Exact Solutions of Fractional Burgers and Cahn-Hilliard Equations Using Extended Fractional Riccati Expansion Method

Abstract

Based on a general fractional Riccati equation and with Jumarie’s modified Riemann-Liouville derivative to an extended fractional Riccati expansion method for solving the time fractional Burgers equation and the space-time fractional Cahn-Hilliard equation, the exact solutions expressed by the hyperbolic functions and trigonometric functions are obtained. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.

The Extended Fractional (DξαG/G)‐Expansion Method and Its Applications to a Space‐Time Fractional Fokas Equation

“…To the best of our knowledge, the solutions obtained in this paper have not been reported in other literatures. The traveling wave transformation used for here ensures that a certain fractional partial differential equation can be converted into another fractional ordinary differential equation, and the solutions of the latter are assumed to possess forms in one certain polynomial in ( ), where ( ) satisfies a given fractional ordinary differential equation denoted by (7), and the degree of the polynomial can be determined by the homogeneous balance principle. It is worthy of further study to extend other existing methods, like the generalized fractional Jacobi elliptic equation-based subequation method [4] used for differential equations, to fractional differential equations.…”

“…Substituting (6) into (5) and using (7) and collecting all terms with the same order of ( / ) together, the left-hand side of (5) is converted into another polynomial in ( / ). Equating each coefficient of this polynomial to zero yields a set of algebraic equations for , , ( = 0, 1, .…”

“…Thus, it is an important and significant task to find more exact solutions of different forms for the FPDEs. In recent decades, many mathematicians and physicists have made significant achievements and also presented some effective methods, for example, the fractional subequation method [1][2][3], the Jacobi elliptic equation method [4], the fractional mapping method [5], the ( / )-expansion method [6], the extended fractional Riccati expansion method [7], the first integral method [8], and the fractional complex transform [9]. Due to these methods, various exact solutions or numerical solutions of FPDEs have been established successfully.…”

Solution of Moving Boundary Space-Time Fractional Burger’s Equation

“…Taking = = −1, 0 = = 1, then, from (17) and 26, the initial-boundary condition is given by ( ) = 1 + tanh ((V − 2) ) , ℎ( ) = 1 + tanh ( ) . (29) Note that in the static case when V → 0, solution (26) becomes kink solution with velocity 2 0 . Figure 1 shows the solution (26) of the forced Burger's equation (24) in the moving frame of reference for a linear moving boundary (28), with V = 100, −100, 0.001, −0.001.…”

“…Journal of Applied Mathematics [28] presented the fractional Riccati expansion method to obtain exact solutions of the space-time fractional Kortewegde Vries equation, the spacetime fractional RLW equation, the space-time fractional Boussinesq equation, and the spacetime fractional Klein-Gordon equation. In addition, Li et al in [29] extended fractional Riccati expansion method for solving the time fractional Burger's equation and the spacetime fractional Cahn-Hilliard equation.…”