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.
Highlights
During recent years, fractional differential equations (FDEs) have attracted much attention because of their potential applications in engineering and applied sciences such as signal processing, materials and mechanics, biology systems, anomalous diffusion, and medical [1,2,3,4]
It is very important to find some proper methods for solving fractional differential equations
Many effective methods have been established to obtain the solutions of FDEs, such as the variational iteration method [5, 6], the finite difference method [7], the fractional complex transform [8, 9], the exponential function method [10], the fractional subequation method [11], the (G/G)-expansion method [12, 13], and the first integral method [14]
Summary
Fractional differential equations (FDEs) have attracted much attention because of their potential applications in engineering and applied sciences such as signal processing, materials and mechanics, biology systems, anomalous diffusion, and medical [1,2,3,4]. Based on Jumarie’s modified Riemann-Liouville derivative and the fractional Riccati equation Dξαφ(ξ) = σ + φ2(ξ), S. Zhang [15] first proposed a new direct method called fractional subequation method in solving nonlinear time fractional biological population model and (4+1)-dimensional space-time fractional Fokas equation. Lu [17] modified the method to derive the rational formal solutions of the spacetime fractional Whitham-Broer-Kaup, the foam drainage equation with time and space-fractional derivatives, and nonlinear time fractional biological population model. In this paper, based on Jumarie’s modified Riemann-Liouville derivative and the general fractional Riccati equation DξαF(ξ) = A + BF(ξ) + CF2(ξ), we will introduce an extended fractional Riccati expansion method to the time fractional Burgers equation and the space-time fractional Cahn-Hilliard equation [19].
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