Numerical Solutions of Time fractional Klein Gordon Equation using Crank-Nicolson Finite Difference Method

Finite difference methods are widely used numerical techniques used to solve partial differential equations observed in many fields, such as science and engineering. This research presents a study on the numerical solutions of the Klein-Gordon equation, which describes anomalous diffusion and wave propagation in quantum fields and possesses a fractional derivative in the Caputo sense. The content of the paper begins by discretizing the region of the problem while taking into account the fundamental characteristics of finite difference methods. Subsequently, the time derivative algorithm, and the other terms, are discretized using the Crank-Nicolson finite difference approach, resulting in a system of algebraic equations. Solving this algebraic equation system yields numerical solutions. The numerical results are calculated for various values of the parameters associated with the equation and fractional order derivatives , leading to the computation of error norms. Graphical findings illustrate the physical behavior of approximation solutions for a variety of fraction order values. Additionally, the stability analysis of the numerical scheme is investigated using von-Neumann stability analysis. The results of this paper will help other researchers studying in the field to apply the presented method to other problems modelling the natural phenomena.