Iterative Methods for Solving Fourth and Sixth Order Time-fractional Cahn-Hillard Equation

TL;DR

This paper introduces new iterative methods to find approximate solutions for complex fourth and sixth order time-fractional Cahn-Hilliard equations, demonstrating their effectiveness and accuracy through examples.

Contribution

It develops and applies the new iterative method (NIM) and q-homotopy analysis method (q-HAM) to solve high-order fractional differential equations.

Findings

Convergent series solutions obtained for TFCH equations.

Methods show simplicity and high accuracy in solving nonlinear fractional equations.

Error estimates provided when exact solutions are available.

Abstract

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 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.

Figures

Tables

Table 1. Table 1 : Case I: Error Analysis for NIM ( U 2 subscript 𝑈 2 U_{2} ) and q-HAM ( U 3 subscript 𝑈 3 U_{3} ) using the Exact Solution ( n = α = μ = 1 , 𝑛 𝛼 𝜇 1 n=\alpha=\mu=1,   h = − 1 ℎ 1 h=-1 ).

		Absolute Error					Absolute Error
$t$	$x$	NIM	q-HAM		$t$	$x$	NIM	q-HAM
0.01	0.0	$1.151971\times {10}^{-7}$	$2.356975\times {10}^{-12}$		0.05	0.0	$1.306675\times {10}^{-5}$	$7.361971\times {10}^{-9}$
	1.0	$1.810671\times {10}^{-7}$	$2.823765\times {10}^{-10}$			1.0	$2.224480\times {10}^{-5}$	$1.736922\times {10}^{-7}$
	2.0	$6.167394\times {10}^{-8}$	$5.749512\times {10}^{-11}$			2.0	$7.794449\times {10}^{-6}$	$3.622408\times {10}^{-8}$
	3.0	$1.165205\times {10}^{-9}$	$3.757261\times {10}^{-11}$			3.0	$1.257660\times {10}^{-7}$	$2.328496\times {10}^{-8}$
0.08	0.0	$4.940148\times {10}^{-5}$	$7.713501\times {10}^{-8}$		0.1	0.0	$9.109940\times {10}^{-5}$	$2.352262\times {10}^{-7}$
	1.0	$8.990891\times {10}^{-5}$	$1.124520\times {10}^{-6}$			1.0	$1.740220\times {10}^{-4}$	$2.722916\times {10}^{-6}$
	2.0	$3.218897\times {10}^{-5}$	$2.387229\times {10}^{-7}$			2.0	$6.321236\times {10}^{-5}$	$5.848640\times {10}^{-7}$
	3.0	$4.548965\times {10}^{-7}$	$1.516340\times {10}^{-7}$			3.0	$8.108096\times {10}^{-7}$	$3.686350\times {10}^{-7}$

Equations

$${\mathcal{D}}_t^{\alpha }u=\mu u_x+(-u_{xx}-u+u^3{)}_{xx},$$

$${\mathcal{D}}_t^{\alpha }u=\mu u{u}_x+(u_{xx}+u-u^3{)}_{xxxx},$$

$$u(x,0)=f(x).$$

$$\Gamma (z)={\int }_0^{\infty }{t}^{z-1}{e}^{-t}dt,$$

$$J^{\alpha }u(x,t)=\frac{1}{\Gamma (\alpha )}{\int }_0^t(t-\mu {)}^{\alpha -1}u(x,\mu )d\mu ,\alpha ,t>0,$$

$$\displaystyle\mathcal{D}^{\alpha}u(x,t)=\left\{\begin{array}[]{lll}u^{(m)}(x,t),\qquad\qquad\alpha=m,\\ \\ J^{m-\alpha}u^{(m)}(x,t),\qquad m-1<\alpha<m,\end{array}\right.$$

$$J^{m-\alpha }{u}^{(m)}(x,t)=\frac{1}{\Gamma (m-\alpha )}{\int }_0^t(t-\xi {)}^{m-\alpha -1}{u}^{(m)}(x,\xi )d\xi ,\alpha ,t>0,$$

$$u(x,t)=f(x,t)+\mathcal{L}(u(x,t))+\mathcal{N}(u(x,t)),$$

$$U_n(x,t)=\sum _{k=0}^n{u}_k(x,t).$$

$$u(x,t)=\underset{n\to \infty }{\lim }{U}_n(x,t)=\sum _{k=0}^{\infty }{u}_k(x,t).$$

$$\mathcal{N}(u(x,t))$$

$$\mathcal{L}(u(x,t))$$

$$\sum _{k=1}^{\infty }\left\{\mathcal{L}\left(\sum _{i=0}^k{u}_i(x,t)\right)-\mathcal{L}\left(\sum _{i=0}^{k-1}{u}_i(x,t)\right)\right\}$$

$$\sum _{k=0}^{\infty }{u}_k(x,t)=f(x,t)+\sum _{k=0}^{\infty }\mathcal{L}(u_k(x,t))+\mathcal{N}(u_0(x,t))+\sum _{k=1}^{\infty }\left\{\mathcal{N}\left(\sum _{j=0}^k{u}_j(x,t)\right)-\mathcal{N}\left(\sum _{j=0}^{k-1}{u}_j(x,t)\right)\right\}.$$

$$u_0$$

$$u_1$$

$$u_2$$

$$u_3$$

$$u_{m+1}$$

$$\mathcal{N}\left({\mathcal{D}}_t^{\alpha }u(x,t)\right)-f(x,t)=0,$$

$$\displaystyle(1-qn)\mathscr{L}\Big{(}\Phi(x,t;q)-u_{0}(x,t)\Big{)}=hq\mathcal{H}(x,t)\Big{(}\mathcal{N}[\mathcal{D}_{t}^{\alpha}\Phi(x,t;q)]-f(x,t)\Big{)},\quad n\geq{1},$$

$$\Phi\Big{(}x,t;0\Big{)}=u_{0}(x,t),\quad\quad\Phi\Big{(}x,t;\frac{1}{n}\Big{)}=u(x,t).$$

$$\Phi (x,t;q)=u_0(x,t)+\sum _{r=1}^{\infty }{u}_r(x,t)q^r,$$

$$u_r(x,t)={\frac{1}{r!}\frac{{\partial }^r\Phi (x,t;q)}{\partial q^r}|}_{q=0}.$$

$$u(x,t)=u_0(x,t)+\sum _{r=1}^{\infty }{u}_r(x,t){\left(\frac{1}{n}\right)}^r.$$

$${\vec{u}}_r=\left\{u_0(x,t),u_1(x,t),\cdots ,u_r(x,t)\right\}.$$

$$\mathcal{L}\left(u_r(x,t)-{\Psi }_r^{\ast }{u}_{r-1}(x,t)\right)=h\mathcal{H}(x,t){\mathcal{R}}_r\left({\vec{u}}_{r-1}\right).$$

$$u_r^{(j)}(0,x)=0,j=0,1,2,3,...,r-1.$$

$${\mathcal{R}}_r\left({\vec{u}}_{r-1}\right)={\frac{1}{(r-1)!}\frac{{\partial }^{r-1}\left(\mathcal{N}[{\mathcal{D}}_t^{\alpha }\Phi (x,t;q)]-f(x,t)\right)}{\partial q^{r-1}}|}_{q=0}.$$

$$u_r(x,t)={\Psi }_r^{\ast }{u}_{r-1}(x,t)+h{J}^{\alpha }\left({\mathcal{R}}_r\left({\vec{u}}_{r-1}(x,t)\right)\right),$$

$$\displaystyle\Psi^{*}_{r}=\left\{\begin{array}[]{lll}0\,\,\,\,\,\,\,\,\,\,\,\,\,r\leqslant 1,\\ \\ n\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise.\end{array}\right.$$

$$u(x,t;n;h)\cong U_r(x,t,n,h)=u_0(x,t)+\sum _{j=1}^r{u}_j(x,t;n;h){\left(\frac{1}{n}\right)}^j,$$

Full text

Keywords: Cahn-Hillard equation, fractional derivatives, new iterative method, q-homotopy analysis method

1 Introduction

The concept of fractional calculus such as fractional derivatives and fractional integral is not new. L’ Hospital, in 1695, wrote a letter to Leibniz, saying “How do we calculate $d^{n}y/dx^{n}$ when $n=1/2$?” that is; “what happen if we consider $n$ to be a fraction?” Leibniz replied to L’ Hospital question saying “$d^{1/2}x$ equal to $x\sqrt{dx:x}$” 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 derivative has a wide applications in the 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, are just a few (for more details, see [9, 4, 11, 5, 6, 7, 10, 14, 12, 13, 8]).

Linear and nonlinear partial fractional differential equations has play a vital role in various fields of engineering and natural sciences. Among such PDEs, we have Cahn-Hillard equation named after Cahn and Hilliard in 1958 [15]. This equation plays an essential role in understanding a number of fascinating physical phenomena for instance, in spinodal decomposition, phase ordering dynamics, and also describes vital qualitative distinctive attribute of two-phase systems connected with phase separation processes (see [16, 15, 17, 18] for a detailed discussion). Because of its real world applications in these various fields mention above, researchers have investigated the mathematical and numerical solutions of this equation [21, 22, 20, 23, 19, 18, 24, 25].

Solving partial differential equations with fractional derivatives is often more difficult than the classical PDEs, for it operator is defined by integral. In the recent year, researchers have develop some approximate methods in solving both linear and nonlinear PDEs with fractional derivatives, such as adomian decomposition method, (ADM) [21, 25, 26], variational iteration method, (VIM) [27, 28], generalized differential transform method, (GDTM) [29, 30], permuturbation iteration transformation method, (PITM) [31], homotopy-perturbation method, (HPM) [32, 25], and residual power series method (RPSM) [33, 34].

In this paper we consider the time-fractional Cahn-Hilliard equations of the forth and sixth order given, respectively, by

[TABLE]

and

[TABLE]

with the initial condition

[TABLE]

Here, the parameters $\alpha$ ($0<\alpha\leq 1$) stands for the order of the time-fractional derivatives, satisfying $\mu\geq 0$. Our aim is to obtain solutions in the form of recurrence relations, using the new iteration method (NIM), which is base on the decomposition the non-linearity term [37, 36, 35, 38] and q-homotopy analysis method (q-HAM), a modification of the homotopy analysis method (HAM) [39, 40, 41, 42].

Definition 1.1**.**

The Gamma function is defined as [43]

[TABLE]

where $\Re(z)>0$.

Definition 1.2**.**

The Riemann-Liouville fractional integral operator of order $\alpha$ ($\alpha\geq 0$) of a function $u(x,t)\in C_{\mu}$, $\mu\geq-1$, denoted by $J^{\alpha}u(x,t)$ (with respect to “t”) is defined as [3, 44],

[TABLE]

where $J^{0}u(x,t)=u(x,t)$. Then the following properties holds for function $u(x,t)$ as follows:

1. a.

$\displaystyle J^{\alpha}J^{\beta}u(x,t)=J^{\alpha+\beta}u(x,t)$** 2. b.

$\displaystyle J^{\alpha}J^{\beta}u(x,t)=J^{\beta}J^{\alpha}u(x,t)$** 3. c.

$\displaystyle J^{\alpha}t^{\uptau}=\frac{\Gamma(\uptau+1)}{\Gamma(\uptau+1+\alpha)}t^{\uptau+\alpha}$.

Here, $\alpha$, $\beta\ \geq 0$ and $\uptau>-1.$

Definition 1.3**.**

The (left sided) Caputo fractional derivative of a function $u(x,t)$ of order $\alpha$ (with respect to “t”), denoted by $\mathcal{D}^{\alpha}u(x,t)$, where $m-1<\alpha<m,$ and $u(x,t)\in C_{l}^{m}$, $m\in\mathbb{N}$ is defined as [3, 44]

[TABLE]

where

[TABLE]

satisfies the following defined properties:

1. a.

$\mathcal{D}^{\alpha}[\sigma\ u(x,t)+\rho\ w(x,t)]=\sigma\mathcal{D}^{\alpha}u(x,t)+\rho\mathcal{D}^{\alpha}w(x,t),$*  $\sigma,\ \rho\in\mathbb{R}$,* 2. b.

$\mathcal{D}^{\alpha}J^{\alpha}u(x,t)=u(x,t)$, 3. c.

$J^{\alpha}\mathcal{D}^{\alpha}u(x,t)=u(x,t)-\sum_{j=0}^{m-1}u^{j}(x,0)\ \frac{t^{j}}{j!}$.

2 Analysis of Approximate Methods

In this subsection, we give a brief description of the new iterative method (NIM) and the q-homotopy analysis method (q-HAM).

2.1 Fundamentals of the NIM

Consider the following functional equation,

[TABLE]

where $\mathscr{L}$ and $\mathcal{N}$ are respectively the linear and nonlinear operator from a Banach space $\mathcal{B}$ to itself, $f(x,t)$ is the known function and $u(x,t)$ is the unknown function. Let

[TABLE]

We seek for a solution $u(x,t)$ of Eq. (11) has a series form

[TABLE]

Eq. (12) converges absolutely and uniformly to a unique solution if the operators $\mathscr{L}$ and $\mathcal{N}$ are contractive [37, 38]. The decomposed nonlinear operator $\mathcal{N}$ is give as

[TABLE]

In the same manner, the linear operator $\mathscr{L}$ can be decomposed as

[TABLE]

Since $\mathscr{L}$ is a linear operator, then from Eq. (15) we obtain

[TABLE]

Considering Eqs. (12)-(16), from Eq. (11) we have

[TABLE]

Then, from Eq. (17), we define the iterations

[TABLE]

2.2 Idea of the q-HAM

Consider the nonlinear differential equation

[TABLE]

where $f(x,t)$ and $u(x,t)$ are respectively the known and unknown functions, $\mathcal{N}$ is the nonlinear operator, and $\mathcal{D}_{t}^{\alpha}$ 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

[TABLE]

where $q$ ($0\leq q\leq\frac{1}{n}$) is the embedded parameter, $h$ ($h\neq 0$) an auxiliary parameter, $\mathscr{L}$ is the auxiliary linear operator, and a non-zero auxiliary function denoted by $\mathcal{H}(x,t).$ For $q=0,\,\frac{1}{n}$ respectively, we obtain from Eq. (20) the following

[TABLE]

When $q$ rises from [math] to $\frac{1}{n}$, the solution $\Phi(x,t;q)$ ranges from the initial guess $u_{0}(x,t)$ to the solution $u(x,t)$. If $u_{0}(x,t)$, $h$, $\mathscr{L}$, and $\mathcal{H}(x,t)$ are chosen appropriately, then the solution $\Phi(x,t;q)$ of Eq. (20) is valid as long as $0\leq q\leq\frac{1}{n}$. Hence, we obtain the Taylor series expansion for $\Phi(x,t;q)$ as

[TABLE]

where

[TABLE]

If we choose $u_{0}$, $h,$ $\mathscr{L}$, and $\mathcal{H}(x,t)$ properly so that the Eq. (22) converges at $q=\frac{1}{n}$, then from Eq. (21) we obtain

[TABLE]

We define the vector $\vec{u}_{r}$ as follows:

[TABLE]

By differentiating Eq. (20) $r$-times (with respect to $``q"$), substituting $q=0,$ and then divide it by $r!$, we obtained what is known as the $r^{th}$-order deformation equation [45, 42],

[TABLE]

subject to the initial conditions

[TABLE]

where

[TABLE]

Therefore, we have the solution to the system for $r\geq 1$ as

[TABLE]

with

[TABLE]

The series solutions by q-HAM are

[TABLE]

which gives the appropriate solutions in terms of convergence parameters $n$ and $h$.

3 Solutions of Fourth-Order TFCH Equation

In this section, we presents the application of the above mentioned methods to obtain approximate solution of the fourth-order TFCH Eq. (1) subject to different initial conditions.

3.1

Case I: Consider the TFCH Eq. (1),

[TABLE]

with the initial condition

[TABLE]

The exact solution of the IVP (35)-(36), when $\alpha,\,\mu=1,$ is

[TABLE]

**NIM solution:

**Applying $J^{\alpha}$ to both sides of Eq. (35), then the IVP (35)-(36) is equivalent to the integral equation:

[TABLE]

where,

[TABLE]

We now obtain components of the series solution using NIM recurrent relation in Eq. (18) successively as follows:

[TABLE]

[TABLE]

Using the same procedure, expression for $u_{m}(x,t),$ $m=3,4,5,...$ can be obtained. The expression of the series solutions given by this new iteration method (NIM) can be written in the form

[TABLE]

Thus, Eq. (38) gives an approximate solution to the IVP (35)-(36).

**q-HAM solution:

**To apply the q-HAM, we rewrite TFCH Eq. (1) as

[TABLE]

Applying q-HAM to Eq. (39), we obtain from Eq. (28),

[TABLE]

We can now obtain the components of the solution using q-HAM recurrent relation in Eq. (29), using Eqs. (33) and (40) successively as follows:

[TABLE]

[TABLE]

[TABLE]

Using the same procedure, expression for $u_{m}(x,t),$ $m=4,5,6,...$ can be obtained. The expression of the series solutions given by q-HAM can be written in the form

[TABLE]

Thus, Eq. (41) gives an approximate solutions to the IVP (35)-(36) in terms of convergence parameters $h$ and $n$. In the case when $n=\alpha=1,$ we choose $h=-1,$ and obtain from Eq. (41) the expression

[TABLE]

which can be expressed in the closed form of the exact solution

[TABLE]

Remark 3.1**.**

This agrees with the solution obtained using ADM and HPM in [25].

Case II: Consider the TFCHE Eq. (1),

[TABLE]

with the initial condition

[TABLE]

**NIM solution:

**Applying $J^{\alpha}$ to both sides of Eq. (44), then the IVP (44)-(45) is equivalent to the integral equation:

[TABLE]

where,

[TABLE]

We now obtain components of the series solution using NIM recurrent relation in Eq. (18) successively as follows:

[TABLE]

[TABLE]

Using the same procedure, expression for $u_{m}(x,t),$ $m=3,4,5,...$ can be obtained. The expression of the series solutions given by NIM can be written in the form

[TABLE]

Thus, Eq. (46) gives an approximate solution to the IVP (44)-(45).

**q-HAM solution:

**Consider Eq. (39), we obtain the components of the solution using q-HAM recurrent relation in Eq. (29), using Eqs. (33) and (40) successively as follows:

[TABLE]

[TABLE]

[TABLE]

Using the same procedure, expression for $u_{m}(x,t),$ $m=4,5,6,...$ can be obtained. The expression of the series solutions given by q-HAM can be written in the form

[TABLE]

Thus, Eq. (47) give an approximate solution to the IVP (44)-(45) in terms of convergence parameters $h$ and $n$.

3.2 Numerical Results for TFCH Equation of Fourth-Order

Here, we check how accurate these two methods are for solving time-fractional Cahn-Hillard Eq. (1) with different initial conditions as shown in section 3 Case I and II. In Figs. 1 to 7, one can acknowledge how closely the approximation series solution obtained by these two methods and the exact solution. In Tab. 1, error analysis was done for the case when the exact solution is known.

4 Solutions of Sixth-Order TFCH Equation

In this section, we presents the application of the above mentioned methods to obtain approximate solution of the sixth-order TFCH Eq. (2) subject to different initial conditions.

4.1

Case I: Consider the TFCH equation Eq. (2),

[TABLE]

with the initial condition

[TABLE]

**NIM solution:

**Applying $J^{\alpha}$ to both sides of Eq. (48), then the IVP (48)-(49) is equivalent to the integral equation:

[TABLE]

where,

[TABLE]

We now obtain components of the series solution using NIM recurrent relation in Eq. (18) successively as follows:

[TABLE]

[TABLE]

Using the same procedure, expression for $u_{m}(x,t),$ $m=3,4,5,...$ can be obtained. The expression of the series solutions given by this new iteration (NIM) can be written in the form

[TABLE]

Thus, subsection 4.1 gives an approximate solution to the IVP (48)-(49).

**q-HAM solution:

**To apply the q-HAM, we rewrite TFCH Eq. (1) as,

[TABLE]

Applying q-HAM to Eq. (51), we obtain from Eq. (28),

[TABLE]

We can now obtain the components of the solution using q-HAM recurrent relation in Eq. (29), using Eqs. (33) and (52) successively as follows:

[TABLE]

[TABLE]

[TABLE]

Using the same procedure, expression for $u_{m}(x,t),$ $m=4,5,6,...$ can be obtained. The expression of the series solutions given by q-HAM can be written in the form

[TABLE]

Thus, Eq. (53) gives an approximate solutions to the IVP (48)-(49) in terms of convergence parameter $h$ and $n$.

Case II: Consider the TFCH Eq. (2),

[TABLE]

with the initial condition

[TABLE]

**NIM solution:

**Applying $J^{\alpha}$ to both sides of Eq. (54), then the IVP (54)-(55) is equivalent to the integral equation:

[TABLE]

where,

[TABLE]

We now obtain components of the series solution using NIM recurrent relation in Eq. (18) successively as follows:

[TABLE]

[TABLE]

Using the same procedure, expression for $u_{m}(x,t),$ $m=3,4,5,...$ can be obtained. Thus, the expression of the series solutions given by this new iteration (NIM) can be written in the form

[TABLE]

Thus, Eq. (56) gives an approximate solution to the IVP (54)-(55).

**q-HAM solution:

**Consider Eq. (51), we obtain the components of the solution using q-HAM recurrent relation in Eq. (29), using Eqs. (33) and (52) successively as follows:

[TABLE]

[TABLE]

[TABLE]

Using the same procedure, expression for $u_{m}(x,t),$ $m=4,5,6,...$ can be obtained. The expression of the series solutions given by q-HAM can be written in the form

[TABLE]

Thus, Eq. (57) give an approximate solution to the IVP (54)-((55)) in terms of convergence parameters $h$ and $n$.

4.2 Numerical Results for TFCH Equation of Sixth-Order

This section present the numerical simulation of each case to demonstrate the effectiveness of the two iterative methods used for solving TFCH Eq. (2) subject to different initial conditions. Figs. 8 and 9 show the NIM $U_{2}$- and q-HAM $U_{3}$- solutions for $\alpha=1$ and $\mu=0$ which are graphically almost indistinguishable.

Remark 4.1**.**

In Figs. 8 to 15, we observed that the fraction factor \Big{(}\frac{1}{n}\Big{)}^{i} and the parameter $h$ highly increase the convergence of the chances of the q-HAM.

5 Concluding remarks

In conclusion, we have studied iterative methods of constructing approximate solutions to the time-fractional nonlinear Cahn-Hilliard equations of fourth and sixth-order with different initial conditions. We used NIM and q-HAM to obtain approximate series solutions. As shown in our examples, the two iterative methods does not required any transformations or perturbation. Therefore, these methods are considerably efficient, powerful, and easy to implement when compared to other numerical methods in constructing approximated solutions to the linear and nonlinear 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 provides a good approximate solutions, even is some cases we can obtain the exact solutions.