On Comparison for Approximate Solutions of Modified Time Caputo Fractional Kawahara Equations in Shallow Water Theory by Using Some Techniques

Abstract

In the theory of shallow water wave equations, the Kawahara equation and modified Kawahara equation are introduced to represent the solitary-wave propagation. In this paper, we use the residual power series method and Aboodh transform to provide a new technique, the Aboodh Residual Power Series Method (ARPSM). By this technique and the Caputo fractional operator, we calculate the coefficients of the power series of the modified Kawahara equation, which will serve as the approximate solution. For providing approximate analytical and numerical solutions of the modified Kawahara equation, we first consider the Modified Time Caputo Fractional Kawahara Equation (MTCFKE) and then use ARPSM in two cases: with the polynomial initial condition and with the perfect condition of MTCFKE. To show the capability, reliability, and efficiency of ARPSM, we describe ARPSM's approximate analytical solutions numerically and graphically and compare these solutions with other solutions obtained by using two methods: the homotopy analysis technique and the natural transform decomposition technique.