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Revista mexicana de física

Print version ISSN 0035-001X

Rev. mex. fis. vol.65 n.1 México Jan./Feb. 2019  Epub Nov 09, 2019

 

Investigación

Analytical solution of the time fractional diffusion equation and fractional convection-diffusion equation

V.F. Morales-Delgadoa 

J.F. Gómez-Aguilarb 

M.A. Taneco-Hernándeza 

aFacultad de Matematicas. Universidad Autónoma de Guerrero, Av. Lazaro Cárdenas S/N, Cd. Universitaria, Chilpancingo, Guerrero, México.

bCONACyT - Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, México, e-mail: jgomez@cenidet.edu.mx.

Abstract

In this paper, we obtain analytical solutions for the time-fractional diffusion and time-fractional convection-diffusion equations. These equations are obtained from the standard equations by replacing the time derivative with a fractional derivative of order α. Fractional operators of type Liouville-Caputo, Atangana-Baleanu-Caputo, fractional conformable derivative in Liouville-Caputo sense, and Atangana-Koca-Caputo were used to model diffusion and convection-diffusion equation. The Laplace and Fourier transforms were applied to obtain analytical solutions for the fractional order diffusion and convection-diffusion equations. The solutions obtained can be useful to understand the modeling of anomalous diffusion, subdiffusive systems and super-diffusive systems, transport processes, random walk and wave propagation phenomenon.

Keywords: Fractional calculus; Mittag-Leffler kernel; fractional conformable derivative; diffusion equation

PACS: 02.30.Uu; 04.20.Jb; 05.40.Fb; 05.60.-k

1. Introduction

Recent studies in science and engineering demonstrated that the dynamics of many systems may be described more accurately by means of differential equations of non-integer order. The diffusion equation is a partial differential equation that portrays density dynamics in a material subject to diffusion [1,2]. The convection-diffusion equation explains the flow of heat, particles, oil reservoir simulations, transport of mass and energy, global weather production, or other physical quantities in conditions where there are both diffusion and convection or advection [3-5]. Fractional diffusion equations are largely used in describing abnormal slowlydiffusion phenomenon, and fractional diffusion equations are always used in describing abnormal convection phenomenon. Time-fractional diffusion is derived by considering continuous time random walk problems, which are in general nonMarkovian processes.

Several definitions, related to fractional order-derivatives have been used in the literature. These definitions include, Riemann-Liouville, Liouville-Caputo, conformable derivatives, Caputo-Fabrizio, Atangana-Baleanu and AtanganaKoca, to mention a few [6]. The choice of fractional differentiation is motivated by the fact that the interaction with the medium is not local but global. The fractional operators can be a useful way to include memory in a dynamical process. A dynamical process that is modelled through fractional order derivatives carries information about its present as well as past states.

In this paper, we consider the time-fractional diffusion and convection-diffusion equations, obtained from the standard equations by replacing the time derivative with fractional derivatives of type Liouville-Caputo, Atangana-Baleanu-Caputo, fractional conformable derivative in Liouville-Caputo sense and Atangana-Koca-Caputo of order α, with 0 < α ≤ 1.

The following fractional diffusion equation is considered

(0Dtαu)(x,t)=μ2x2u(x,t),          t>0, xR, μR+, 0<α1, (1)

u(x,0)=ψ(x), (2)

where µ is the diffusion coefficient.

The fractional convection-diffusion equation considered is [4]

(0Dtαu)(x,t)=-ϵηxu(x,t)+μ2x2u(x,t)+Q(x,t)cρ,

t>0, 0<α1, (3)

u(x,0)=ψ(x), xR, μR+, (4)

where µ = λ/cρ is the diffusion coefficient, 𝜖 is the porosity, η is the velocity, λ is the thermal conductivity, c is the specific heat, ρ is the mass density, and Q(x,t) is the source term.

2. Basic Tools

The Liouville-Caputo (C) fractional operator of order α is defined as [7]

aCDtαu(t,x)={dndtnu(x,t), α=nN,1Γ(n-α)at(t-z)n-α-1nznu(x,z)dz, n-1<α<nN. (5)

Where C0Dtα is the Liouville-Caputo fractional operator of order α with respect to t.

Atangana and Baleanu considered the generalized Mittag-Leffer function as the kernel of differentiation. This kernel is non-singular and nonlocal and preserves the benefits of the above fractional operators. Replaced the exponential kernel with the generalized Mittag-Leffler function, we obtain the fractional operator of type Atangana-Baleanu in Liouville-Caputo sense (ABC) of order α defined as follows [8]

(aABCDt(n+α)u)(x,t)=1g(α)at Eα(-g(α)(t-z)α)n+1uzn+1(x,z)dz, n-1<α<nN. (6)

where n∈ℕ and g(α) is a normalization function that depends of α, which satisfies that, g(0) = g(1) = 1.

Let 0 < α ≤ 1 and n ∈ ℕ, the Laplace transforms of the Liouville-Caputo and Atangana-Baleanu-Caputo fractional operators are given by

L[0CDt(n+α)u(x,t)](x,s)=1sn-α(snL[u(x,t)]-sn-1u(x,0)--u(n-1)(x,0)). (7)

L[0ABCDt(n+α)u(x,t)](x,s)=1g(α) 1s1-α sn+1L[u(x,t)]-snu(x,0)-sn-1u˙(x,0)-u(n)(x,0)s+g(α). (8)

Khalil in [9] gives a new definition of derivative called “conformable derivative”. Let f : [a,∞) −→ ℝ. The conformable derivative of f(t) is given by

aDtαf(t)=limϵ0f(t+ϵt1-α)-f(t)ϵ, (9)

for all t > 0, α ∈ (0,1). If f(t) is α-differentiable in some (0), α > 0, and limϵ0+fαt exists, then define fα(0)=limϵ0+f(α)(t)

The left conformable integral is given by

aItαf(t)=atf(x)(x-a)1-αdx, xa, 0<α1, (10)

Iterating n-times the integral (10) and replacing the integer n, for β ∈C, with Re(β) > 0, we define the following fractional conformable integral

aβItαf(t)=1Γ(β) at ((t-a)α-(x-a)αα)β-1×f(x)(x-a)1-α dx. (11)

Considering the definition given by Eq. (11) we get the left fractional conformable derivative in the Liouville-Caputo sense. Let Re(β) ≥ 0, n = [Re(β)] + 1, fCα,αn α,b, fCα,bnα,b. Then the left fractional conformable derivative in the Liouville-Caputo sense is given by [10]

 ac βDtαf(t)=1Γ(n-β)at((t-a)α-(x-a)αα)n-β-1×aDxαf(x)(x-a)1-α dx= an-βItα( anDtαf(t)), (12)

The Atangana-Koca fractional derivative in LiouvilleCaputo sense (AKC) is given by [11,12]

(aAKCDtαu)(x,t)=1g(α)×at Eα,βγ,q(-g(α)(t-z)α)uz(x,z)dz, (13)

where g(α) is a normalization function as in the previous cases.

Let 0 < α ≤ 1, the Laplace transform of the AtanganaKoca fractional-order derivative is given as

L{0AKCDtαu(x,t)}(x,s)=1g(α)(1-g(α))q×(s-nαL[u(x,t)]-s-nα-1u(x,0)). (14)

Given a function u(x) ∈ L 1(ℝ), the Fourier transform is given by

u^(k)=(Fxu(x))(k):=-eikx u(x)dx, (15)

and the inverse Fourier transform of u(x) is given by

Fk-1(Fxu(k))(x):=12π×-e-ikx (Fxu(x))(k)dk. (16)

3. Fractional diffusion equations

In this paper, we solved the diffusion and convectiondiffusion equation considering fractional operators of type Liouville-Caputo, Atangana-Baleanu-Caputo, fractional conformable derivative in Liouville-Caputo sense and AtanganaKoca-Caputo.

Diffusion Equation.

In the Liouville-Caputo sense we have the following diffusion equation

(0CDtαu)(x,t)=μ2x2u(x,t),

t>0, xR, μR+, 0<α1, (17)

u(x,0)=ψ(x), (18)

where µ is the diffusion coefficient.

Solution. Applying the Laplace transform to Eq. (17) and taking the condition (18) we get

sα(Ltu)(x,s)-sα-1ψ(x)=μ2x2(Ltu)(x,s). (19)

Applying the Fourier transform in the left hand of the Eq. (19) we have

Fx{sα(Ltu)(x,s)-sα-1ψ(x)}(k,s)=sαu^(k,s)-sα-1Ψ(k), (20)

and for the right hand of the Eq. (19) we have

Fx{μ2x2(Ltu)}(k,s)=μ2x2(FxLtu)(k,s)=μ(-ik)2u^(k,s). (21)

Equating Eqs. (20) and (21) the following explicit relation is deduced for u^(k,s)

u^(k,s)=sα-1Ψ(k)sα+μk2. (22)

Now, applying the inverse Laplace and inverse Fourier transforms to Eq. (22) we have

u(x,t)=12π- Eα,1(-μk2tα) Ψ(k)e-ikxdk. (23)

In the Atangana-Baleanu-Caputo sense we have the following diffusion equation

(0ABCDtαu)(x,t)=μ2x2u(x,t),

t>0,xR, μR+, 0<α1, (24)

u(x,0)=ψ(x), (25)

where µ is the diffusion coefficient.

Solution. Applying the Laplace transform to Eq. (24) and taking the condition (25) we get

sα(Ltu)(x,s)-sα-1ψ(x)s+g(α)=μ2x2(Ltu)(x,s). (26)

Applying the Fourier transform to Eq. (26) and simplifying, we have the following relation for u^(k,s)

u^(k,s)=sα-1Ψ(k)sα+μk2(sα+g(α)), (27)

and applying the inverse Fourier transform to Eq. (27) we have

u~(x,s)=(Fk-1(u^))(x,s)=sα-12π×-Ψ(k)sα+μk2(sα+g(α))e-ikxdk. (28)

Finally, applying the inverse Laplace transform to the above equation we get

u(x,t)=12πiϵ-iϵ+iestsα-1ds×12π-Ψ(k)sα+μk2(sα+g(α))e-ikxdk. (29)

Considering the fractional conformable derivative in the Liouville-Caputo sense we have the following diffusion equation

(0c βDtαu)(x,t)=μ2x2u(x,t),

t>0,xR, μR+, 0<α1, (30)

u(x,0)=ψ(x), (31)

where µ is the diffusion coefficient.

Solution. Applying the Laplace transform to Eq. (30) and taking the condition (31) we get

Γ(1-αβ)α-βΓ(1-β)(sαβ(Ltu)(x,s)-sαβ-1ψ(x))=μ2x2(Ltu)(x,s). (32)

Applying the Fourier transform to Eq. (32) and simplifying, we have

u^(k,s)=sαβ-1Ψ(k)sαβ+μk2Γ(1-β)αβΓ(1-αβ). (33)

Now applying the inverse Laplace and inverse Fourier transforms to Eq. (33) we have

u(x,t)=12π- Eαβ,1(-μk2Γ(1-β)Γ(1-αβ)tαβ) ×Ψ(k)e-ikxdk. (34)

In the case when α = 1 the expression (34) matches the solution obtained in the Eq. (23) in the Liouville-Caputo sense.

Considering the Atangana-Koca fractional-order derivative in the Liouville-Caputo sense we have the following diffusion equation

(0AKCDtαu)(x,t)=μ2x2u(x,t),

t>0, xR, μR+, 0<α1, (35)

u(x,0)=ψ(x), (36)

where µ is the diffusion coefficient.

Solution. Applying the Laplace transform to Eq. (35) and taking the condition (36) we get

1a(s-nαL[u(x,t)]-s-nα-1u(x,0))=μ2x2(Ltu)(x,s), (37)

Where α=g(α)(1-g(α))α

Applying the Fourier transform to Eq. (37) and simplifying, we have

u^(k,s)=s-nα-1Ψ(k)s-nα+aμk2. (38)

Now applying the inverse Laplace and inverse Fourier transforms to Eq. (38) we have

u(x,t)=12π- [1-Enα,1(-tnαaμk2)]×Ψ(k)e-ikxdk. (39)

Convection-Diffusion Equation

In the Liouville-Caputo sense we have the following convection-diffusion equation

(0CDtαu)(x,t)=-ϵηxu(x,t)+μ2x2u(x,t)+Q(x,t)cρ, t>0, 0<α1, (40)

u(x,0)=ψ(x), xR, μR+, (41)

where µ = λ/cρ is the diffusion equation.

Solution. Applying the Laplace transform to Eq. (40) and taking the condition (41) we get

sα(Ltu)(x,s)-sα-1ψ(x)=-ϵηx(Ltu)(x,s)+μ2x2(Ltu)(x,s)+Q(x,s)cρ. (42)

Applying the Fourier transform to Eq. (42) and simplifying, we have the following relation for û(k,s)

u^(k,s)=sα-1Ψ(k)sα+(μk2-ϵηik)+1cρQ(k,s)sα+(μk2-ϵηik). (43)

Applying the inverse Laplace transform and the inverse Fourier transforms to Eq. (43) we get

u(x,t)=12π- Eα,1(-(μk2-ϵηik)tα) Ψ(k)e-ikxdk+12πcρ-e-ikxdk12πi×ϵ-iϵ+iQ(k,s)sα+(μk2-ϵηik)estds. (44)

In the Atangana-Baleanu-Caputo sense we have the following convection-diffusion equation

(0ABCDtαu)(x,t)=-ϵηxu(x,t)+μ2x2u(x,t)+Q(x,t)cρ,          t>0,0<α1, (45)

u(x,0)=ψ(x), xR, μR+, (46)

where µ = λ/cρ is the diffusion equation.

Solution. Applying the Laplace transform to Eq. (45) and taking the condition (46) we get

sα(Ltu)(x,s)-sα-1ψ(x)s+g(α)=-ϵηx(Ltu)(x,s)+μ2x2(Ltu)(x,s)+Q(x,s)cρ. (47)

Applying the Fourier transform to Eq. (47) and simplifying, we have the following relation for û(k,s)

u^(k,s)=sα-1Ψ(k)sα+(μk2-ϵηik)(s+g(α))+1cρ(s+g(α))Q(k,s)sα+(μk2-ϵηik)(s+g(α)). (48)

Finally, applying the inverse Fourier transform and the inverse Laplace transform to Eq. (48) we get

u(x,t)=12πiϵ-iϵ+iestsα-1ds12π

×-Ψ(k)sα+(μk2-ϵηik)(s+g(α))e-ikxdk

+12πicρϵ-iϵ+iest(s-g(α))ds12π

×-Q(k,s)sα+(μk2-ϵηik)(s+g(α))e-ikxdk. (49)

Considering the fractional conformable derivative in the Liouville-Caputo sense we have the following convectiondiffusion equation

(0c βDtαu)(x,t)=-ϵηxu(x,t)+μ2x2u(x,t)+Q(x,t)cρ, t>0 0<α1, (50)

u(x,0)=ψ(x), xR, μR+, (51)

where µ is the diffusion coefficient.

Solution. Applying the Laplace transform to Eq. (50) and taking the condition (51) we get

Γ(1-αβ)α-βΓ(1-β)(sαβ(Ltu)(x,s)-sαβ-1ψ(x))=-ϵηx(Ltu)(x,s)+μ2x2(Ltu)(x,s)+Q(x,s)cρ. (52)

Applying the Fourier transform to Eq. (42) and simplifying, we have

u^(k,s)=sαβ-1Ψ(k)sαβ+(μk2-ϵηik)Γ(1-β)αβΓ(1-αβ)+Γ(1-β)cραβΓ(1-αβ)×Q(k,s)sαβ+(μk2-ϵηik)Γ(1-β)αβΓ(1-αβ). (53)

Finally, applying the inverse Laplace transform and the inverse Fourier transform to Eq. (53), we get

u(x,t)=12π-Eαβ,1(-(μk2-ϵηik)Γ(1-β)αβΓ(1-αβ)tαβ)

×Ψ(k)e-ikxdk+Γ(1-β)cραβΓ(1-αβ)12π-e-ikxdk

×12πiϵ-iϵ+iQ(k,s)sαβ+(μk2-ϵηik)Γ(1-β)αβΓ(1-αβ)estds. (54)

In the case when α = 1 the expression (54) matches the solution obtained in the Eq. (44) in the Liouville-Caputo sense.

Considering the Atangana-Koca fractional-order derivative in the Liouville-Caputo sense we have the following convection-diffusion equation

(0AKCDtαu)(x,t)=-ϵηxu(x,t)+μ2x2u(x,t)+Q(x,t)cρ, t>0, 0<α1, (55)

u(x,0)=ψ(x), xR, μR+, (56)

where µ is the diffusion coefficient.

Solution. Applying the Laplace transform to Eq. (55) and taking the condition (56) we get

1b(s-nαL[u(x,t)]-s-nα-1u(x,0))=-ϵηx(Ltu)(x,s)+μ2x2(Ltu)(x,s)+Q(x,s)cρ, (57)

where b=g(α)(1-g(α))α.

Applying the Fourier transform to Eq. (57) and simplifying, we have

u^(k,s)=s-nα-1Ψ(k)s-nα+bμk2-ϵηbik+bcρ Q(k,s)s-nα+bμk2-ϵηbik. (58)

Applying the inverse Laplace transform and the inverse

Fourier transform to Eq. (58) we get

u(x,t)=12π- [1-Enα,1(-tnαbμk2-ϵηbik)] ×Ψ(k)e-ikxdk+b2πcρϵ-iϵ+ie-ikxdk12πi×ϵ-iϵ+iQ(k,s)s-nα+bμk2-ϵηbikestds. (59)

4. Illustrative examples

Figures 1(a-d) show numerical simulations of the Eqs. (44), (49), (54) and (59) for α = 0.85 and α = 0.92-β = 0.83 for the fractional conformable derivative in the Liouville-Caputo sense, these values were chosen arbitrarily.

FIGURE 1 Numerical solutions of Eqs. (44), (49), (54) and (59). In (a) Eq. (44); in (b) Eq. (49); in (c) Eq. (54) and (d) (59), we consider α = 0.85 for the cases (a), (b), (d) and for (c), we consider α = 0.92-β = 0.83 for the fractional conformable derivative in the Liouville-Caputo sense. 

5. Conclusion

In this work we applied fractional-order derivatives of type Liouville-Caputo, Atangana-Baleanu, fractional conformable derivative and Atangana-Koca to obtain analytical solutions for the diffusion and convection-diffusion equation. The fractional equations were solved using the Laplace and Fourier transform. The anomalous diffusion concept is naturally obtained from diffusion equations using the fractional calculus approach. Our results indicate that the kernel involved in the fractional derivative and the fractional-order α has an important influence on the concentration. When memory effects described by the fractional order α are incorporated using fractional time derivatives, the crossover dynamics is richer. The alternative solutions obtained in this paper provide a new theoretical perspective of the diffusion and convection-diffusion phenomena.

Acknowledgments

The authors are grateful to all of the anonymous reviewers for their valuable suggestions. Jose Francisco Gómez Aguilar acknowledges the support provided by CONACyT: catedras CONACyT para jovenes investigadores 2014. José Francisco Gómez Aguilar and Marco Antonio Taneco Hernández acknowledges the support provided by SNI-CONACyT.

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Received: July 18, 2018; Accepted: September 06, 2018

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