Analysis of local fractional Klein-Gordon equations arising in relativistic fractal quantum mechanics

Abstract

In this paper, we present the implementation of a local fractional homotopy perturbation method pertaining to the local fractional natural transform (LFNT) operator for local fractional Klein-Gordon equations (LFKGEs) under distinct fractal initial conditions. The Klein-Gordon equation is a relativistic wave equation which estimates the nature and movement of zero spin particles at high velocities and energies resembling the lightspeed. This equation admits the laws of special relativity and describes zero spin particles with relativistic energy. This work also examines uniqueness and convergence analyses of the obtained solution for a general local fractional partial differential equation via applied method. The numerical simulations are presented for each LFKG equation on the Cantor set. The computational process ensures that the copulation of local fractional homotopy perturbation method and LFNT operator provides series solutions in closed form with fast convergence for local fractional Klein-Gordon equation in a very lucid way. Furthermore, the results have also been compared with solutions obtained by other methods.