Numerical approximations of nonlinear fractional differential difference equations by using modified He-Laplace method

Abstract

In this paper, a numerical algorithm based on a modified He-Laplace method (MHLM) is proposed to solve space and time nonlinear fractional differential-difference equations (NFDDEs) arising in physical phenomena such as wave phenomena in fluids, coupled nonlinear optical waveguides and nanotechnology fields. The modified He-Laplace method is a combined form of the fractional homotopy perturbation method and Laplace transforms method. The nonlinear terms can be easily decomposed by the use of He’s polynomials. This algorithm has been tested against time-fractional differential-difference equations such as the modified Lotka Volterra and discrete (modified) KdV equations. The proposed scheme grants the solution in the form of a rapidly convergent series. Three examples have been employed to illustrate the preciseness and effectiveness of the proposed method. The achieved results expose that the MHLM is very accurate, efficient, simple and can be applied to other nonlinear FDDEs.

Keywords

• Modified He-Laplace method (MHLT)
• Fractional differential-difference equations
• Lotka–Volterra equation
• mKdV equations
• Caputo sense