Nonstandard finite difference scheme associated with harmonic mean averaging for the nonlinear Klein-Gordon equation

In this paper, we demonstrate a modified scheme for solving the nonlinear KleinGordon equation of PDE hyperbolic types. The Klein-Gordon equation is a relativistic wave equation version of the Schrodinger equation, which is widely used in quantum mechanics. Additionally, the nonstandard finite difference scheme has been used extensively to solve differential equations and we have constructed a modified scheme based on the nonstandard finite difference scheme associated with harmonic mean averaging for solving the nonlinear inhomogeneous Klein-Gordon equation where the denominator is replaced by an unusual function. The numerical results obtained have been compared and showed to have a good agreement with results attained using the standard finite difference (CTCS) procedure, which provided that the proposed scheme is reliable. Numerical experiments are tested to validate the accuracy level of the scheme with the analytical results