Research Article  Open Access
Application of Residual Power Series Method to Fractional Coupled Physical Equations Arising in Fluids Flow
Abstract
The approximate analytical solution of the fractional CahnHilliard 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 qhomotopy analysis method (qHAM). 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 tanhcoth method [11], expfunction method [12], MittagLeffler function method [13], differential quadrature method [14], and reproducing kernel Hilbert space method [15, 16].
The Gardner equation [17] (combined KdVmKdV 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 CahnHilliard 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 timefractional Gardner equation [28–30] and timefractional CahnHilliard equation [31–37] of this form,where , , and Numerous methods have been used to solve this equations, for example, qHomotopy analysis method [28], the new version of Fexpansion method [29], reduced differential transform method [30], the generalized tanhcoth 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 CahnHilliard 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 secondorder 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 LaneEmden equation [41], which is a highly nonlinear singular differential equation, in the numerical solution of higherorder regular differential equations [42], in approximate solution of the nonlinear fractional KdVBurgers equation [43], in construct and predict the solitary pattern solutions for nonlinear timefractional 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 loworder to the higherorder 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 timefractional 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 timefractional Gardner and CahnHilliard equations.
4.1. TimeFractional Gardner Equation
Consider the timefractional 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. TimeFractional CahnHilliard Equation
Consider the timefractional CahnHilliard 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
5. Numerical Results
This section deals with the approximate analytical solutions obtained by RPSM for Gardner and CahnHilliard equations. In classical case, Figure 1 and Tables 1 and 2 describe the comparison between RPSM with qHAM [28] and HPM [36]. In fractional case, Figures 2, 3, and 4 describe the geometrical behavior of the solutions obtained by RPSM for different fractional value of the two equations.


(a)
(b)
(a)
(b)
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
6. Conclusions
This work has used the RPSM for finding the solution of the timefractional Gardner and CahnHilliard equations. A very good agreement between the results obtained by the RPSM and qHAM [28] was observed in Figure 1(a) and Table 1. Figure 1(b) and Table 2 indicate that the mentioned method achieves a higher level of accuracy than HPM [36]. Consequently, the work emphasized that the method introduces a significant improvement in this field over existing techniques.
Data Availability
The [approximate solution obtained by qhomotopy analysis method] data used to support the findings of this study have been deposited in the [article] repository ([doi.org/10.1016/j.asej.2014.03.014]) [28]. The [approximate solution obtained by homotopy perturbation method] data used to support the findings of this study have been deposited in the [article] repository ([doi.org/10.1080/10288457.2013.867627]) [36].
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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Copyright © 2018 Anas Arafa and Ghada Elmahdy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.