of Fourth Order and Crank-Nicolson Technique for Numerical Soliton of Klein-Gordon and Sine-Gordon Equations

This article examines the nonlinear hyperbolic Klein-Gordon equation (KGE) and sine-Gordon equation (SGE) with Crank-Nicolson and the finite element method (FEM) based on an improvised quartic order cubic B-spline collocation approach and explores their novel numerical solutions along with computational complexity. This work explains the parameters such as Euclidean 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}, maximum absolute error L infinity\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}, and root-mean-square (RMS) error with computational time cost, experimental order of convergence (EOC), and three conservative laws I1,I2,I3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I}_{1},{I}_{2},{I}_{3}$$\end{document} of mass momentum and energy conservation, respectively. It is demonstrated that the method is unconditionally stable with the von-Neumann process and accurate to convergence of order O (h4+Delta t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h}<^>{4}+\Delta t)$$\end{document}. Finally, four test examples are investigated to support our assertion, and the experimental findings are compared to existing approaches using software tools like MATLAB and MATHEMATICA. 2D and 3D graphical representations of solutions are also presented and compared with the exact solution and results of others.