Construction of Fractional Power Series Solutions to Fractional Boussinesq Equations Using Residual Power Series Method

Abstract

This paper is aimed at constructing fractional power series (FPS) solutions of time-space fractional Boussinesq equations using residual power series method (RPSM). Firstly we generalize the idea of RPSM to solve any-order time-space fractional differential equations in high-dimensional space with initial value problems inRn. Using RPSM, we can obtain FPS solutions of fourth-, sixth-, and 2nth-order time-space fractional Boussinesq equations inRand fourth-order time-space fractional Boussinesq equations inR2andRn. Finally, by numerical experiments, it is shown that RPSM is a simple, effective, and powerful method for seeking approximate analytic solutions of fractional differential equations.

Iterative methods for solving fourth‐ and sixth‐order time‐fractional Cahn‐Hillard equation

“…Solving partial differential equations with fractional derivatives is often more difficult than solving the classical type, for its operator is defined by integral. In the recent year, researchers have developed some iterative methods for solving the nonlinear fractional differential equations, such as Adomian decomposition method, variational iteration method, homotopy decomposition method, differential transform method, permuturbation iteration transformation method, homotopy‐perturbation method, homotopy analysis method, exp‐function method, wavelet method, Khater method, and residual power series method …”

“…In the recent year, researchers have developed some iterative methods for solving the nonlinear fractional differential equations, such as Adomian decomposition method, 21,28,29 variational iteration method, [30][31][32] homotopy decomposition method, 33 differential transform method, 34,35 permuturbation iteration transformation method, 36 homotopy-perturbation method, 28,37 homotopy analysis method, [38][39][40] exp-function method, [41][42][43] wavelet method, 44 Khater method, 45 and residual power series method. 46,47 In this paper, we consider the time-fractional Cahn-Hilliard (TFCH) equations of the fourth and sixth order given, respectively, as follows:…”

Homotopy Series Solutions to Time-Space Fractional Coupled Systems

“…There are many effective methods to solve this problem, like Adomian decomposition method [8][9][10], variation iteration method [11], differential transform method [12], residual power series method [13,14], iteration method [15], homotopy perturbation method [16], homotopy analysis method [17], and so on. Furthermore, for the nonlinear problem, the multiple exp-function method [18,19], the transformed rational function method [20][21][22], and invariant subspace method [23,24] are three systematical approaches to handle the nonlinear terms.…”

Applications of General Residual Power Series Method to Differential Equations with Variable Coefficients

“…RPS has been extended to many partial differential equations (PDE), especially to fractional partial differential equations (FPDE), such as time-fractional dispersive PDE [15,16], time-fractional KdV-Burgers equations [17], homogeneous time-fractional wave equation [18], and time-space fractional Boussinesq equations [19]. In the present paper, we will apply GRPS to a series of PDE with variable coefficients, including fourth-order parabolic equations, fractional heat equation, and fractional wave equation.…”