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.

1. Introduction

Fractional differential equations (FDEs) have found applications in many problems in physics and engineering [1, 2]. Since most of the nonlinear FDEs cannot be solved exactly, approximate and numerical methods must be used. 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], exp-function method [12], Mittag-Leffler function method [13], differential quadrature method [14], and reproducing kernel Hilbert space method [15, 16].

The Gardner equation [17] (combined KdV-mKdV equation) is a useful model for the description of internal solitary waves in shallow water,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]. It also describes a variety of wave phenomena in plasma and solid state [22, 23].

The Cahn-Hilliard equation [24] is one type of partial differential equations (PDEs) and was first introduced in 1958 as a model for process of phase separation of a binary alloy under the critical temperature [25],This equation is related to a number of interesting physical phenomena like the spinodal decomposition, phase separation, and phase ordering dynamics. On the other hand it becomes important in material sciences [26, 27].

The aim of this paper is to study the time-fractional Gardner equation [28–30] and time-fractional Cahn-Hilliard equation [31–37] of this form,where , , and Numerous methods have been used to solve this equations, for example, q-Homotopy analysis method [28], the new version of F-expansion method [29], reduced differential transform method [30], the generalized tanh-coth method [38], the generalized Kudryashov method [39], extended fractional Riccati expansion method [31], subequation method [32], homotopy analysis method [33], the Adomian decomposition method [34], improved expansion method [35], homotopy perturbation method [36], and variational iteration method [37]. We solve Cahn-Hilliard equation and Gardner equation by RPSM.

The RPSM was first devised in 2013 by the Jordanian mathematician Omar Abu Arqub as an efficient method for determining values of coefficients of the power series solution for first and the second-order fuzzy differential equations [40]. The RPSM is an effective and easy to construct power series solution for strongly linear and nonlinear equations without linearization, perturbation, or discretization. In the last few years, the RPSM has been applied to solve a growing number of nonlinear ordinary and PDEs of different types, classifications, and orders. It has been successfully applied in the numerical solution of the generalized Lane-Emden equation [41], which is a highly nonlinear singular differential equation, in the numerical solution of higher-order regular differential equations [42], in approximate solution of the nonlinear fractional KdV-Burgers equation [43], in construct and predict the solitary pattern solutions for nonlinear time-fractional dispersive PDEs [44], and in predicting and representing the multiplicity of solutions to boundary value problems of fractional order [45]. The RPSM distinguishes itself from various other analytical and numerical methods in several important aspects [46]. Firstly, the RPSM does not need to compare the coefficients of the corresponding terms and a recursion relation is not required. Secondly, the RPSM provides a simple way to ensure the convergence of the series solution by minimizing the related residual error. Thirdly, the RPSM is not affected by computational rounding errors and does not require large computer memory and time. Fourthly, the RPSM does not require any converting while switching from the low-order to the higher-order and from simple linearity to complex nonlinearity; as a result, the method can be applied directly to the given problem by choosing an appropriate initial guess approximation.

2. Fundamental Concepts

Definition 1 (see [43]). The Caputo time-fractional derivatives of order of is defined as

Definition 2 (see [47, 48]). A power series representation of the formwhere is called fractional power series about

Theorem 3 (see [47, 48]). Suppose that has a fractional power series representation at of the form If are continuous on , then coefficients will take the form

Definition 4 (see [43]). A power series representation of the form is called a multiple fractional power series about

Theorem 5 (see [43, 44]). Suppose that has a multiple fractional Power series representation at of the form If are continuous on , then coefficients will take the form

Corollary 6 (see [44]). Suppose that has a multiple fractional Power series representation at of the form If are continuous on , then will take the form

3. Basic Idea of RPSM

To give the approximate solution of nonlinear fractional order differential equations by means of the RPSM, we consider a general nonlinear fractional differential equation:where is nonlinear term and is a linear term. Subject to the initial conditionThe RPSM proposes the solution for (14) as a fractional power series about the initial point ,Next we let denote the kth truncated series of ,The 0th RPS approximate solution of isEquation (17) can be written asWe define the residual function for (14)Therefore, the kth residual function is

As in [40, 41], since the fractional derivative of a constant in the Caputo sense is zero and the fractional derivatives of and are matching at for each ; that is, ,

To determine we consider in (19) and substitute it into (21), applying the fractional derivative in both sides, , and finally we solve

4. Applications

To illustrate the basic idea of RPSM, we consider the following two time-fractional Gardner and Cahn-Hilliard equations.

4.1. Time-Fractional Gardner Equation

Consider the time-fractional homogeneous Gardner equationSubject to the initial ConditionThe exact solution when isWe define the residual function for (23) astherefore, for the kth residual function ,To determine , we consider in (27)But from (19) at ,

Now depending on the result of (22) In the case of k=1, we have ,To determine , we consider in (27)But from (19) at ,Applying on both sides and solving the equation , then we getThe solution in series form is given by

4.2. Time-Fractional Cahn-Hilliard Equation

Consider the time-fractional Cahn-Hilliard equationSubject to the initial conditionThe exact solution when isWe define the residual function for (39) astherefore, for the kth residual function ,To determine , we consider in (43)From (19) at ,If we put , thenSimilarity, to determine , we substituteinto (43) where ,Solving the equation , we find thatTo determine , we substituteinto (43) where k=3,Applying on both sides and then solving the equation , we getThe solution in series form is given by