An efficient technique for time fractional Klein-Gordon equation based on modified Laplace Adomian decomposition technique via hybridized Newton-Raphson Scheme arises in relativistic fractional quantum mechanics

In this paper, a hybridized Newton-Raphson method with convergence order (2+1) is coupled with modified Laplace Adomian decomposition technique (LADT) to obtain approximate soliton of Time Fractional Klein Gordon Equation (TFKGE) in Caputo environment and associated with relativistic fractional quantum mechanics. Our approach involves utilising an effective Laplacian and inverse Laplacian method to approximate the soliton of the proposed equation. The comprehensive investigation of convergence and stability for the current system improves the theoretical notion. The corresponding graphs of the exact and approximate solutions, as well as the error norms L2 and L∞ for each test problem, are provided to validate the numerical results of the problem. The present result is compared with the analytical solution and existing solutions in the literature to illustrate the effectiveness and applicability of our proposed method.