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

This paper presents analytical‐approximate solutions of the time‐fractional Cahn‐Hilliard (TFCH) equations of fourth and sixth order using the new iterative method (NIM) and q‐homotopy analysis method (q‐HAM). We obtained convergent series solutions using these two iterative methods. The simplicity and accuracy of these methods in solving strongly nonlinear fractional differential equations is displayed through the examples provided. In the case where exact solution exists, error estimates are also investigated.


INTRODUCTION AND PRELIMINARIES
The concept of fractional calculus such as fractional derivative and fractional integral is not new. L' Hospital, in 1695, wrote a letter to Leibniz, saying "How do we calculate d n dx n when n = 1 2 ?" That is, "what will happen if we consider n to be a fraction?" Leibniz replied to L' Hospital question saying "d 1∕2 x equal to x √ dx ∶ x. In actual fact, the reply is an apparent paradox, from this apparent paradox, one day, useful result might be drawn." [1][2][3] Later, researchers discovered that fractional calculus has a wide range of applications in various fields of natural sciences and engineering such as found in control theory of dynamical systems, signal and image processing, financial modeling, nanotechnology, viscoelasticity, random walk, anomalous transport, and anomalous diffusion, just to name a few (for more details, see previous studies [4][5][6][7][8][9][10][11][12][13][14][15][16] ).
Nonlinear partial differential equations (NPDEs) have played a vital role in various fields of engineering and natural sciences. Among such NPDEs, we have Cahn-Hillard equation named after Cahn and Hilliard in 1958. 17 This equation plays an essential role in understanding several fascinating physical phenomena, for instance, in spinodal decomposition and phase ordering dynamics. It also describes vital qualitative distinctive attribute of two-phase systems connected with phase separation processes (see previous studies [17][18][19][20] for a detailed discussion). As a result of its real-world applications in the various fields mention above, researchers have investigated the mathematical and numerical solutions of this equation. [20][21][22][23][24][25][26][27][28] 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, 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: and with the initial condition Here, 0 < ≤ 1 stands for the order of the fractional derivative and parameter ≥ 0. Our aim is to obtain solutions in the form of recurrence relations, using the new iteration method (NIM), which is based on the decomposition the nonlinearity term [48][49][50][51] and q-homotopy analysis method (q-HAM), a modification of the homotopy analysis method. [52][53][54][55][56] Definition 1. The Gamma function is defined as 57 where ℜ(z) > 0.
satisfies the following defined properties:

ANALYSIS OF APPROXIMATE METHODS
In this section, we give a brief description of the NIM and the q-HAM.

Fundamentals of the NIM
Consider the following functional equation where ℒ and  are, respectively, the linear and nonlinear operator from a Banach space  to itself, (x, t) is a known function and u(x, t) is the unknown function. Let We seek for a solution u(x, t) of Equation (8) in the form of a series given as Thus, Equation (9) converges absolutely and uniformly to a unique solution if the operators ℒ and  are contractive. 48,51 The decomposed nonlinear operator  is given as In the same manner, the linear operator ℒ can be decomposed as From Equation (12), we have Considering Equations (9) to (13), from Equation (8), we have Then, from Equation (14), we define the iterations u 0 = (x),

Idea of the q-HAM
Consider the nonlinear differential equation where (x, t) and u(x, t) are, respectively, the known and unknown functions,  is the nonlinear operator, and  t is the conformable fractional derivative with respect to "t," In order to generalize the concept of homotopy method, we construct the zeroth-order deformation equation given as where q (0 ≤ q ≤ 1 n ) is the embedded parameter, (h ≠ 0) an auxiliary parameter, ℒ is the auxiliary linear operator, and (x, t) is a nonzero auxiliary function. For q = 0, 1 n , respectively, we obtain from Equation (17) the following: When q rises from 0 to 1 n , the solution Φ(x, t; q) ranges from the initial guess u 0 (x, t) to the solution u(x, t). If u 0 , h, ℒ , and (x, t) are chosen appropriately, then the solution Φ(x, t; q) of Equation (17) is valid as long as 0 ≤ q ≤ 1 n . Hence, we obtain the Taylor series expansion for Φ(x, t; q) as where If we choose u 0 , h, ℒ , and (x, t) properly so that Equation (19) converges at q = 1 n , then from Equation (18), we obtain We define the vector ⃗ u r as follows: By differentiating Equation (17)r-times (with respect to }}qε), substituting q = 0, and then multiply it by 1 r! , we obtained what is known as the r th -order deformation equation 52,54,60 subject to the initial conditions where and From Equation (23) with r ≥ 1, one can get Finally, the series solutions by q-HAM is presented by which gives the appropriate solution in terms of convergence parameters n and h.

SOLUTIONS OF FOURTH-ORDER TFCH EQUATION
In this section, we present the application of the above mentioned methods to obtain approximate solution of the fourth-order TFCH Equation (1) subject to different initial conditions.

Case I
Consider the general form of TFCH Equation (1) as with the initial condition The exact solution of Equation (29) when = 1 and = 1 is NIM solution: Applying J to both sides of Equation (29), then Equations (29) and (30) are equivalent to the integral equation .
We obtain components of the series solution using NIM recurrent relation in Equation (15) successively as follows: Following the same procedure, expression for u m (x, t), m = 3, 4, 5, ... can be obtained. The expression of the series solution given by NIM can be written in the form Thus, Equation (32) gives an approximate solution to problem (29). q-HAM solution: To apply the q-HAM, we rewrite Equation (29) as Applying q-HAM to Equation (33), we obtain from Equation (25) the expression Using q-HAM recurrent relation in Equation (27). Then from Equations 26 and 34, we obtain the following: Following the same procedure, expression for u m (x, t), m = 4, 5, 6, ... can be obtained. The expression of the series solution given by q-HAM can be written in the form Thus, Equation (35) gives an approximate solution to problem (29) in terms of convergence parameters h and n. In the case when n = = 1, we choose h = −1, and obtain from Equation (35), the expression which can be expressed in the closed-form of the exact solution (31) when = 1.
Remark 1. This agrees with the solution obtained using ADM and HPM in Ugurlu and Kaya. 28

Case II
We consider the general form of TFCH Equation (1) as with the initial condition NIM solution: Applying J to both sides of Equation (37), then Equations (37) and 38 are equivalent to the integral equation .
We obtain components of the series solution using NIM recurrent relation in Equation (15) successively as follows: Following the same procedure, expressions for u m (x, t), m = 3, 4, 5, ... can be obtained. The expression of the series solution given by NIM can be written in the form Thus, Equation (39) gives an approximate solution to problem (37).

q-HAM solution:
To apply the q-HAM, we rewrite Equation (37) as Using q-HAM recurrent relation in Equation (27). Then from Equations (26) and (34), we obtain the following Thus, Equation (41) gives an approximate solution to problem (37) in terms of convergence parameters h and n.

Numerical results for TFCH equation of fourth order
Here, we check how accurate these two methods are for solving time-fractional Cahn-Hillard Equation (1) with different initial conditions as shown in cases I and II of Section 3. In Figures 1 to 11, one can acknowledge how closely the approx- imation series solution obtained by these two methods and the exact solution. In Table 1, error estimate is done for the case when the exact solution ( = 1 and = 1) is known. Abbreviations: NIM, new iterative method; q-HAm, q-homotopy analysis method.

SOLUTIONS OF SIXTH-ORDER TFCH EQUATION
In this section, we present the application of the above mentioned methods to obtain approximate solution of the sixth-order TFCH Equation (2) subject to different initial conditions.

Case I
Consider the general form of TFCH Equation (2) as with the initial condition NIM solution: Applying J to both sides of Equation (42), then Equations (42) and (43) are equivalent to the integral equation .
We obtain components of the series solution using NIM recurrent relation in Equation (15) successively as follows: Following the same procedure, expressions for u m (x, t), m = 3, 4, 5, ... can be obtained. The expression of the series solution given by NIM can be written in the form Thus, Equation (45) gives an approximate solution to problem (42).

q-HAM solution:
To apply the q-HAM, we rewrite Equation (42) as Applying q-HAM to Equation (45), we obtain from Equation (25) the expression Using q-HAM recurrent relation in Equation (27). Then from Equations (26) and (46), we obtain the following: Following the same procedure, expression for u m (x, t), m = 3, 4, 5, ... can be obtained. The expression of the series solution given by q-HAM can be written in the form Thus, Equation (47) gives an approximate solution to problem (42) in terms of convergence parameter h and n.

Case II
Consider the general form of TFCH Equation (2) as with the initial condition NIM solution: Applying J to both sides of Equation (48), then Equations (48) and (49) are equivalent to the integral equation .

Numerical results for TFCH equation of sixth order
Here, we present the numerical simulation of cases I and II of Section 4 to demonstrate the effectiveness of the two iterative methods used for solving TFCH Equation (2) subject to different initial conditions. Figures 12 to 22 show that the U 2solutions obtained by these two iterative methods are graphically and numerically indistinguishable.

CONCLUDING REMARKS
In conclusion, we have studied iterative methods for constructing approximate solutions to the time-fractional nonlinear Cahn-Hilliard equations of fourth and sixth order using different initial conditions. We used NIM and q-HAM to obtain approximate series solution and present the graphical representation of the obtained results for different fractional order. We observed that the fraction factor 1 n and the parameter h highly increase the convergence of the chances of the q-HAM. As shown in our examples, the two iterative methods do not require any transformations or perturbations. Therefore, these methods are considerably efficient, powerful, and easy to implement when compared with other numerical methods for constructing approximated solutions to the linear and nonlinear fractional differential equations. Our aim in this paper is not to conclude that one method is better than the other, but rather conclude that both methods provide a good approximate solution even in some cases, we can obtain the exact solutions.
This work does not have any conflicts of interest.