ABSTRACT
In this chapter authors have considered a special class of reaction-diffusion equations viz. the Klein-Gordon equation. The key work of this chapter is to propose a tool to find the numerical solution of fractional nonlinear Klein-Gordon equation (FNKGE) with some given boundary and initial conditions. In this numerical technique the operational matrix based on orthogonal Laguerre polynomials is applied to fractional nonlinear Klein-Gordon equations and converted this equation in to an algebraic systems which further simplified by Newton method and gives the desired numerical solution of our considered problem. Here the fractional order derivative is taken in accordance of Caputo sense. Several numerical examples are examined to show the efficiency and validity of our proposed numerical techniques. In addition, the stability and convergence analysis of the proposed numerical algorithm is given to show the effectiveness of the scheme. Numerical discussions and graphical presentations ensure that the introduced method is easy to apply and reliable for fractional order systems.
