The Traveling Wave Solutions of Space-Time Fractional Differential Equation Using Fractional Riccati Expansion Method

Abstract

In this paper, we firstly give a counterexample to indicate that the chain rule d d x x u D u D α α σ ξ ξ = is lack of accuracy. After that, we put forward the fractional Riccati expansion method. No need to use the chain rule, we apply this method to fractional KdV-type and fractional Telegraph equations and obtain the tangent and cotangent functions solutions of these fractional equations for the first time.

Traveling wave solutions of the generalized Burgers-Huxley Equation

“…In addition, the execution of these methods has become more vital. There have been several ways for finding accurate traveling wave solutions investigated and developed, including modified kudryashov method [2], The homotopy analysis method [7], extended tanh method [14], first integral method [16], sine-cosine method [18], dynamical system method [19], modified simple equation method [20], improved kudryashov method [8], tanh method [13], fractional sub-equation method [4] , 𝐺′ 𝐺 -expansion method [3], F-expansion method [12] , homotopy analysis method [10], variational iteration method [9], fractional Riccati expansion method [6], Sumudu transform method [11] and so on.…”

Exponential--Weierstrass type, exponential--Jacobi type and solitary type solutions to some conformable fractional equations

“…Therefore, it is an acceptable choice to use CFD to solve the conformable fractional partial differential equations (CFPDEs). As a matter of fact,the CFD (1.1) with its properties (1.4), (1.5) and (1.6) were successfully used to solve some CFPDEs by means of the Riccati equation expansion [3,15],the sine-cosine method [4], the exp (φ(ε))-expansion [9], the modified Kudryashov method [9,12] and the Jacobi elliptic function expansion method [18], etc. However,we find that all these existing direct methods cannot be used to find two types of exact traveling wave solutions of CFPDEs when their original equations have the non-integrable background.The first type solution,namely the exponential-Weierstrass type solution,is expressed by the product of exponential function and Weierstrass elliptic function.…”

“…is a free parameter. The space-time fractional telegraph equation[15] D 2α t u − D 2α x u + D α t u + µu + νu 3 = 0. (3.32) Taking (2.8) with β = α into (3.32) we obtain the following ODE ω 2 − 1 u ′′ + ωu ′ + µu + νu 3 = 0.…”