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.
A high-order uniform Cartesian grid compact finite difference scheme for the Goursat problem is developed. The basic idea of high-order compact schemes is to find the compact approximations to the derivatives terms by differentiating centrally the governing equations. Our compact scheme will approximate the derivative terms by involving the higher terms and reducing the number of grid points. The compact finite difference scheme is given for general form of the Goursat problem in uniform domain and illustrates the performance by applying a linear problem. Numerical experiments have been conducted with the new scheme and encouraging results have been obtained. In this paper we present the compact finite difference scheme for the Goursat problem. With the aid of computational software the scheme was programmed for determining the relative errors of linear Goursat problem.