Application of Residual Power Series Method to Fractional Coupled Physical Equations Arising in Fluids Flow

Anas Arafa,Ghada Elmahdy

doi:10.1155/2018/7692849

Anas Arafa, Ghada Elmahdy

Open Access

https://doi.org/10.1155/2018/7692849

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Abstract

The approximate analytical solution of the fractional Cahn-Hilliard and Gardner equations has been acquired successfully via residual power series method (RPSM). The approximate solutions obtained by RPSM are compared with the exact solutions as well as the solutions obtained by homotopy perturbation method (HPM) and q-homotopy analysis method (q-HAM). Numerical results are known through different graphs and tables. The fractional derivatives are described in the Caputo sense. The results light the power, efficiency, simplicity, and reliability of the proposed method.

Highlights

Fractional differential equations (FDEs) have found applications in many problems in physics and engineering [1, 2]
The Gardner equation [17] is a useful model for the description of internal solitary waves in shallow water, ut + 6uux ± 6u2ux + uxxx = 0. Those two models will be classified as positive Gardner equation and negative Gardner equation depending on the sign of the cubic nonlinear term [18, 19]
Gardner equation is widely used in various branches of physics, such as plasma physics, fluid physics, and quantum field theory [20, 21]

Summary

Introduction

Fractional differential equations (FDEs) have found applications in many problems in physics and engineering [1, 2]. Some of the recent analytical methods for solving nonlinear problems include the Adomian decomposition method [3, 4], variational iteration method [5], homotopy perturbation method [6, 7], homotopy analysis method [8, 9], spectral collocation method [10], the tanh-coth method [11], expfunction method [12], Mittag-Leffler function method [13], differential quadrature method [14], and reproducing kernel Hilbert space method [15, 16]. Gardner equation is widely used in various branches of physics, such as plasma physics, fluid physics, and quantum field theory [20, 21]. It describes a variety of wave phenomena in plasma and solid state [22, 23]

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