- Email: [email protected]
Analysis of the fractional diffusion equations described by Atangana-Baleanu-Caputo fractional derivative
Chaos, Solitons and Fractals 127 (2019) 158–164
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Analysis of the fractional diffusion equations described by Atangana-Baleanu-Caputo fractional derivative Ndolane Sene a,∗, Karima Abdelmalek b a
Laboratoire Lmdan, Département de Mathématiques de la Décision, Université Cheikh Anta Diop de Dakar, Faculté des Sciences Economiques et Gestion, Dakar Fann BP 5683, Senegal b Laboratory LAMIS, Department of Mathematics and Informatiques, University Larbi Tebessi, Tebessa 12002, Algeria
a r t i c l e
i n f o
Article history: Received 14 May 2019 Revised 20 June 2019 Accepted 26 June 2019
Keywords: Fractional diffusion equations Mean square displacement Atangana-Baleanu fractional derivative operator
a b s t r a c t In this paper, we analyze two types of diffusion processes obtained with the fractional diffusion equations described by the Atangana-Baleanu-Caputo (ABC) fractional derivative. The mean square displacement (MSD) concept has been used to discuss the types of diffusion processes obtained when the order of the fractional derivative take certain values. Many types of diffusion processes exist and depend to the value of the order of the used fractional derivatives: the fractional diffusion equation with the subdiffusive process, the fractional diffusion equation with the superdiffusive process, the fractional diffusion equation with the ballistic diffusive process and the fractional diffusion equation with the hyper diffusive process. Here we use the Atangana-Baleanu fractional derivative and analyze the subdiffusion process obtained when the order of ABC α is into (0,1) and the normal diffusion obtained in the limiting case α = 1. The Laplace transform of the Atangana-Baleanu-Caputo fractional derivative has been used for getting the mean square displacement of the fractional diffusion equation. The central limit theorem has been discussed too, and the main results illustrated graphically. © 2019 Published by Elsevier Ltd.
1. Introduction In the literature, many researchers prove the applicability of the Atangana-Baleanu-Caputo derivative in real word problems. Notably in physics [30,33], in epidemiology, in statistics and probability [8–10,16], in science and engineering, and in many other fields. Recently, it was introduced in the literature the discrete forms of the fractional derivatives with nonsingular kernels, and many applications were proposed with these new issues. Many investigations related the discrete form of the Atananga-Baleanu fractional derivative can be found in [1–4]. Many applications of the fractional order derivatives in the fractional diffusion equations were recently performed. Nowadays, there exist many types of fractional diffusion equations. We enumerate some of them. The fractional diffusion reaction equation [29], the fractional subdiffusion equation [12], the fractional dispersion equation [18] and many others. Many investigations on the fractional diffusion equations concern the analytical solutions [28,31,32], the semi-analytical solutions [29], the approximate solutions [11,13] and the numerical solutions [18]. Many methods ex-
∗
Corresponding author. E-mail addresses: [email protected] (N. Sene), [email protected] (K. Abdelmalek). https://doi.org/10.1016/j.chaos.2019.06.036 0960-0779/© 2019 Published by Elsevier Ltd.
ist in the studies of the solutions of the fractional diffusion equations. We enumerate certain of them. In [20], Kader has proposed the numerical solution of the fractional diffusion equation using the Caputo fractional derivative. In [16], Henry et al. have proposed an introduction of modeling the fractional diffusion equation described by the Caputo fractional derivative. The statistical properties of the fractional diffusion equations were also provided in this paper [16]. In [21], Tasbozan has proposed the numerical solution fractional diffusion equation for force-free case. In [15], Hashemi has solved the time fractional diffusion equation using a lie group integrator. In [19], Al-Refai et al. have proposed a complete analysis of the fractional diffusion equations with Atangana-Baleanu-Caputo fractional derivative. In [28,30], Sene has introduced the solution of the fractional diffusion equations and the stokes’ equation using the Laplace and Fourier transform methods. For recent investigations on the subdiffusive and superdiffusive processes, see in Owolabi et al. [23–27]. Motivated by the fact to understand, what are the types of diffusion generated by the new fractional order derivatives, in this paper, we propose to study the probability aspect of the fractional diffusion equations. We particularly use the AtanangaBaleanu-Caputo fractional derivative. The objective of the paper is to study the Mean Square Displacement of the fractional diffusion equation to analyze the diffusion processes generated by the
N. Sene and K. Abdelmalek / Chaos, Solitons and Fractals 127 (2019) 158–164
Atangaga-Baleanu-Caputo fractional order derivative. Note we use MSD [8–10] to analyze the following diffusion processes: normal diffusion process, subdiffusion process. Other diffusion processes will be recalled as superdiffusion process, hyperdiffusion process, ballistic process, Richardson diffusion process. This paper makes a connection between fractional calculus and statistical physics. In Section 2, we propose the definitions of the fractional derivatives operators with Mittag-Leffler kernels known as the Atangana-Baleanu-Caputo fractional derivative and the AtanganaBaleanu-Riemann fractional derivative. In Section 3, we propose constructive equations. In Section4, we study the existence and the uniqueness of the solution of the proposed model. In Section 5, we propose the procedure for getting the analytical solution. In Section 6, we propose the calculation of the MSD. In Section 7, we discuss the MSD and analyze the diffusion processes. In Section 8, we focus the application of the central limit Theorem in our diffusion model. We give the conclusions and perspectives in Section 9. 2. Derivative operators with non-singular kernels In this section, we recall the definitions of the existing fractional derivative operators, their Laplace transforms and also introduce the Mittag-Leffler function. We begin with the Mittag-Leffler function. The Mittag Leffler function with two parameters is represented as the following series
Eα ,β (z ) =
∞ k=0
zk , (α k + β )
(1)
where α > 0, β ∈ R and z ∈ C. The Atangana-Baleanu-Riemann operator [6,7] for a given function u, of order α ∈ (0, 1] is defined by
DABR α u (y, t ) =
B (α ) d 1 − α dt
0
t
u ( y ( s ), s )Eα −
α
1−α
(t − s )α ds,
(2)
for all t > 0, and the function (. . . ) represents the Euler Gamma function. The Atangana-Baleanu-Caputo derivative [6,7] for a function u, of order α ∈ (0, 1] is defined as the following form
DABC α u (y, t ) =
B (α ) 1−α
t 0
u ( y ( s ), s )Eα −
α
1−α
(t − s )α ds,
(3)
for all t > 0, where the function (. . . ) is Euler Gamma function. The Riemann-Liouville integral [11] for a given function u, of order α ∈ (0, 1] is represented as the form
IαRL u(y, t ) =
1
(α )
a
t
(t − s )α−1 u(y(s ), s )ds,
(4)
for all t > 0, where the function (. . . ) represents the Gamma function. The Atangana-Baleanu integral [6,7] for a given function u, of order α ∈ (0, 1] is defined as the form
IαAB u(y, t ) =
1−α α RL u(y, t ) + I u(y, t ), B (α ) B (α ) α
(5)
for all t > 0. We recall the Laplace transform of the Atangana-BaleanuCaputo derivative [6,7], and the Atangana-Baleanu-Riemann derivative represented as the following forms
L DABC α u (s ) =
B(α ) sα L{u}(s ) − sα −1 u(0 ) , 1−α sα + 1−αα
Due to space limitation, we do not recall the other fractional derivatives. The readers can refer to the following references for more information related to the Riemann-Liouville fractional derivative [14,17], the Caputo-Liouville fractional derivative [14,17], the certain generalized fractional derivatives [17], discrete version of the fractional derivatives can be found in [1–4]. 3. Formalistic model In this section, we present formalistic model introduced by Hristov in [11]. The model is obtained using Fick’s first and second equations. Note Fick’s equations are used in many problems to modeled the fractional diffusion equations. The Fick first equation when the diffusion coefficient is unit is described by
q=−
∂u . ∂y
(8)
Fick’s first equation measures the flux into the system. It represents the density in the diffusion equation and is the subject of many works in the diffusion equation. The second Fick’s equation when we use the Atangana-Baleanu-Caputo fractional derivative is given by ABC t Dα u
=−
∂q . ∂y
(9)
The fractional diffusion equation in Hristov formalistic sense, when we use Atangana-Baleanu-Caputo fractional derivative, is expressed in the following form ABC t Dα u
=
∂ 2u . ∂ y2
(10)
In this paper, the statistical aspect of the fractional diffusion equation will be investigated. For this issue, a statistical boundary condition is used. That is u(x, 0 ) = δ (x ) and u(±∞, t ) = 0. Note that in our study, the following condition is held for the distribution
+∞ −∞
u(x, t )dx = 1.
(11)
The following formula can obtain the mean square displacement of the fractional diffusion equation
2 x
=
+∞ −∞
x2 u(x, t )dx = lim − q→0
d2 u¯ (q, s ) , dq2
(12)
where u¯ represents the Laplace transformation of the function u. The variable q and s came respectively form the Fourier, and the Laplace transform. Note the MSD [8–10] is linear in time. We will use it to analyze the diffusion of the fractional differential equation described by the Atangana-Baleanu-Caputo fractional derivative. • Firstly, we propose the analytical solution of the fractional differential equation Eq. (10) under statistical boundary conditions described above. • Secondly, we get the mean square displacement of the fractional diffusion equation using Eq. (12). • At last, we discuss the nature of the processes using the order α. 4. Existence of the solution using picard-Lindelof method
(6)
In this section, we use Picard-Lindelof method for proving the existence and the uniqueness of the solution. The sketch of the proof is not new and can be found in [5,22]. The main objective is to define Picard-Lindelof operator and to show this operator is well definite and is a contraction. From the assumption, v is bounded the function expressed by
(7)
(y, t, v ) =
Here L represents the classical Laplace transform.
B ( α ) s α L {u } ( s ) L DABR , α u (s ) = 1 − α sα + 1−αα
159
∂ 2 v(y, t ) . ∂ y2
(13)
160
N. Sene and K. Abdelmalek / Chaos, Solitons and Fractals 127 (2019) 158–164
defines a contraction. That is
(y, t, v ) − (y, t, u ) ≤ v − u,
(14)
where we define the following norm
f (t ) =
sup t ∈[t −a,t +a]
| f (t )|.
1−α α (y, t, v ) + B (α ) B(α )(α )
t 0
(t − s )α−1 (y, t, v )ds. (16)
We prove the Picard’s operator is well posed. For that, we use the norm defined at the beginning of this section; we have the following relations
1 − α (y, t, v ) Pv(t ) − v(x, 0 ) =
B (α )
t
α + (t − s )α−1 (y, t, v )ds
B(α )(α ) 0 1−α ≤ (y, t, v ) B (α ) t α + (t − s )α−1 (y, t, v )ds B(α )(α ) 0 1 − α aα ≤ + (17) (y, t, v ), B (α ) B(α )(α )
From which we conclude the Picard’s operator P is well posed. The last step is the Picard’s, and Lindelof method consists of proving the operator P defines a contraction. We have the following relation
1 − α (y, t, v ) Pv(t ) − Pu(t ) =
B (α )
t
α α −1 + t − s ( y, t, v ds ( ) )
B(α )(α ) 0
1 − α −
(y, t, u ) B (α )
t
α α −1 + (t − s ) (y, t, u )ds
B(α )(α ) 0 1 − α α a + Pv(t ) − Pu(t ) ≤ B (α ) B(α )(α ) (18) (y, t, v ) − (y, t, u ), From wich under the assumption
1 − α B (α )
+
aα B(α )(α )
< 1,
In this section, we described the method of finding the solution of the fractional diffusion equation described by AtanganaBaleanu-Caputo fractional derivative. The fractional differential equation which we consider in this section is defined by
∂ 2u . ∂ y2
sα −1 B (α ) × = u˜ (q, s ). 2 2 B ( α ) + ( 1 − α )q sα + B(α )+α(q1−α )q2
(19)
(20)
Eq. (20) will be used in many times in our studies. The equation is fundamental to determine the MSD of the fractional diffusion equation described by the Atangana-Baleanu-Caputo fractional derivative. For simplification in the expression, let’s
η=
B (α ) and B ( α ) + ( 1 − α )q2
ξ=
α q2 . B ( α ) + ( 1 − α )q2
Applying both the inverse of the Laplace transform and the Fourier transform to both sides of equation Eq. (20), we obtain
u(x, t ) =
2
π
∞
0
η sin(qx )Eα (−ξ t α )dq.
(21)
We can observe this solution is in good agreement with the classical solution obtained when α = 1. To see that, we make some analysis. From Eq. (20), the Fourier sine transformation and the Laplace transform of the classical diffusion equation obtained with α = 1 is expressed as the following form
u˜ (q, s ) =
1 . s + q2
(22)
Applying both the inverse of Laplace transform and the Fourier sine transform, we have the following relationships
u(x, t ) = =
2
π 2
∞
0
π
∞
exp −q2 t sin(qx )dq
√ 2 exp − q t sin(qx )dq
0
x2 = √ exp − 4 t 4π t 1
,
(23)
It is straightforward √ to see the function u verifies a density distribution. Let’s σ = 2t . Thus the function u can be rewritten as the following form
x2 1 u(x, σ 2 ) = √ exp − 2σ 2 2π σ 2
We can observe u ≥ 0 and satisfies a Gaussian density distribution. Furthermore, for confirmation, we can see the following calculations +∞
−∞
u(x, t )dx =
+∞ −∞
=2 = 1.
5. Solution procedure of fractional diffusion equation
(y, t ) =
B(α ) sα u˜ (q, s ) − sα −1 = −q2 u˜ (q, s ) 1−α sα + 1−αα
the operator P is a contraction. Using Banach fixed Theorem, we can conclude the solution exists and is unique. Finally, our problem in this paper is well definite.
ABC t Dα u
B(α ) sα u˜ (q, s ) − sα −1 u(q, 0 ) = −q2 u˜ (q, s ) 1−α sα + 1−αα
(15)
Let’s the following Picard’s operator Pv: K → K, where K represents functional spaces of continuous functions given by K = [t − a1 , t + a2 ] × [t − b1 , t + b2 ]. In our case, the explicit form of the Picard’s operator is given by the following
P v(t ) =
under boundary condition u(x, 0 ) = δ (x ) and u(±∞, t ) = 0. Furthermore the condition Eq. (11) is satisfied. We apply both the Fourier sine transformation and the Laplace transform to both sides of Eq. (19), we obtain
+∞ 0
x2 exp − √ 4t 4π t 1
dx
2
x 1 exp − √ 4 t 4π t
dx (24)
In Fig. 5, we depict the Gaussian profile for different values of
σ 2 . The profiles decrease when the number σ 2 increases, Fig. 5,
(see arrow direction). It is mean when the time of the diffusion process is long the density of the material decreases. In other words, the solution u converge to zero. We note an applatissement of the Gaussian profile. That means the Person value of the diffusion is less than 3 when the time increase rapidly.
N. Sene and K. Abdelmalek / Chaos, Solitons and Fractals 127 (2019) 158–164
161
Fig. 1. Gaussian profil with increasing σ 2 .
Fig. 2. Gaussian profil with decreasing σ 2 .
In Fig. 5, we depict the Gaussian profile for different values of
σ 2 . The profiles increase when the number σ 2 decreases, see ar-
row direction. It is mean when the time of the diffusion process is low the density of the material increases. In other words, the solution u do not converge to zero. We note non-applatissement of the Gaussian profile. That means the Person value of the diffusion is superior to 3 at the beginning of the diffusion process. 6. Mean square displacement of fractional diffusion equation
calculations. Let’s the function
w(q ) = −2B(α )qsα −1 [(1 − α )sα + α ], it derivative is given by the following relation
w (q ) = −2B(α )sα −1 [(1 − α )sα + α ] Let’s the function
v (q ) =
2 B ( α ) + ( 1 − α )q2 sα + α q2 ,
it derivative is given by the following relation In this section, we analyze the fractional diffusion equation described by the Atangana Baleanu-Caputo fractional derivative. We use the Mean Square Displacement to do our studies. This tool is used in Santos works related to the fractional diffusion equations described by other fractional derivatives. For getting the MSD, it is better and more useful to use Eq. (20). Using the result of Eq. (20), (q,s ) the function du¯dq in term of Laplace transform is given by 2α −1
+ 2B(α )α qsα −1
2B(α )(1 − α )qs du¯ (q, s ) = − 2 dq B ( α ) + ( 1 − α )q2 sα + α q2 2B(α )qsα −1 [(1 − α )sα + α ]
= −
B ( α ) + ( 1 − α )q2 sα + α q2
We calculate the function
d2 u¯ (q,s ) dq2
2 .
v (q ) = 4[(1 − α )qsα + α q] B(α ) + (1 − α )q2 sα + α q2 . From which we obtain the followiing relationships
v ( 0 ) = 0. We obtain the following relationships
d2 u¯ (q, s ) w ( q )v ( q ) − w ( q )v ( q ) = . dq2 v2 ( q )
(26)
The MSD in term of the Laplace transform is given by the following expression
(25)
in term of the Laplace trans-
form using the Eq. (25). We have the following relationship after
2 x
(s ) = lim − ABC
2 x
ABC
q→0
(s ) = −
d2 u¯ (q, s ) dq2
w ( 0 )v ( 0 ) − w ( 0 )v ( 0 ) v2 ( 0 )
162
N. Sene and K. Abdelmalek / Chaos, Solitons and Fractals 127 (2019) 158–164
Fig. 3. MSD when β = 1; 2; 3.
2 x
(s ) = ABC
2 x
ABC
(s ) =
2B(α )sα −1 [(1 − α )sα + α ][B(α )sα ]
2
4 [B ( α ) s α ]
2 (1 − α ) 2α + . B ( α )s B(α )s1+α
(27)
Applying the inverse of Laplace transform to both sides of Eq. (27), thus the MSD of the fractional diffusion equation described by the Atangana-Baleanu fractional derivative is given by
2 x
ABC
(t ) =
2 (1 − α ) 2α α + t B (α ) B (α )
= 2(IαAB )(1 ).
(28)
We Conclude the Mean Square Displacement of the formalistic fractional diffusion equation described by the Mittag-Leffler fractional derivative is equivalent to the Atangana-Baleanu fractional integral of the unit. A vital remark about Atangana Baleanu-Caputo fractional derivative is when the order α = 1, we recover the MSD of the classical diffusion equation expressed in the following form
2 x
classical
(t ) = 2t.
(29)
A second important remark is when the diffusion process is long. In other words, for a long time, (t → ∞) the MSD of the fractional diffusion equation described by the Atangana-BaleanuCaputo derivative can be expressed in a more simple form. That is
2 x
(t ) ≈ ABC
2α α t . B (α )
(30)
In Fig. 7, we represent the behavior of the MSD for the normal diffusion β = 1, the subdiffusion β = 0.5, the superdiffusion β = 1.5 and the hyperdiffusion β = 2.5 processes. In Fig. 7, we represent the behavior of the MSD for the normal diffusion α = 1, the subdiffusion process for the different values of the orders α = 0.25; 0.50; 0.75; 0.85, when the use the AtanganaBaleanu-Caputo fractional derivative. We note when the order α increase the subdiffusion converge to a normal diffusion. In Fig. 7, we compare the MSD for the subdiffusion processes β = α = 0.5 obtained with ordinary derivative and the AtanganaBaleanu-Caputo fractional derivative. In Fig. 7, we represent the behavior of the MSD for the normal diffusion α = 1, and the behavior of the subdiffusion processes for different values of the orders α = 0.25; 0.50; 0.75; 0.85. We consider a long time, and we use the Atangana-Baleanu-Caputo fractional derivative. 8. Central limit theorem and atangana-Baleanu-Caputo derivative In this section, we focus on how we apply the Central Limit Theorem. We stipulate the random variable X admit a Gaussian distribution obtained with Atangana-Baleanu-Caputo derivative. We first calculate the mean of the random variable X in term of Laplace and Fourier transform, we have the following expression
x ABC (s ) = lim i q→0
(31)
Using Eq. (20), we have the following expression
7. Discusion about the mean square displacement In this section, we discuss the diffusion processes depending on the order of the Atangana-Baleanu-Caputo fractional derivative. For the types of diffusion processes, we begin the discussion with the mean square displacement classically given by x2 ≈ tβ . From which we have different types of diffusion processes. When the order β = 1, the diffusion process is called normal. For β = 2, it corresponds to the ballistic diffusive process. When β < 1, we have the subdiffusion process. For the orders into 1 < β < 2, the diffusion process is called the superdiffusive. And for the value of the order superior to the condition 2 < β < 3, we have a hyper diffusive process. For β = 3, we recover the Richardson diffusion process. In Fig. 7, we represent the behavior of the MSD for the normal diffusion β = 1, the ballistic diffusion β = 2, and the Richardson process β = 3.
du¯ (q, s ) . dq
x ABC (s ) = lim i q→0
du¯ (q, s ) = 0 ⇒ x ABC (t ) = 0. dq
(32)
The following expression gives the variance of the random variable X
2 (1 − α ) 2α α σ 2 = x2 ABC (t ) − x 2ABC (t ) = + t . B (α ) B (α )
(33)
Let’s N the random variable X with the same Gaussian distribution, the same mean x ABC and the same variance σ 2 . The application of the central limit theorem stipulate the sum of the N random variable X has the mean value in the form N x ABC = 0 and the variance value in the form
Nσ 2 =
2N ( 1 − α ) 2N α α + t , B (α ) B (α )
and the distribution is also Gaussian.
(34)
N. Sene and K. Abdelmalek / Chaos, Solitons and Fractals 127 (2019) 158–164
Fig. 4. MSD when β = 0.5; 1; 1.5; 2.5;.
Fig. 5. MSD for subdiffusion with Atangana-Baleanu-Caputo derivative α = 0.25; 0.5; 0.75; 0.85;.
Fig. 6. Comparaison between MSD for Normal subdiffusion and MSD with Atangana-Baleanu-Caputo derivative.
163
164
N. Sene and K. Abdelmalek / Chaos, Solitons and Fractals 127 (2019) 158–164
Fig. 7. MSD for subdiffusion with Atangana-Baleanu-Caputo derivative α = 0.25; 0.5; 0.75; 0.85; 1.
9. Conclusion In this paper, we have analyzed the subdiffusion process obtained when α is in (0,1) and the normal diffusion obtained when α = 1 for the fractional diffusion equation. We mean square displacement plays an essential role in the studies and appear a good compromise which we can use to determine the types of diffusion processes. The statistical aspect of the fractional diffusion equation was provided in this paper, and the central limit theorem was illustrated in the context of fractional calculus. This paper contributes to the application of the fractional calculus in real life problems. Future investigation can be done with the MSD to determine the types of diffusion processes for other classes of the fractional diffusion equations with arbitrary order; some papers will be done in this direction. References [1] Abdeljawad T. Different type kernel h-fractional differences and their fractional h-sums. Chaos Soli Fract 2018;116:146–56. [2] Abdeljawad T, Al-Mdallal QM. Discrete Mittag-Leffler kernel type fractional difference initial value problems and gronwall’s inequality. J Comput Appl Math 2018;339:218–30. [3] Abdeljawad T, Baleanu D. Discrete fractional differences with non-singular discrete Mittag-Leffler kernels. Adv Diff Equat 2016;232. [4] Abdeljawad T, Baleanu D. On fractional derivatives with generalized Mittag-Leffler kernels. Adv Diff Equat 2018;468. [5] Alkahtani BST, Koca I. A new numerical scheme applied on re-visited nonlinear model of predator-prey on derivative with non-local and non-singular kernel. Disct contin dyn syst Ser-S 2020;3(13). [6] Atangana A, Baleanu D. New fractional derivatives with nonlocal and nonsingular kernel: theory and application to heat transfer model; 2016. arXiv:160203408. [7] Atangana A, Koca I. Chaos in a simple nonlinear system with Atangana Baleanu derivatives with fractional order. Chaos Solit Fractal 2016;89: 447–454. [8] Santos MD, Gomez IS. A fractional Fokker–planck equation for non-singular kernel operators. J Stat Mech Theory Exp 2018;2018:123205. [9] Santos MD. Fractional prabhakar derivative in diffusion equation with non-static stochastic resetting. Physics 2019;1:40–58. [10] Santos MD. Non-gaussian distributions to random walk in the context of memory kernels. Fractal Fract 2018;2:20. [11] Hristov J. On the Atangana-Baleanu derivative and its relation to the fading memory concept: the diffusion equation formulation. Trend Theory ApplFractDerivat Mittag-Leffler kernel 2019. [12] Hristov J. Approximate solutions to fractional subdiffusion equations. Eur Phys J Spec Topic 2011;193(1):229–43.
[13] Hristov J. Fourth-order fractional diffusion model of thermal grooving: integral approach to approximate closed form solution of the mullins model. Math Modell Natural Phenomena 2018;13(6). [14] Hristov J. A transient flow of a non-newtonian fluid modelled by a mixed time-space derivative: an improved integral-balance approach. In: Mathematical methods in engineering. Springer; 2019. p. 153–74. [15] Hashemi MS, Baleanu D, Haghighi MP. Solving the time fractional diffusion equation using a lie group integrator. Thermal Sci 2015;19:77–83. [16] Henry BI, Langlands TAM, Straka P. An Introduction to Fractional Diffusion. World Sci Rev; 2009. [17] Fahd J, Abdeljawad T. A modified laplace transform for certain generalized fractional operators. Result Nonlinear Anal 2018;2:88–98. [18] Meerschaert MM, Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations. J Comput Appl Math 2004;172:65– 67. [19] Al-Refai M, Abdeljawad T. Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel. Adv Diff Eqs 2017;315:2017. [20] Khader MM. On the numerical solutions for the fractional diffusion equation. Commun Nonlinear Sci Numer Simulat 2011;16:2535–42. [21] Tasbozan O, Esen A, Yagmurlu NM, Ucar Y. A numerical solution to fractional diffusion equation for force-free case. Abs Appl Anal 2013(6). [22] Owolabi KM, Atangana A. On the formulation of adams-bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems. Chaos 2019;29:023111. [23] Owolabi KM, Atangana A. Analysis of mathematics and numerical pattern formation in superdiffusive fractional multicomponent system. Advan Dif Equt 2019;29:023111. [24] Owolabi KM, Atangana A. Numerical simulation of noninteger order system in subdiffusive, diffusive, and superdiffusive scenarios. J Comp Nonl Dyan 2017;12(3):031010. [25] Owolabi KM, Atangana A. Modeling and formation of spatiotemporal paterns of fractional predation system in subdiffusion and superdiffusion scenarios. Euro Phys J Plus 2018;133(2):43. [26] Owolabi KM, Atangana A. Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations. Chaos, Soliton Fractals 2018;111:119–27. [27] Owolabi KM. Numerical analysis and pattern formation process for space fractional superdiffusive systems. Disct Contin Dyn Syst Ser-S 2019;12(3):543–66. [28] Sene N. Analytical solutions of Hristov diffusion equations with non-singular fractional derivatives. choas 2019;29:023112. [29] Sene N. Homotopy perturbation ρ -laplace transform method and its application to the fractional diffusion equation and the fractional diffusion-reaction equation. Fractal Fract 2019;3(14). [30] Sene N. Stokes first problem for heated flat plate with Atangana Baleanu fractional derivative. Chaos Solit Fractals 2018;117:68–75. [31] Fall AN, Ndiaye SN, Sene N. Black-scholes option pricing equations described by the Caputo generalized fractional derivative. Chaos Solit Fractals 2019(125):108–18. [32] Sene, N. Analytical solutions and numerical schemes of certain generalized fractional diffusion models. Eur Phys J Plus 2019;134:199. [33] Sene N, Gomez-Aguilar JF. Analytical solutions of electrical circuits considering certain generalized fractional derivatives. Eur Phys J Plus 2019;134:260.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Analysis of the fractional diffusion equations described by Atangana-Baleanu-Caputo fractional derivative Ndolane Sene a,∗, Karima Abdelmalek b a
Laboratoire Lmdan, Département de Mathématiques de la Décision, Université Cheikh Anta Diop de Dakar, Faculté des Sciences Economiques et Gestion, Dakar Fann BP 5683, Senegal b Laboratory LAMIS, Department of Mathematics and Informatiques, University Larbi Tebessi, Tebessa 12002, Algeria
a r t i c l e
i n f o
Article history: Received 14 May 2019 Revised 20 June 2019 Accepted 26 June 2019
Keywords: Fractional diffusion equations Mean square displacement Atangana-Baleanu fractional derivative operator
a b s t r a c t In this paper, we analyze two types of diffusion processes obtained with the fractional diffusion equations described by the Atangana-Baleanu-Caputo (ABC) fractional derivative. The mean square displacement (MSD) concept has been used to discuss the types of diffusion processes obtained when the order of the fractional derivative take certain values. Many types of diffusion processes exist and depend to the value of the order of the used fractional derivatives: the fractional diffusion equation with the subdiffusive process, the fractional diffusion equation with the superdiffusive process, the fractional diffusion equation with the ballistic diffusive process and the fractional diffusion equation with the hyper diffusive process. Here we use the Atangana-Baleanu fractional derivative and analyze the subdiffusion process obtained when the order of ABC α is into (0,1) and the normal diffusion obtained in the limiting case α = 1. The Laplace transform of the Atangana-Baleanu-Caputo fractional derivative has been used for getting the mean square displacement of the fractional diffusion equation. The central limit theorem has been discussed too, and the main results illustrated graphically. © 2019 Published by Elsevier Ltd.
1. Introduction In the literature, many researchers prove the applicability of the Atangana-Baleanu-Caputo derivative in real word problems. Notably in physics [30,33], in epidemiology, in statistics and probability [8–10,16], in science and engineering, and in many other fields. Recently, it was introduced in the literature the discrete forms of the fractional derivatives with nonsingular kernels, and many applications were proposed with these new issues. Many investigations related the discrete form of the Atananga-Baleanu fractional derivative can be found in [1–4]. Many applications of the fractional order derivatives in the fractional diffusion equations were recently performed. Nowadays, there exist many types of fractional diffusion equations. We enumerate some of them. The fractional diffusion reaction equation [29], the fractional subdiffusion equation [12], the fractional dispersion equation [18] and many others. Many investigations on the fractional diffusion equations concern the analytical solutions [28,31,32], the semi-analytical solutions [29], the approximate solutions [11,13] and the numerical solutions [18]. Many methods ex-
∗
Corresponding author. E-mail addresses: [email protected] (N. Sene), [email protected] (K. Abdelmalek). https://doi.org/10.1016/j.chaos.2019.06.036 0960-0779/© 2019 Published by Elsevier Ltd.
ist in the studies of the solutions of the fractional diffusion equations. We enumerate certain of them. In [20], Kader has proposed the numerical solution of the fractional diffusion equation using the Caputo fractional derivative. In [16], Henry et al. have proposed an introduction of modeling the fractional diffusion equation described by the Caputo fractional derivative. The statistical properties of the fractional diffusion equations were also provided in this paper [16]. In [21], Tasbozan has proposed the numerical solution fractional diffusion equation for force-free case. In [15], Hashemi has solved the time fractional diffusion equation using a lie group integrator. In [19], Al-Refai et al. have proposed a complete analysis of the fractional diffusion equations with Atangana-Baleanu-Caputo fractional derivative. In [28,30], Sene has introduced the solution of the fractional diffusion equations and the stokes’ equation using the Laplace and Fourier transform methods. For recent investigations on the subdiffusive and superdiffusive processes, see in Owolabi et al. [23–27]. Motivated by the fact to understand, what are the types of diffusion generated by the new fractional order derivatives, in this paper, we propose to study the probability aspect of the fractional diffusion equations. We particularly use the AtanangaBaleanu-Caputo fractional derivative. The objective of the paper is to study the Mean Square Displacement of the fractional diffusion equation to analyze the diffusion processes generated by the
N. Sene and K. Abdelmalek / Chaos, Solitons and Fractals 127 (2019) 158–164
Atangaga-Baleanu-Caputo fractional order derivative. Note we use MSD [8–10] to analyze the following diffusion processes: normal diffusion process, subdiffusion process. Other diffusion processes will be recalled as superdiffusion process, hyperdiffusion process, ballistic process, Richardson diffusion process. This paper makes a connection between fractional calculus and statistical physics. In Section 2, we propose the definitions of the fractional derivatives operators with Mittag-Leffler kernels known as the Atangana-Baleanu-Caputo fractional derivative and the AtanganaBaleanu-Riemann fractional derivative. In Section 3, we propose constructive equations. In Section4, we study the existence and the uniqueness of the solution of the proposed model. In Section 5, we propose the procedure for getting the analytical solution. In Section 6, we propose the calculation of the MSD. In Section 7, we discuss the MSD and analyze the diffusion processes. In Section 8, we focus the application of the central limit Theorem in our diffusion model. We give the conclusions and perspectives in Section 9. 2. Derivative operators with non-singular kernels In this section, we recall the definitions of the existing fractional derivative operators, their Laplace transforms and also introduce the Mittag-Leffler function. We begin with the Mittag-Leffler function. The Mittag Leffler function with two parameters is represented as the following series
Eα ,β (z ) =
∞ k=0
zk , (α k + β )
(1)
where α > 0, β ∈ R and z ∈ C. The Atangana-Baleanu-Riemann operator [6,7] for a given function u, of order α ∈ (0, 1] is defined by
DABR α u (y, t ) =
B (α ) d 1 − α dt
0
t
u ( y ( s ), s )Eα −
α
1−α
(t − s )α ds,
(2)
for all t > 0, and the function (. . . ) represents the Euler Gamma function. The Atangana-Baleanu-Caputo derivative [6,7] for a function u, of order α ∈ (0, 1] is defined as the following form
DABC α u (y, t ) =
B (α ) 1−α
t 0
u ( y ( s ), s )Eα −
α
1−α
(t − s )α ds,
(3)
for all t > 0, where the function (. . . ) is Euler Gamma function. The Riemann-Liouville integral [11] for a given function u, of order α ∈ (0, 1] is represented as the form
IαRL u(y, t ) =
1
(α )
a
t
(t − s )α−1 u(y(s ), s )ds,
(4)
for all t > 0, where the function (. . . ) represents the Gamma function. The Atangana-Baleanu integral [6,7] for a given function u, of order α ∈ (0, 1] is defined as the form
IαAB u(y, t ) =
1−α α RL u(y, t ) + I u(y, t ), B (α ) B (α ) α
(5)
for all t > 0. We recall the Laplace transform of the Atangana-BaleanuCaputo derivative [6,7], and the Atangana-Baleanu-Riemann derivative represented as the following forms
L DABC α u (s ) =
B(α ) sα L{u}(s ) − sα −1 u(0 ) , 1−α sα + 1−αα
Due to space limitation, we do not recall the other fractional derivatives. The readers can refer to the following references for more information related to the Riemann-Liouville fractional derivative [14,17], the Caputo-Liouville fractional derivative [14,17], the certain generalized fractional derivatives [17], discrete version of the fractional derivatives can be found in [1–4]. 3. Formalistic model In this section, we present formalistic model introduced by Hristov in [11]. The model is obtained using Fick’s first and second equations. Note Fick’s equations are used in many problems to modeled the fractional diffusion equations. The Fick first equation when the diffusion coefficient is unit is described by
q=−
∂u . ∂y
(8)
Fick’s first equation measures the flux into the system. It represents the density in the diffusion equation and is the subject of many works in the diffusion equation. The second Fick’s equation when we use the Atangana-Baleanu-Caputo fractional derivative is given by ABC t Dα u
=−
∂q . ∂y
(9)
The fractional diffusion equation in Hristov formalistic sense, when we use Atangana-Baleanu-Caputo fractional derivative, is expressed in the following form ABC t Dα u
=
∂ 2u . ∂ y2
(10)
In this paper, the statistical aspect of the fractional diffusion equation will be investigated. For this issue, a statistical boundary condition is used. That is u(x, 0 ) = δ (x ) and u(±∞, t ) = 0. Note that in our study, the following condition is held for the distribution
+∞ −∞
u(x, t )dx = 1.
(11)
The following formula can obtain the mean square displacement of the fractional diffusion equation
2 x
=
+∞ −∞
x2 u(x, t )dx = lim − q→0
d2 u¯ (q, s ) , dq2
(12)
where u¯ represents the Laplace transformation of the function u. The variable q and s came respectively form the Fourier, and the Laplace transform. Note the MSD [8–10] is linear in time. We will use it to analyze the diffusion of the fractional differential equation described by the Atangana-Baleanu-Caputo fractional derivative. • Firstly, we propose the analytical solution of the fractional differential equation Eq. (10) under statistical boundary conditions described above. • Secondly, we get the mean square displacement of the fractional diffusion equation using Eq. (12). • At last, we discuss the nature of the processes using the order α. 4. Existence of the solution using picard-Lindelof method
(6)
In this section, we use Picard-Lindelof method for proving the existence and the uniqueness of the solution. The sketch of the proof is not new and can be found in [5,22]. The main objective is to define Picard-Lindelof operator and to show this operator is well definite and is a contraction. From the assumption, v is bounded the function expressed by
(7)
(y, t, v ) =
Here L represents the classical Laplace transform.
B ( α ) s α L {u } ( s ) L DABR , α u (s ) = 1 − α sα + 1−αα
159
∂ 2 v(y, t ) . ∂ y2
(13)
160
N. Sene and K. Abdelmalek / Chaos, Solitons and Fractals 127 (2019) 158–164
defines a contraction. That is
(y, t, v ) − (y, t, u ) ≤ v − u,
(14)
where we define the following norm
f (t ) =
sup t ∈[t −a,t +a]
| f (t )|.
1−α α (y, t, v ) + B (α ) B(α )(α )
t 0
(t − s )α−1 (y, t, v )ds. (16)
We prove the Picard’s operator is well posed. For that, we use the norm defined at the beginning of this section; we have the following relations
1 − α (y, t, v ) Pv(t ) − v(x, 0 ) =
B (α )
t
α + (t − s )α−1 (y, t, v )ds
B(α )(α ) 0 1−α ≤ (y, t, v ) B (α ) t α + (t − s )α−1 (y, t, v )ds B(α )(α ) 0 1 − α aα ≤ + (17) (y, t, v ), B (α ) B(α )(α )
From which we conclude the Picard’s operator P is well posed. The last step is the Picard’s, and Lindelof method consists of proving the operator P defines a contraction. We have the following relation
1 − α (y, t, v ) Pv(t ) − Pu(t ) =
B (α )
t
α α −1 + t − s ( y, t, v ds ( ) )
B(α )(α ) 0
1 − α −
(y, t, u ) B (α )
t
α α −1 + (t − s ) (y, t, u )ds
B(α )(α ) 0 1 − α α a + Pv(t ) − Pu(t ) ≤ B (α ) B(α )(α ) (18) (y, t, v ) − (y, t, u ), From wich under the assumption
1 − α B (α )
+
aα B(α )(α )
< 1,
In this section, we described the method of finding the solution of the fractional diffusion equation described by AtanganaBaleanu-Caputo fractional derivative. The fractional differential equation which we consider in this section is defined by
∂ 2u . ∂ y2
sα −1 B (α ) × = u˜ (q, s ). 2 2 B ( α ) + ( 1 − α )q sα + B(α )+α(q1−α )q2
(19)
(20)
Eq. (20) will be used in many times in our studies. The equation is fundamental to determine the MSD of the fractional diffusion equation described by the Atangana-Baleanu-Caputo fractional derivative. For simplification in the expression, let’s
η=
B (α ) and B ( α ) + ( 1 − α )q2
ξ=
α q2 . B ( α ) + ( 1 − α )q2
Applying both the inverse of the Laplace transform and the Fourier transform to both sides of equation Eq. (20), we obtain
u(x, t ) =
2
π
∞
0
η sin(qx )Eα (−ξ t α )dq.
(21)
We can observe this solution is in good agreement with the classical solution obtained when α = 1. To see that, we make some analysis. From Eq. (20), the Fourier sine transformation and the Laplace transform of the classical diffusion equation obtained with α = 1 is expressed as the following form
u˜ (q, s ) =
1 . s + q2
(22)
Applying both the inverse of Laplace transform and the Fourier sine transform, we have the following relationships
u(x, t ) = =
2
π 2
∞
0
π
∞
exp −q2 t sin(qx )dq
√ 2 exp − q t sin(qx )dq
0
x2 = √ exp − 4 t 4π t 1
,
(23)
It is straightforward √ to see the function u verifies a density distribution. Let’s σ = 2t . Thus the function u can be rewritten as the following form
x2 1 u(x, σ 2 ) = √ exp − 2σ 2 2π σ 2
We can observe u ≥ 0 and satisfies a Gaussian density distribution. Furthermore, for confirmation, we can see the following calculations +∞
−∞
u(x, t )dx =
+∞ −∞
=2 = 1.
5. Solution procedure of fractional diffusion equation
(y, t ) =
B(α ) sα u˜ (q, s ) − sα −1 = −q2 u˜ (q, s ) 1−α sα + 1−αα
the operator P is a contraction. Using Banach fixed Theorem, we can conclude the solution exists and is unique. Finally, our problem in this paper is well definite.
ABC t Dα u
B(α ) sα u˜ (q, s ) − sα −1 u(q, 0 ) = −q2 u˜ (q, s ) 1−α sα + 1−αα
(15)
Let’s the following Picard’s operator Pv: K → K, where K represents functional spaces of continuous functions given by K = [t − a1 , t + a2 ] × [t − b1 , t + b2 ]. In our case, the explicit form of the Picard’s operator is given by the following
P v(t ) =
under boundary condition u(x, 0 ) = δ (x ) and u(±∞, t ) = 0. Furthermore the condition Eq. (11) is satisfied. We apply both the Fourier sine transformation and the Laplace transform to both sides of Eq. (19), we obtain
+∞ 0
x2 exp − √ 4t 4π t 1
dx
2
x 1 exp − √ 4 t 4π t
dx (24)
In Fig. 5, we depict the Gaussian profile for different values of
σ 2 . The profiles decrease when the number σ 2 increases, Fig. 5,
(see arrow direction). It is mean when the time of the diffusion process is long the density of the material decreases. In other words, the solution u converge to zero. We note an applatissement of the Gaussian profile. That means the Person value of the diffusion is less than 3 when the time increase rapidly.
N. Sene and K. Abdelmalek / Chaos, Solitons and Fractals 127 (2019) 158–164
161
Fig. 1. Gaussian profil with increasing σ 2 .
Fig. 2. Gaussian profil with decreasing σ 2 .
In Fig. 5, we depict the Gaussian profile for different values of
σ 2 . The profiles increase when the number σ 2 decreases, see ar-
row direction. It is mean when the time of the diffusion process is low the density of the material increases. In other words, the solution u do not converge to zero. We note non-applatissement of the Gaussian profile. That means the Person value of the diffusion is superior to 3 at the beginning of the diffusion process. 6. Mean square displacement of fractional diffusion equation
calculations. Let’s the function
w(q ) = −2B(α )qsα −1 [(1 − α )sα + α ], it derivative is given by the following relation
w (q ) = −2B(α )sα −1 [(1 − α )sα + α ] Let’s the function
v (q ) =
2 B ( α ) + ( 1 − α )q2 sα + α q2 ,
it derivative is given by the following relation In this section, we analyze the fractional diffusion equation described by the Atangana Baleanu-Caputo fractional derivative. We use the Mean Square Displacement to do our studies. This tool is used in Santos works related to the fractional diffusion equations described by other fractional derivatives. For getting the MSD, it is better and more useful to use Eq. (20). Using the result of Eq. (20), (q,s ) the function du¯dq in term of Laplace transform is given by 2α −1
+ 2B(α )α qsα −1
2B(α )(1 − α )qs du¯ (q, s ) = − 2 dq B ( α ) + ( 1 − α )q2 sα + α q2 2B(α )qsα −1 [(1 − α )sα + α ]
= −
B ( α ) + ( 1 − α )q2 sα + α q2
We calculate the function
d2 u¯ (q,s ) dq2
2 .
v (q ) = 4[(1 − α )qsα + α q] B(α ) + (1 − α )q2 sα + α q2 . From which we obtain the followiing relationships
v ( 0 ) = 0. We obtain the following relationships
d2 u¯ (q, s ) w ( q )v ( q ) − w ( q )v ( q ) = . dq2 v2 ( q )
(26)
The MSD in term of the Laplace transform is given by the following expression
(25)
in term of the Laplace trans-
form using the Eq. (25). We have the following relationship after
2 x
(s ) = lim − ABC
2 x
ABC
q→0
(s ) = −
d2 u¯ (q, s ) dq2
w ( 0 )v ( 0 ) − w ( 0 )v ( 0 ) v2 ( 0 )
162
N. Sene and K. Abdelmalek / Chaos, Solitons and Fractals 127 (2019) 158–164
Fig. 3. MSD when β = 1; 2; 3.
2 x
(s ) = ABC
2 x
ABC
(s ) =
2B(α )sα −1 [(1 − α )sα + α ][B(α )sα ]
2
4 [B ( α ) s α ]
2 (1 − α ) 2α + . B ( α )s B(α )s1+α
(27)
Applying the inverse of Laplace transform to both sides of Eq. (27), thus the MSD of the fractional diffusion equation described by the Atangana-Baleanu fractional derivative is given by
2 x
ABC
(t ) =
2 (1 − α ) 2α α + t B (α ) B (α )
= 2(IαAB )(1 ).
(28)
We Conclude the Mean Square Displacement of the formalistic fractional diffusion equation described by the Mittag-Leffler fractional derivative is equivalent to the Atangana-Baleanu fractional integral of the unit. A vital remark about Atangana Baleanu-Caputo fractional derivative is when the order α = 1, we recover the MSD of the classical diffusion equation expressed in the following form
2 x
classical
(t ) = 2t.
(29)
A second important remark is when the diffusion process is long. In other words, for a long time, (t → ∞) the MSD of the fractional diffusion equation described by the Atangana-BaleanuCaputo derivative can be expressed in a more simple form. That is
2 x
(t ) ≈ ABC
2α α t . B (α )
(30)
In Fig. 7, we represent the behavior of the MSD for the normal diffusion β = 1, the subdiffusion β = 0.5, the superdiffusion β = 1.5 and the hyperdiffusion β = 2.5 processes. In Fig. 7, we represent the behavior of the MSD for the normal diffusion α = 1, the subdiffusion process for the different values of the orders α = 0.25; 0.50; 0.75; 0.85, when the use the AtanganaBaleanu-Caputo fractional derivative. We note when the order α increase the subdiffusion converge to a normal diffusion. In Fig. 7, we compare the MSD for the subdiffusion processes β = α = 0.5 obtained with ordinary derivative and the AtanganaBaleanu-Caputo fractional derivative. In Fig. 7, we represent the behavior of the MSD for the normal diffusion α = 1, and the behavior of the subdiffusion processes for different values of the orders α = 0.25; 0.50; 0.75; 0.85. We consider a long time, and we use the Atangana-Baleanu-Caputo fractional derivative. 8. Central limit theorem and atangana-Baleanu-Caputo derivative In this section, we focus on how we apply the Central Limit Theorem. We stipulate the random variable X admit a Gaussian distribution obtained with Atangana-Baleanu-Caputo derivative. We first calculate the mean of the random variable X in term of Laplace and Fourier transform, we have the following expression
x ABC (s ) = lim i q→0
(31)
Using Eq. (20), we have the following expression
7. Discusion about the mean square displacement In this section, we discuss the diffusion processes depending on the order of the Atangana-Baleanu-Caputo fractional derivative. For the types of diffusion processes, we begin the discussion with the mean square displacement classically given by x2 ≈ tβ . From which we have different types of diffusion processes. When the order β = 1, the diffusion process is called normal. For β = 2, it corresponds to the ballistic diffusive process. When β < 1, we have the subdiffusion process. For the orders into 1 < β < 2, the diffusion process is called the superdiffusive. And for the value of the order superior to the condition 2 < β < 3, we have a hyper diffusive process. For β = 3, we recover the Richardson diffusion process. In Fig. 7, we represent the behavior of the MSD for the normal diffusion β = 1, the ballistic diffusion β = 2, and the Richardson process β = 3.
du¯ (q, s ) . dq
x ABC (s ) = lim i q→0
du¯ (q, s ) = 0 ⇒ x ABC (t ) = 0. dq
(32)
The following expression gives the variance of the random variable X
2 (1 − α ) 2α α σ 2 = x2 ABC (t ) − x 2ABC (t ) = + t . B (α ) B (α )
(33)
Let’s N the random variable X with the same Gaussian distribution, the same mean x ABC and the same variance σ 2 . The application of the central limit theorem stipulate the sum of the N random variable X has the mean value in the form N x ABC = 0 and the variance value in the form
Nσ 2 =
2N ( 1 − α ) 2N α α + t , B (α ) B (α )
and the distribution is also Gaussian.
(34)
N. Sene and K. Abdelmalek / Chaos, Solitons and Fractals 127 (2019) 158–164
Fig. 4. MSD when β = 0.5; 1; 1.5; 2.5;.
Fig. 5. MSD for subdiffusion with Atangana-Baleanu-Caputo derivative α = 0.25; 0.5; 0.75; 0.85;.
Fig. 6. Comparaison between MSD for Normal subdiffusion and MSD with Atangana-Baleanu-Caputo derivative.
163
164
N. Sene and K. Abdelmalek / Chaos, Solitons and Fractals 127 (2019) 158–164
Fig. 7. MSD for subdiffusion with Atangana-Baleanu-Caputo derivative α = 0.25; 0.5; 0.75; 0.85; 1.
9. Conclusion In this paper, we have analyzed the subdiffusion process obtained when α is in (0,1) and the normal diffusion obtained when α = 1 for the fractional diffusion equation. We mean square displacement plays an essential role in the studies and appear a good compromise which we can use to determine the types of diffusion processes. The statistical aspect of the fractional diffusion equation was provided in this paper, and the central limit theorem was illustrated in the context of fractional calculus. This paper contributes to the application of the fractional calculus in real life problems. Future investigation can be done with the MSD to determine the types of diffusion processes for other classes of the fractional diffusion equations with arbitrary order; some papers will be done in this direction. References [1] Abdeljawad T. Different type kernel h-fractional differences and their fractional h-sums. Chaos Soli Fract 2018;116:146–56. [2] Abdeljawad T, Al-Mdallal QM. Discrete Mittag-Leffler kernel type fractional difference initial value problems and gronwall’s inequality. J Comput Appl Math 2018;339:218–30. [3] Abdeljawad T, Baleanu D. Discrete fractional differences with non-singular discrete Mittag-Leffler kernels. Adv Diff Equat 2016;232. [4] Abdeljawad T, Baleanu D. On fractional derivatives with generalized Mittag-Leffler kernels. Adv Diff Equat 2018;468. [5] Alkahtani BST, Koca I. A new numerical scheme applied on re-visited nonlinear model of predator-prey on derivative with non-local and non-singular kernel. Disct contin dyn syst Ser-S 2020;3(13). [6] Atangana A, Baleanu D. New fractional derivatives with nonlocal and nonsingular kernel: theory and application to heat transfer model; 2016. arXiv:160203408. [7] Atangana A, Koca I. Chaos in a simple nonlinear system with Atangana Baleanu derivatives with fractional order. Chaos Solit Fractal 2016;89: 447–454. [8] Santos MD, Gomez IS. A fractional Fokker–planck equation for non-singular kernel operators. J Stat Mech Theory Exp 2018;2018:123205. [9] Santos MD. Fractional prabhakar derivative in diffusion equation with non-static stochastic resetting. Physics 2019;1:40–58. [10] Santos MD. Non-gaussian distributions to random walk in the context of memory kernels. Fractal Fract 2018;2:20. [11] Hristov J. On the Atangana-Baleanu derivative and its relation to the fading memory concept: the diffusion equation formulation. Trend Theory ApplFractDerivat Mittag-Leffler kernel 2019. [12] Hristov J. Approximate solutions to fractional subdiffusion equations. Eur Phys J Spec Topic 2011;193(1):229–43.
[13] Hristov J. Fourth-order fractional diffusion model of thermal grooving: integral approach to approximate closed form solution of the mullins model. Math Modell Natural Phenomena 2018;13(6). [14] Hristov J. A transient flow of a non-newtonian fluid modelled by a mixed time-space derivative: an improved integral-balance approach. In: Mathematical methods in engineering. Springer; 2019. p. 153–74. [15] Hashemi MS, Baleanu D, Haghighi MP. Solving the time fractional diffusion equation using a lie group integrator. Thermal Sci 2015;19:77–83. [16] Henry BI, Langlands TAM, Straka P. An Introduction to Fractional Diffusion. World Sci Rev; 2009. [17] Fahd J, Abdeljawad T. A modified laplace transform for certain generalized fractional operators. Result Nonlinear Anal 2018;2:88–98. [18] Meerschaert MM, Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations. J Comput Appl Math 2004;172:65– 67. [19] Al-Refai M, Abdeljawad T. Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel. Adv Diff Eqs 2017;315:2017. [20] Khader MM. On the numerical solutions for the fractional diffusion equation. Commun Nonlinear Sci Numer Simulat 2011;16:2535–42. [21] Tasbozan O, Esen A, Yagmurlu NM, Ucar Y. A numerical solution to fractional diffusion equation for force-free case. Abs Appl Anal 2013(6). [22] Owolabi KM, Atangana A. On the formulation of adams-bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems. Chaos 2019;29:023111. [23] Owolabi KM, Atangana A. Analysis of mathematics and numerical pattern formation in superdiffusive fractional multicomponent system. Advan Dif Equt 2019;29:023111. [24] Owolabi KM, Atangana A. Numerical simulation of noninteger order system in subdiffusive, diffusive, and superdiffusive scenarios. J Comp Nonl Dyan 2017;12(3):031010. [25] Owolabi KM, Atangana A. Modeling and formation of spatiotemporal paterns of fractional predation system in subdiffusion and superdiffusion scenarios. Euro Phys J Plus 2018;133(2):43. [26] Owolabi KM, Atangana A. Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations. Chaos, Soliton Fractals 2018;111:119–27. [27] Owolabi KM. Numerical analysis and pattern formation process for space fractional superdiffusive systems. Disct Contin Dyn Syst Ser-S 2019;12(3):543–66. [28] Sene N. Analytical solutions of Hristov diffusion equations with non-singular fractional derivatives. choas 2019;29:023112. [29] Sene N. Homotopy perturbation ρ -laplace transform method and its application to the fractional diffusion equation and the fractional diffusion-reaction equation. Fractal Fract 2019;3(14). [30] Sene N. Stokes first problem for heated flat plate with Atangana Baleanu fractional derivative. Chaos Solit Fractals 2018;117:68–75. [31] Fall AN, Ndiaye SN, Sene N. Black-scholes option pricing equations described by the Caputo generalized fractional derivative. Chaos Solit Fractals 2019(125):108–18. [32] Sene, N. Analytical solutions and numerical schemes of certain generalized fractional diffusion models. Eur Phys J Plus 2019;134:199. [33] Sene N, Gomez-Aguilar JF. Analytical solutions of electrical circuits considering certain generalized fractional derivatives. Eur Phys J Plus 2019;134:260.