An Improvised Cubic B-Spline-Based Collocation Approach for Solving Nonlinear K-G Equations

This manuscript explores the complex Klein-Gordon (K-G) equation that is a fundamental relativistic wave equation in nature. This equation is integral to scalar field interactions within various physics disciplines, including quantum field theory, relativistic quantum mechanics, and plasma as well as condensed matter physics. The integration of an advanced nonlinear term significantly complicates the numerical simulation of the K-G equation’s solutions. This paper introduces a novel approach for approximating solutions to the K-G equation through a col-location technique, leveraging the computational power of third-degree B-splines. For temporal discretization, the Crank-Nicolson discretization is employed, while the Rubin Graves method is utilized to manage the equation’s nonlinearity. The manuscript outlines a structured procedure for computing and categorizing the Klein-Gordon equation’s approximate solutions. It encompasses the examination of notable specific instances, such as solitary and periodic waves, and delves into the analysis of certain characteristics that could be instrumental in future explorations of the nonlinear dynamics related to this partial differential equation. By benchmarking our results against known exact solutions or those derived from other numerical strategies, we have substantiated the proposed method’s precision and effectiveness across a variety of test scenarios.