Numerical computation of Klein–Gordon equations arising in quantum field theory by using homotopy analysis transform method

AbstractIn this paper, we present a reliable algorithm based on the homotopy analysis transform method (HATM) to solve the linear and nonlinear Klein–Gordon equations. The Klein–Gordon equation is the equation of motion of a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. It describes the quantum amplitude for finding a point particle in various places, the relativistic wave function, but the particle propagates both forwards and backwards in time. The HATM is a combined form of the Laplace transform method and homotopy analysis method. The method provides the solution in the form of a rapidly convergent series. Some numerical examples are used to illustrate the preciseness and effectiveness of the proposed method. The results show that the HATM is very efficient, simple and can be applied to other nonlinear problems

Numerical Solutions for the Time and Space Fractional Nonlinear Partial Differential Equations

We implement relatively analytical techniques, the homotopy perturbation method, and variational iteration method to find the approximate solutions for time and space fractional Benjamin-Bona Mahony equation. The fractional derivatives are described in the Caputo sense. These methods are used in applied mathematics to obtain the analytic approximate solutions for the nonlinear Bejamin-Bona Mahoney (BBM) partial fractional differential equation. We compare between the approximate solutions obtained by these methods. Also, we present the figures to compare between the approximate solutions. Also, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. We use the improved -expansion function method to find exact solutions of nonlinear fractional BBM equation