Analysis and numerical simulation of fractional order Cahn–Allen model with Atangana–Baleanu derivative

Chaos, Solitons and Fractals 124 (2019) 134–142

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

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Analysis and numerical simulation of fractional order Cahn–Allen model with Atangana–Baleanu derivative Amit Prakash∗, Hardish Kaur Department of Mathematics, National Institute of Technology, Kurukshetra 136119, India

a r t i c l e

i n f o

Article history: Received 8 December 2018 Revised 6 May 2019 Accepted 11 May 2019

Keywords: Cahn–Allen model Atangana–Baleanu derivative Homotopy perturbation technique Laplace transform Fixed-point theorem

a b s t r a c t In this work, we present a fractional model of Cahn–Allen equation associated with newly introduced Atangana–Baleanu (AB) derivative of fractional order which uses Mittag–Leffler function as the nonsingular and non-local kernel. The existence and uniqueness of this modified fractional model are discussed by employing the fixed-point postulate. An efficient scheme homotopy perturbation transform technique (HPTT) which is an amalgamation of homotopy perturbation technique with Laplace transform is used to examine this time-fractional phase-field model numerically. Also, convergence and error analysis of the proposed technique is presented. The numerical simulations are analyzed graphically as well as in tabulated form. © 2019 Published by Elsevier Ltd.

1. Introduction During the last decades, the development of fast and highly efficient methods for fractional differential equations have emerged. The reason behind this fast development is that most complex phenomena such as memory dependent phenomena, mechanical properties of materials, anomalous diffusion, groundwater flow problems, control theory are better described using fractional derivatives and equations. Also, mathematical models based on fractional order derivatives are more realistic and efficiently describe a variety of natural phenomena. The Riemann-Liouville fractional derivative and Caputo’s fractional derivative are the most commonly used fractional derivatives which have some limitations as the definitions of these well-known fractional derivatives involve singular kernel and this weakness sometimes affects the physical problems. In 2015, Caputo and Fabrizio [1] proposed a new definition of fractional operator with a non-singular kernel based on the exponential function. This definition has the capability of describing the heterogeneities and structures with different scales which cannot be well portrayed by fractional models with a singular kernel. The concept of Caputo–Fabrizio (CF) operator was used by many researchers to investigate various fractional mathematical models as groundwater flow [2], El Nino-Southern oscillation model [3], fractional epidemiological model for computer viruses [4], groundwater pollution equation [5], evolution equations [6] etc. Recently, ∗

Corresponding author. E-mail address: [email protected] (A. Prakash).

https://doi.org/10.1016/j.chaos.2019.05.005 0960-0779/© 2019 Published by Elsevier Ltd.

Atangana and Baleanu [7], in order to solve the problem of fractional derivative [8,9] with non-singular and non-local kernel suggested a new derivative based on the generalized Mittag–Leffler function. The interest in this new approach with non-singular kernel was originated from the prospect that there is a class of nonlocal systems, which efficiently describe the material heterogeneities and fluctuations of different scales which cannot be well described by classical local theories or by fractional local models with a singular kernel. The AB derivative has been implemented by researchers in a number of ways to model real-world situations as Gomez Aguilar et al. [10] discussed alternate solutions of electromagnetic waves in dielectric media by applying AB fractional operator. In [11], Alkahtani studied the dynamics of Chua’s circuit model by newly established AB derivative and obtained new Chaotic behaviors. Owolabi [12] analyzed the modeling and simulation of an ecological system with two-step Adams–Bashforth method via AB derivative. Being motivated by the ongoing research in this area, we study the Cahn–Allen equation (CAE) which is a very important mathematical model written as follows

Dξ u ( τ , ξ ) − u τ τ ( τ , ξ ) + u 3 ( τ , ξ ) − u ( τ , ξ ) = 0 , τ > 0 , with the initial condition

u ( τ , 0 ) = f ( τ ). In recent times, it has been widely studied due to its connection with several physical motivated problems like phase separation, crystal growth, image analysis. Here, we analyze the new fractional modified form of the nonlinear Cahn–Allen equation ex-

A. Prakash and H. Kaur / Chaos, Solitons and Fractals 124 (2019) 134–142

pressed as ABC

AB α a Iτ

(ψ (η ) ) =

Dαξ u(τ , ξ ) − uτ τ (τ , ξ ) + u3 (τ , ξ ) − u(τ , ξ )

= 0, where

τ > 0 , 0 < α ≤ 1 , u ( τ , 0 ) = f ( τ ), ABC Dα u (τ , ξ )

ξ

is Atangana–Baleanu fractional derivative op-

erator in Caputo’s sense with respect to time variable ξ , α is a parameter describing the order of temporal fractional derivation. For α = 1, the nonlinear Cahn–Allen equation of fractional order converts to the standard nonlinear CAE. The Cahn–Allen equation which arises in several scientific applications such as mathematical biology, quantum mechanics and plasma physics, describes the process of phase separation in iron alloys [13], including the orderdisorder transition. Various numerical techniques such as Haar wavelet method [14], homotopy analysis transform method [15], homotopy analysis method [16], residual power series method [17], first integral method [18] have been used to investigate the Cahn– Allen equation. A comparative study between Atangana–Baleanu and Caputo–Fabrizio fractional operators for Allen Cahn model is done by Algahtani [19] and Crank Nicholson scheme is used for numerical simulations. The present work aims to extend the Cahn–Allen equation of reaction-diffusion to the scope of fractional calculus using AB derivative [20,21] with fractional order and to investigate the modified model numerically via homotopy perturbation technique using Laplace transform of Atangana–Baleanu derivative with fractional order. Homotopy perturbation method, firstly introduced by He [22] is a powerful approach for handling numerous nonlinear problems and the Laplace transform has proved to be an effective mechanism for solving several linear and nonlinear differential equations. The combination of semi-analytical techniques with Laplace transform [23–25] is proved to be time-saving and reduces computational work to solve nonlinear fractional models which are applicable in many fields of science and engineering.

1−α α ψ (τ ) + B (α ) B(α )(α ) τ ≥ 0.

135

 τ a

ψ (s )(τ − s )α−1 ds,

Theorem 2.1 [7]. The Laplace transform of the Atangana–Baleanu derivative of the function ψ (τ ) in the Riemann and Caputo sense is given by

L

ABR

L

ABC

0



B(α ) sα L{ψ (τ )}(s ) . 1 − α sα + 1−αα



B(α ) sα L{ψ (τ )}( p) − sα −1 ψ (0 ) , 1−α sα + 1−αα

Dατ (ψ (τ ) ) (s ) =

And 0

Dατ (ψ (τ ) ) (s ) =

respectively. Theorem 2.2 [7]. For a function ψ ∈ C[a, b], the result given below holds

ABC α  a Dτ (ψ (τ ) ) < B(α ) ψ (τ ), 1−α where ψ (τ ) = maxa≤τ ≤b |ψ (τ )|.

Further, the Atangana–Baleanu Caputo derivative fulfills the Lipschitz condition ABC α a Dτ

α (ψ1 (τ ) ) − ABC a Dτ (ψ2 (τ ) ) < μψ1 (τ ) − ψ2 (τ ).

3. Existence and uniqueness analysis Consider the following fractional model of Cahn–Allen equation ABC

Dαξ u(τ , ξ ) − uτ τ (τ , ξ ) + u3 (τ , ξ ) − u(τ , ξ )

= 0, τ > 0, 0 < α ≤ 1, u(τ , 0 ) = f (τ ),

2. Atangana–Baleanu fractional derivative

where

ABC Dα u (τ , ξ )

ξ

(1) (2)

is Atangana–Baleanu fractional derivative oper-

In this section, some important concepts and results of AB derivative and integrals [2,7] are discussed.

ator in Caputo’s sense with respect to time variable ξ . We present the existence of the solution by using the fixed-point theorem. Firstly, transform Eq. (1) into the integral equation and using the Definition 2.3 as given by Atangana and Baleanu [7], we get

Definition 2.1. Let ψ ∈ H1 (a, b), b > a, α ∈ [0, 1] be differentiable, then the Atangana–Baleanu derivative of order α in Caputo sense is given by

u (τ , ξ ) − u (τ , 0 ) =

ABC α a Dτ

(ψ (τ ) ) =

B (α ) 1−α

 τ a

  α ψ  (s )Eα − (τ − s )α ds, 1−α

where, the function B is a normalization of the function satisfies B ( 0 ) = B ( 1 ) = 1. Noting that

(1 − α ) uτ τ ( τ , ξ ) − u3 ( τ , ξ ) + u ( τ , ξ ) B (α )  ξ α + (ξ − s )α−1 B(α )(α ) 0

× uτ τ (τ , s ) − u3 (τ , s ) + u(τ , s ) ds. (3)

For simplicity, let



ϕ ( τ , ξ , u ) = uτ τ ( τ , ξ ) − u3 ( τ , ξ ) + u ( τ , ξ ) , then Eq. (3) becomes

Eα (t α ) =

∞ 

ϑ =0

t αϑ . (αϑ + 1 )

u (τ , ξ ) − u (τ , 0 ) =

Definition 2.2. If ψ b), b > a, α ∈ [0, 1] and is nondifferentiable, then the Atangana– Baleanu derivative of order α in Riemann-Liouville sense is given as ∈ H1 (a,

ABR α a Dτ

(ψ (τ ) ) =

B (α ) d 1 − α dτ

 τ a

  α ψ (s )Eα − (τ − s )α ds. 1−α

Definition 2.3. Consider 0 < α < 1, the fractional integral operator of order α in the Atangana–Baleanu sense is written as

×

 ξ 0

α (1 − α ) (ϕ (τ , ξ , u ) ) + B (α ) B(α )(α )

(ξ − s )α−1 (ϕ (τ , s, u ) )ds.

Now we prove that ϕ (τ , ξ , u) satisfies the Lipschitz condition provided that the function u(τ , ξ ) has an upper bound. So, if u is bounded above then, we have

τ , ξ , u) − ϕ (τ , ξ , ρ ) ϕ (  = u τ τ ( τ , ξ ) − ρτ τ ( τ , ξ ) + u 3 ( τ , ξ )  −ρ 3 (τ , ξ ) + u(τ , ξ ) − ρ (τ , ξ ).

136

A. Prakash and H. Kaur / Chaos, Solitons and Fractals 124 (2019) 134–142

By applying the triangular inequality of norm on the above equation, we come to the following result:

ϕ (τ ,ξ , u) − ϕ (τ , ξ , ρ) ≤ (uτ τ − ρτ τ )(τ , ξ )

+  u3 − ρ 3 (τ , ξ ) + (u − ρ )(τ , ξ ) 

 = (uτ τ − ρτ τ ) + (u − ρ ) u2 + uρ + ρ 2  + (u − ρ )

≤ λ2 (u − ρ ) + a2 + b2 + ab (u − ρ ) + (u − ρ )

≤ λ2 + a2 + b2 + ab + 1 (u − ρ ).

Proof. If u(τ , ξ ) is bounded, ϕ (τ , ξ , u) satisfies the Lipschitz condition and using Eq. (7), we deduce



ψn (τ , ξ ) ≤

ϕ (τ , ξ , u) − ϕ (τ , ξ , ρ ) ≤ μ(u − ρ ). Hence, the Lipschitz condition is satisfied. If in addition 0 <

Therefore, we have

 1 − α Hn (τ , ξ ) =   B(α ) {ϕ (τ , ξ , u) − ϕ (τ , ξ , un−1 )}

  ξ  α α −1 + (ξ − s ) (ϕ (τ , s, u) − ϕ (τ , s, un−1 ))ds  B(α )(α ) 0 1−α ≤ μu(τ , ξ ) − un−1 (τ , ξ ) B (α ) α + μξ α u(τ , ξ ) − un−1 (τ , ξ ). B(α )(α + 1 )

To examine the existence of the solution, we construct the iterative formula as

(4)

Repeating the process, we get



with the initial condition

Hn (τ , ξ ) ≤

u(τ , 0 ) = f (τ ) = u0 (τ , ξ ).

It is noticeable that un (τ , ξ ) = From Eq. (5), we get

n

i=0



Hn (τ , ξ ) ≤

(5)

(6)

  1 − α   ψn (τ , ξ ) ≤   B(α ) {ϕ (τ , ξ , un−1 ) − ϕ (τ , ξ , un−2 )}    ξ   α α −1  + ξ − s ϕ τ , s, u − ϕ τ , s, u ds ( ) ( ( ) ( ) ) n −1 n −2  B(α )(α ) 0  1−α ≤ μun−1 (τ , ξ ) − un−2 (τ , ξ ) B (α )  ξ α + μ (ξ − s )α−1 un−1 (τ , s ) − un−2 (τ , s )ds. B(α )(α ) 0

Since the kernel ϕ (τ , ξ , u) satisfies the Lipschitz condition, we arrive at the following result:

+

α

B(α )(α )

1−α μψn−1 (τ , ξ ) B (α )  ξ μ (ξ − s )α−1 ψn−1 (τ , s )ds

(7)

0

Theorem 3.1. The solution of the given model exists if we can find ξ 0 which satisfies the following condition



1−α α μ+ μξ0 α B (α ) B(α )(α + 1 )

< 1.

1−α α + ξ0 α B (α ) B(α )(α + 1 )

μn+1 a.

n+1 μn+1 a.

Theorem 3.2. The fractional model given by Eq. (1) has a unique solution if the condition given below holds



1−

Further, on employing the triangular inequality of norm on Eq. (6), it yields

ψn (τ , ξ ) ≤

n+1

As n → ∞, we get Hn (τ , ξ ) → 0. Hence the existence of the solution is proved. 

ψi ( τ , ξ ).

ψn (τ , ξ ) = un (τ , ξ ) − un−1 (τ , ξ ).

1−α α + ξα B (α ) B(α )(α + 1 )

Then at ξ = ξ0 , we have

The difference between succeeding terms is represented as

ψn (τ , ξ ) = un (τ , ξ ) − un−1 (τ , ξ ) 1−α = {ϕ (τ , ξ , un−1 ) − ϕ (τ , ξ , un−2 )} B (α )  ξ α + (ξ − s )α−1 (ϕ (τ , s, un−1 ) B(α )(α ) 0 − ϕ (τ , s, un−2 ) )ds.

u(τ , 0 ).

u(τ , ξ ) − u(τ , 0 ) = un (τ , ξ ) − Hn (τ , ξ ).

λ2 + a2 + b2 + ab + 1 ≤ 1, then it is also a contraction.

(1 − α ) ( ϕ ( τ , ξ , un ) ) B (α )  ξ α + (ξ − s )α−1 (ϕ (τ , s, un ) )ds, B(α )(α ) 0

n

 Therefore, the function un (τ , ξ ) = ni=0 ψi (τ , ξ ) exists and is smooth. Next, we show that this function is the solution of fractional nonlinear Cahn–Allen equations. Suppose

Taking μ = λ2 + a2 + b2 + ab + 1, where u and ρ are bounded functions such that u ≤ a, ρ ≤ b , then

un+1 (τ , ξ ) =

1−α α μ+ μξ0 α B (α ) B(α )(α + 1 )

α (1 − α ) μ− μξ α B (α ) B(α )(α + 1 )

> 0.

(8)

Proof. To prove the uniqueness of the solution for the mathematical model given by Eq. (1), we suppose that there exists another solution u1 (τ , ξ ) for given fractional model, then

u ( τ , ξ ) − u1 ( τ , ξ ) =

(1 − α ) ( ϕ ( τ , ξ , u ) − ϕ ( τ , ξ , u1 ) ) B (α )  ξ α + (ξ − s )α−1 B(α )(α ) 0 × (ϕ (τ , s, u ) − ϕ (τ , s, u1 ) )ds.

Taking norm

(1 − α ) u(τ , ξ ) − u1 (τ , ξ ) ≤ ϕ (τ , ξ , u) − ϕ (τ , ξ , u1 ) B (α )  ξ    α α −1  + (ξ − s ) (ϕ (τ , s, u ) − ϕ (τ , s, u1 ))ds  B(α )(α )  0 (1 − α ) ≤ μu(τ , ξ ) − u1 (τ , ξ ) B (α ) α + μξ α u(τ , ξ ) − u1 (τ , ξ ). B(α )(α + 1 ) This gives



α (1 − α ) μ− μξ α ≤ 0. u(τ , ξ ) − u1 (τ , ξ ) 1 − B (α ) B(α )(α + 1 ) (9)

A. Prakash and H. Kaur / Chaos, Solitons and Fractals 124 (2019) 134–142

If the condition in Eq. (8) holds, then combining the Eqs. (8) and (9), we have

u(τ , ξ ) − u1 (τ , ξ ) = 0.

p2 : u2 (τ , ξ ) = −L−1

Then

p3 : u3 (τ , ξ ) = −L−1

u(τ , ξ ) = u1 (τ , ξ ), 

which verifies the uniqueness of the solution. 4. Analysis of the proposed scheme

ABC

Dαξ u(τ , ξ ) + Ru(τ , ξ ) + Nu(τ , ξ ) = f (τ , ξ ), 0 < α ≤ 1

(10)

with the condition

u(τ , 0 ) = f (τ ), where

(11)

ABC Dα u (τ , ξ )

ξ

represents Atangana–Baleanu derivative of or-

der α of u(τ , ξ ) in Caputo’s sense, R stands for the linear differential operator, N represents the nonlinear differential operator and f(τ , ξ ) is the source term. Firstly, exerting Laplace transform operator (denoted throughout this study by L) w.r.t time variable ξ on Eq. (10) and after simplifying with the help of Theorem 2.1, it yields



u (τ , 0 ) sα + α ( 1 − sα ) L[u(τ , ξ )] − + s sα L[Ru(τ , ξ ) + Nu(τ , ξ ) − f (τ , ξ )] = 0.



(12)

Taking inverse Laplace transform on both sides of Eq. (12), it gives

u (τ , ξ ) = L

 −1 u (τ , 0 ) s

−L

+

 sα + α 1 − sα  ( ) sα

 sα + α 1 − sα  ( ) −1 sα



L[ f (τ , ξ )]



L[Ru(τ , ξ ) + Nu(τ , ξ )] . (13)

∞ 

sα

 sα + α 1 − sα  ( )

pi ui (τ , ξ ),

sα



L{[Ru1 (τ , ξ )] + H1 (u )} ,



L{[Ru2 (τ , ξ )] + H2 (u )} ,

in this pattern,

 s α + α 1 − s α  ( ) sα



L{[Run (τ , ξ )] + Hn (u )} .

Therefore, the series solution is given by

N u(τ , ξ ) = lim p→1 limN→∞ pi ui (τ , ξ ) i=0 N = lim ui (τ , ξ ). N→∞

(16)

i=0

Theorem 4.1. Let ui (τ , ξ ) and u(τ , ξ ) be defined in Banach space (C[0, 1], .). If ∃ 0 < γ < 1, such that ui+1 (τ , ξ ) ≤  γ ui (τ , ξ ), ∀i ∈ N then the HPTT solution ∞ i=0 ui (τ , ξ ) converges to the solution u(τ , ξ ) of the time-fractional CAE (1) with AB fractional derivative. Proof. Let {si } be a sequence of partial sums of the series (16) and we require to show that si (τ , ξ ) is a Cauchy’s sequence in the Banach space. Then,

si+1 − si  = ui+1  ≤ γ ui  ≤ γ 2 ui−1  ≤ . . . ≤ γ i+1 u0 . For any i, j ∈ N, i ≥ j,

  

 si − s j  = (si − si−1 ) + (si−1 − si−2 ) + . . . + s j+1 − s j    ≤ (si − si−1 ) + (si−1 − si−2 ) + . . . + (s j+1 − s j ) ≤ γ i u0 (τ , ξ ) + γ i−1 u0 (τ , ξ ) + . . . + γ j+1 u0 (τ , ξ )





≤ γ j+1 1 + γ + γ 2 + . . . + γ i− j−1 u0 (τ , ξ )

Express u(τ , ξ ) by an infinite series of components given as

u (τ , ξ ) =

 sα + α 1 − sα  ( )

pn+1 : un+1 (τ , ξ ) = −L−1

Consider the nonlinear differential equation of order α in Atangana–Baleanu sense

137

(14)

≤

1 − γ i− j j+1 γ u0 (τ , ξ ), 1−γ

i=0

Since 0 < γ < 1, we have 1 − γ i− j < 1, then

and the nonlinear term can be decomposed as

Nu(τ , ξ ) =

∞ 

pi Hi (u ),

(15)

i=0

where Hi (u) represents the homotopy polynomial and given in the following form:

Hi (u ) =

 ∞  1 ∂i j N p u , i = 0, 1, 2, 3, . . . , j j=0 i! ∂ pi p=0

i=0

pi ui (τ , ξ ) = L−1



−p L

 u τ, 0 ( ) s

+

 sα + α 1 − sα  ( ) sα

   sα + α 1 − sα  ∞  ( ) −1 L

sα

R

pi ui (τ , ξ ) +

i=0



∞ 

    j  γ j+1   ui (τ , ξ ) ≤ u0 (τ , ξ ). u(τ , ξ ) − 1   ( −γ) i=0

 pi Hi (u )

.

i=0

Equating on both sides the coefficients of like powers of p, it gives the following components

p0 : u0 (τ , ξ ) = L−1

 u τ, 0 ( )

p1 : u1 (τ , ξ ) = −L−1

s

+

 sα + α 1 − sα  ( ) sα

 sα + α 1 − sα  ( ) sα

So, si − s j  → 0 as i, j → ∞ as u0 is bounded. Thus {si } is a Cauchy sequence in the Banach space and hence convergent.  Therefore, ∃ u(τ , ξ ) ∈ B such that ∞  i=0 ui (τ , ξ ) = u (τ , ξ ).

truncated error of the HPTT solution (16) of the time-fractional CAE with AB derivative (1) is estimated as

L[ f (τ , ξ )]



(17)

Theorem 4.2. If there exists 0 < γ < 1 in such a way that ui+1 (τ , ξ ) ≤ γ ui (τ , ξ ), ∀i ∈ N, then the maximum absolute

using Eqs. (14) and (15) in Eq. (13), we get ∞ 

j+1   s i − s j  ≤ γ u0 (τ , ξ ). 1−γ



L[ f (τ , ξ )] ,



L{[Ru0 (τ , ξ )] + H0 (u )} ,

Proof. By using Eq. (17) and from Theorem 4.1, as i → ∞, si → u(τ , ξ ), we get

    j  γ j+1   ui (τ , ξ ) ≤ u0 (τ , ξ ). u(τ , ξ ) −   (1 − γ ) i=0

138

A. Prakash and H. Kaur / Chaos, Solitons and Fractals 124 (2019) 134–142



5. Applications of AB derivative with fractional order to the time-fractional CAE

u2 ( τ , ξ ) =

In this section, we examine the nonlinear Cahn–Allen equations involving the AB derivative with fractional order by using homotopy perturbation technique combined with Laplace transform.

×

1



1+e

√τ 2

,

(18)

The exact solution as derived in [16,26] is 1+e

( √τ

1

2

−

3ξ ) 2

1





s 1+e

√τ 2

 +

 sα + α 1 − sα   ( )

u3 ( τ , ξ ) =



×

(19) Exerting the inverse of Laplace transform on Eq. (19), we have 1

u (τ , ξ ) =



1+e

√τ 2

 + L

−1

 sα + α 1 − sα   ( )

3



L uτ τ − u + u

sα

.

(20)

pn un (τ , ξ )

n=0

=

1





1+e + pL

√τ 2

 + pL

−1

   sα + α 1 − sα  ∞  ( ) L

sα

   sα + α 1 − sα  ∞  ( ) −1 sα

L −

pn Hn (u ) +

n=0

∞ 

ττ

pn un (τ , ξ )

,

n=0

3

Evaluating the components of homotopy polynomials, the first few components are given as

H0 (u ) = u0 3 ,



√τ

 5





81 e



√τ 2



4 1+e





45 e

+

9 e



√τ 2

 3

⎫ ⎪ ⎪ ⎪ ⎬



1+e

 2

√τ 2

 3

+





2 1+e

√τ 2

√τ 2

 3  6

√τ 2

 3

 4



⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

 4

⎪ ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ √τ ⎪ 27e 2 ⎪ +  τ  6 ⎪ ⎪ √ ⎭ 2 1+e 2 2 1+e

√τ 2

117 e



− 27e 

2



√τ 2

4 1+e

 6

√τ





 5

√τ 2





261 e

√τ

2



(1 − α )3 + α (1 − α ) 1 + α + 2α 2 3 α 2 ( 1 − α )ξ 2 α (2α + 1)

⎫ ⎪ ⎪ ⎪ ⎬

ξα (α + 1 )

α 3 (2α + 1)ξ 3α + (3α + 1)

  5 , ⎪ ⎪ √τ ⎪ ⎪ ⎪ ⎪ ⎩4 1 + e 2 ⎭

following the same procedure, the remaining iterates un (τ , ξ ) can be calculated easily. Hence, the numerical solution is written as

u ( τ , ξ ) = u0 ( τ , ξ ) + u1 ( τ , ξ ) + u2 ( τ , ξ ) + · · · .

1



1+e

− √τ

,

(21)

2

The exact solution as derived in [14] is 1+e

1 . (− √τ − 32ξ ) 2

Working on the same steps as in the first model problem, and evaluating the components of homotopy polynomials as

H1 (u ) = 3u0 2 u1 ,

H0 (u ) = u0 3 ,

H2 (u ) = 3u0 2 u2 + 3u1 2 u0 ,

H1 (u ) = 3u0 2 u1 ,

Solving the above equations, we have the following successive components of the solution

H2 (u ) = 3u0 2 u2 + 3u1 2 u0 ,

1



1+e



 2

3α (1 − α )2 ξ α 3α (1 − α )ξ 2α α 3 ξ 3α + + (α + 1 ) (2α + 1) (3α + 1)

⎧   2 ⎪ √τ ⎪ ⎪ 27 e 2 ⎨

u (τ , 0 ) =

n=0

u0 ( τ , ξ ) =

√τ 2

Example 5.2. In the second model problem, we consider the fractional model (1) with the initial condition given as

p Hn (u ) = u . n

×

p un ( τ , ξ )





9 e

⎪ 1+e 2 ⎪ ⎪ ⎪   2 ⎪ ⎪ √τ ⎪ ⎪ 81 e 2 ⎪ ⎪ ⎪ +  τ  5 − ⎪ ⎩ 2 1+e √2

+

 

where ∞ 

−



n

n=0





⎧   ⎪ √τ ⎪ ⎪ 27e 2 ⎪   2 − ⎪ √τ ⎪ ⎪ 8 1+e 2 ⎪ ⎪ ⎪   4 ⎪ ⎪ ⎨ 90 e √τ2

−

Working on the aforesaid procedure of the proposed scheme, Eq. (20) becomes ∞

9e

(1 − α )3 +

L uτ τ − u3 + u .

sα

√τ 2



.

Firstly, applying the Laplace transform operator on the Cahn– Allen equations with AB fractional derivative, we have

L[u(τ , ξ )] =



  2 −   3 +   4 ⎪ ⎪ √τ √τ √τ ⎪ ⎪ ⎪ ⎪ 2 1+e 2 2 1+e 2 ⎩4 1 + e 2 ⎭ ⎧ ⎫ ⎪ ⎪   ⎪ ⎪ τ ⎪ ⎪

⎨ √ ⎬ α 2 2α 2 2 α 1 − α ξ α ξ 9 e ( ) − ( α − 1 )2 + +   4 , (α + 1 ) (2α + 1) ⎪ ⎪ ⎪ √τ ⎪ ⎪ ⎪ ⎩2 1 + e 2 ⎭

Example 5.1. In the first model problem, we consider the fractional model (1) with the initial condition given as

u (τ , 0 ) =

⎧ ⎪ ⎪ ⎪ ⎨

2α (1 − α )ξ α α 2 ξ 2α + (α + 1 ) (2α + 1)

(α − 1 )2 +

√τ

we get the following successive components of solution

,

αξ α u1 ( τ , ξ ) = 1 − α + (α + 1 )

1

u0 ( τ , ξ ) =

2





3e

√τ 2

2 1+e



√τ 2



1+e

  2 ,



− √τ

,

2

αξ α u1 ( τ , ξ ) = 1 − α + (α + 1 )





3e

− √τ

2 1+e



2



− √τ

2

 2 ,

A. Prakash and H. Kaur / Chaos, Solitons and Fractals 124 (2019) 134–142

u2 ( τ , ξ ) =

( α − 1 )2 +

×

⎧ ⎪ ⎪ ⎪ ⎨



9e

2 α ( 1 − α )ξ α α 2 ξ 2α + (α + 1 ) (2α + 1) 

9 e

− √τ 2

 2





9 e

− √τ 2

⎫ ⎪ ⎪ ⎪ ⎬

 3

  2 −   3 +   4 ⎪ ⎪ − √τ − √τ − √τ ⎪ ⎪ ⎪ ⎪ 2 2 2 2 1+e 2 1+e ⎩4 1 + e ⎭



2α (1 − α )ξ α α 2 ξ 2α + (α + 1 ) (2α + 1)

(α − 1 )2 +

−

u3 ( τ , ξ ) =





− √τ 2

139

(1 − α )3 +

⎧ ⎪ ⎪ ⎪ ⎨

− √τ 2

27e







−

+





 2 −

)

− √τ 2







81 e

− √τ 2





4 1+e

 4

− √τ 2

⎪ ⎪ ⎪ ⎩2 1 + e



 5 +

− √τ

2





 5 −







 3 +

− √τ 2











− √τ 2



4 1+e  5

− √τ 2

117 e 2 1+e

261 e

− √τ 2

1+e

 2

 2

− √τ 2

45 e

− √τ 2

81 e

2 1+e





− √τ 2



1+e

9e







− √τ 2

 4

2

− √τ 2

90 e



3α ( 1 − α ) ξ α 3α (1 − α )ξ 2α α 3 ξ 3α + + (α + 1 ) (2α + 1) (3α + 1)



⎪ ⎪ ⎪ ⎩ 8 (1 + e

⎧ ⎪ ⎪

⎪ ⎨



 3

− √τ

− √τ 2



2 1+e



 6 +

27e

2 1+e

2

− (1 − α )3 + α (1 − α ) 1 + α + 2α 2







− √τ 2

− √τ 2



 4



 6 −

,

⎪ ⎪ ⎪ ⎭

 3

− √τ 2

27e

⎫ ⎪ ⎪ ⎪ ⎬

 4



− √τ 2

 6

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

3α 2 (1 − α )ξ 2α ξα + (α + 1 ) (2α + 1)

⎧  2 ⎫  ⎪ ⎪ − √τ ⎪ ⎪ 2 ⎪ 27 e ⎪

⎨ ⎬ α 3 (2α + 1)ξ 3α +   5 , (3α + 1) ⎪ ⎪ − √τ ⎪ ⎪ ⎪ ⎪ 2 ⎩4 1 + e ⎭

following the same procedure, the remaining iterates un (τ , ξ ) can be calculated easily. Hence, the numerical solution is written as

u ( τ , ξ ) = u0 ( τ , ξ ) + u1 ( τ , ξ ) + u2 ( τ , ξ ) + . . . 6. Numerical simulation results Fig. 1(a–c) represent the exact solution, approximate solution and absolute error respectively for Eq. (1) with condition (18). The graphical representation clearly shows the accuracy of the proposed numerical approach. The above graphical results are derived by 3rd order approximation and the accuracy can be enhanced by increasing the order of approximation. It can be noticed from Fig. 2 that model (1) depends notably to the fractional order. Figs. 2 and 3 demonstrates the nature of u(τ , ξ ) w.r.t time for varying fractional order in AB and Caputo’s sense respectively for Example 5.1 and we clearly observe that as α increases u(τ , ξ ) decreases. From Tables 1 and 2, it is clear that the results obtained by HPTT using AB fractional derivative are very near to the exact solution. We conclude from the graphical representations that at α = 1, Caputo fractional derivatives and AB derivative

Fig. 1. (a) Exact solution when α = 1, for Example 5.1. (b) Approx. solution when α = 1, for Example 5.1. (c) Absolute error when α = 1, for Example 5.1.

140

A. Prakash and H. Kaur / Chaos, Solitons and Fractals 124 (2019) 134–142 Table 1 A comparative study between approximate solution and the exact solution for u(τ , ξ ) for different values of ξ when τ = 1 and α = 1, for Example 5.1.

ξ

Exact solution

Approximate solution

|uexa. (τ , ξ ) − uapp. (τ , ξ )|

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

0.3305703067 0.3309023313 0.3312345245 0.3315668859 0.3318994153 0.3322321125 0.3325649770 0.3328980089 0.3332312077 0.3335645733

0.3305703066 0.3309023318 0.3312345256 0.3315668885 0.3318994204 0.3322321213 0.3325649910 0.3328980298 0.3332312376 0.3335646142

7.62 × 10−12 3.81 × 10−10 1.12 × 10−9 2.62 × 10−9 5.09 × 10−9 8.74 × 10−9 1.40 × 10−8 2.09 × 10−8 2.98 × 10−8 4.09 × 10−8

Table 2 A comparative study between approximate solution and exact solution for u(τ , ξ ) for different values of ξ when τ = 1 and α = 1, for Example 5.2.

Fig 2. Numerical behavior of versus time when in AB sense, for Example 5.1.

ξ

Exact solution

Approximate solution

|uexa. (τ , ξ ) − uapp. (τ , ξ )|

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

0.6700932362 0.6704247541 0.6707561025 0.6710872810 0.6714182899 0.6717491282 0.6720797963 0.6724102936 0.6727406202 0.6730707758

0.6700932361 0.6704247542 0.6707561034 0.6710872835 0.6714182946 0.6717491368 0.6720798101 0.6724103144 0.6727406497 0.6730708160

1.0 × 10−10 1.0 × 10−10 9.0 × 10−10 2.50 × 10−9 4.70 × 10−9 8.60 × 10−9 1.38 × 10−8 2.08 × 10−8 2.95 × 10−8 4.02 × 10−8

Fig. 3. Numerical behavior of u(τ , ξ ) versus time ξ when τ = 1 in Caputo’s sense, for Example 5.1.

with fractional order show exactly the same nature. Both fractional models are reduced to the classical model for α = 1. Fig. 5(a– c) represent the exact solution, approximate solution and absolute error respectively for Eq. (1) with condition (21). Figs. 6 and 7 demonstrates the nature of u(τ , ξ ) w.r.t time for varying fractional order in AB and Caputo sense respectively for Example 5.2. We clearly observe from graphical representations that fractional order significantly affects the distance and on changing the value of α , we get very interesting results. In Fig. 7, initially on increasing the value of α distance decreases but after some time on increasing the value of α , the value of distance increases. Figs. 4

Fig. 4. Comparison of the approximate solution and the exact solution when τ = 1, α = 1, for Example 5.1.

and 8 represent the graphical comparison of the approximate solution obtained by using AB derivative with fractional order, Caputo fractional derivative and the exact solution when τ = 1, α = 1 for Examples 5.1 and 5.2, respectively. It is noticeable that for smaller values of time the numerical solution derived by both fractional derivatives coincides with the exact solution and the corresponding error increases as time increases.

A. Prakash and H. Kaur / Chaos, Solitons and Fractals 124 (2019) 134–142

141

Fig. 6. Numerical behavior of u(τ , ξ ) versus time ξ when τ = 1 in AB sense, for Example 5.2.

Fig. 7. Numerical behavior of u(τ , ξ ) versus time ξ when τ = 1 in Caputo’s sense, for Example 5.2.

7. Concluding remarks

Fig. 5. (a) Exact solution when α = 1, for Example 5.2. (b) Approx. solution when α = 1, for Example 5.2. (c) Absolute error when α = 1, for Example 5.2.

One of the key objectives of this work is to check the potential of AB derivative with a fractional order to study the time-fractional Cahn–Allen equation to account for the memory effect of certain materials or the sub diffusive process in heterogeneously distributed media. It can be analyzed from the numerical results that the fractional phase-field models describe these processes more accurately than an integer order model. The HPTT is adapted to efficiently deal with the significantly increased memory require-

142

A. Prakash and H. Kaur / Chaos, Solitons and Fractals 124 (2019) 134–142

Fig. 8. Comparison of the approximate solution and the exact solution when τ = 1, α = 1, for Example 5.2.

ments and computational complications which arise because of the nonlocal behavior of the time-fractional models. Finally, we arrive at the conclusion that the investigated fractional model of Cahn– Allen equation and other similar dynamical models associated with Atangana–Baleanu fractional derivative are very effective to analyze the natural phenomena and the suggested scheme is very innovative and strong computational approach with high accuracy to solve nonlinear fractional models. Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Prog Fract Differ Appl 2015;1:73–85. [2] Atangana A, Baleanu D. Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer. J Eng Mech 2016:1943–7889 10,1061/(ASCE) EM0 0 01091.

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