New Iteration Methods for Time-Fractional Modified Nonlinear Kawahara Equation

We put side by side the methodology of two comparatively new analytical techniques to get to the bottom of the system of nonlinear fractional modified Kawahara equation. The technique is described and exemplified with a numerical example. The dependability of both methods and the lessening in computations give these methods a wider applicability. In addition, the computations implicated are very simple and undemanding. 1. Introduction Within the scope of fractional calculus in the recent decade several scholars have modeled physical and engineering problems. Respective scholar while dealing with real world problems found out that it is worth describing these phenomena with the idea of derivatives with fractional order. While searching the literature, we found out that, this concept of noninteger order derivative not only has been intensively used but also has played an essential role in assorted branches of sciences including but not limited to hydrology, chemistry, image processing, electronics and mechanics; the applicability of this philosophy can be found in [1–10]. In the foregone respective decennial, the research of travelling-wave solutions for nonlinear equations has played a crucial character in the examination of nonlinear physical phenomena. Nonlinear wave phenomena of dispersion, dissipation, diffusion, reaction, and convection are very important in nonlinear wave equations. Concepts like solitons, peakons, kinks, breathers, cusps, and compactons have now been thoroughly investigated in the scientific literature [11–13]. Various powerful mathematical methods such as the inverse scattering method, bilinear transformation [14], the tanh-sech method [15, 16], extended tanh method [16], Exp-function method [17–19], sine-cosine method [20] Adomian decomposition method [21], Exp-function method [22], homotopy perturbation method [23] have been proposed for obtaining exact and approximate analytical solutions. The purpose of this paper is to examine the approximated solution of the nonlinear fractional modified Kawahara equation, using the relatively new analytical method, the Homotopy decomposition method (HDM), and the Sumudu transform method. The fractional partial differential equations under investigation here are given below as subject to the initial condition The outstanding of this paper is prearranged as follows. In Section 2 we present a succinct history of the fractional derivative order and their properties. We present the basic ideal of the HDM and the STM for solving high order nonlinear fractional partial differential equations. We