A relatively new iterative Laplace transform method, which combines two methods; the iterative method and the Laplace transform method, is applied to obtain the numerical solutions of fractional Fokker-Planck equations. The method gives numerical solutions in the form of convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results show that the approach is easy to implement and straightforward when applied to space-time fractional Fokker-Planck equations. The method provides a promising tool for solving space-time fractional partial differential equations.
1. Introduction
Fractional calculus has attracted much attention for its potential applications in various scientific fields such as fluid mechanics, biology, viscoelasticity, engineering, and other areas of science [1–4]. So it becomes important to find some efficient methods for solving fractional differential equations. A great deal of effort has been spent on constructing of the numerical solutions and many effective methods have been developed such as fractional wavelet method [5–8], fractional differential transform method [9], fractional operational matrix method [10, 11], fractional improved homotopy perturbation method [12, 13], fractional variational iteration method [14, 15], and fractional Laplace Adomian decomposition method [16, 17].
In 2006, Daftardar-Gejji and Jafari proposed a new iterative method to seek numerical solutions of nonlinear functional equations [18, 19]. By now, the iterative method has been used to solve many nonlinear differential equations of integer and fractional order [20] and fractional boundary value problem [21]. Most recently, Jafari et al. firstly applied Laplace transform in the iterative method and proposed a new direct method called iterative Laplace transform method [22] to search for numerical solutions of a system of fractional partial differential equations. The method is based on Laplace transform, iterative method, Caputo fractional derivative, and symbolic computation. By using this method, Jafari and Seifi successfully obtained the numerical solutions of two systems of space-time fractional differential equations. It has been shown that, with this method, one can discover some solutions found by the existing methods such as homotopy perturbation method, Laplace Adomian decomposition method, and variational iterative method [23].
It is well known that the choice of an appropriate ansatz is of great importance when a method is applied to search for numerical solutions of nonlinear partial differential equations. In the present paper, we will use the iterative Laplace transform method to solve space-time fractional Fokker-Planck equations. The fractional derivatives described here are in the Caputo sense.
Fokker-Planck equation has been applied in various natural science fields such as quantum optics, solid-state physics, chemical physics, theoretical biology, and circuit theory. It is firstly proposed by Fokker and Planck to characterize the Brownian motion of particles [24]. The general form of Fokker-Planck equation is as follows:
(1)∂u∂t=[-∂∂xA(x)+∂2∂x2B(x)]u(x,t)
with the initial condition
(2)u(x,0)=f(x),
where A(x),B(x)>0 are called diffusion coefficient and drift coefficient, respectively. If A(x),B(x)>0 depend on u(x,t) and the time t, then (1) becomes the following generalized nonlinear form [25]:
(3)∂u∂t=[-∂∂xA(x,t,u)+∂2∂x2B(x,t,u)]u(x,t).
The space-time fractional Fokker-Planck is as follows [26]:
(4)Dtαu=[-DxβA(x,t,u)+Dx2βB(x,t,u)]u(x,t)
which is the generalized fractional form of (3). Here Dtα(·),Dxβ(·),Dx2β(·) are the Caputo fractional derivative with respect to t and x defined in Section 2. When α=β=1, (4) reduces to (3).
The rest of this paper is organized as follows. In Section 2, we review some basic definitions of Caputo fractional derivative and Laplace transform. In Section 3, we describe the iterative Laplace transform method for solving fractional partial differential equations. In Section 4, we give three applications of the method to Fokker-Planck equations. In Section 5, some conclusions and discussions are given.
2. PreliminariesDefinition 1.
The Caputo fractional derivative [27, 28] of function u(x,t) is defined as
(5)Dtαu(x,t)=1Γ(m-α)∫0t(t-η)m-α-1u(m)(x,η)dη,hhhhhhhhhhhhhhhhhhhm-1<α≤m,m∈N,
where Γ(·) denotes the gamma function.
Definition 2.
The Laplace transform of f(t) is defined as [27, 28]
(6)F(s)=ℒ[f(t)]=∫0∞e-stf(t)dt.
Definition 3.
Laplace transform of Dtαu(x,t) is given as [27, 28]
(7)ℒ[Dtαu(x,t)]=sαℒ[u(x,t)]-∑k=0m-1u(k)(x,0)sα-1-k,hhhhhhhhhhhhhhhhhhhhhhhhhim-1<α≤m,
where u(k)(x,0) is the k-order derivative of u(x,t) at t=0.
Further information about fractional derivative and its properties can be found in [27–29].
3. The Iterative Laplace Transform Method
To illustrate the basic idea of the iterative Laplace transform method [22], we consider the general space-time fractional partial differential equation with initial conditions of the form
(8)Dtαu=𝒜(u,Dxβu,Dx2βu,…),m-1<α≤m,n-1<β≤n,m,n∈N,
with initial value conditions
(9)u(k)(x,0)=hk(x),k=0,1,…,m-1,
where 𝒜(u,Dxβu,Dx2βu,…) is a linear or nonlinear operator of u,Dxβu,Dx2βu,…, and u=u(x,t) is the unknown function that will be determined later.
Taking Laplace transfer of both sides of (8) results in
(10)sαℒ[u(x,t)]-∑k=0m-1sα-1-ku(k)(x,0)=ℒ[𝒜(u,Dxβu,Dx2βu,…)].
Equivalently,
(11)ℒ[u(x,t)]=∑k=0m-1s-1-ku(k)(x,0)+s-αℒ[𝒜(u,Dxβu,Dx2βu,…)].
Operating with Laplace inverse (denoted by ℒ-1 throughout the present paper) on both sides of (11) gives
(12)u(x,t)=ℒ-1[∑k=0m-1s-1-ku(k)(x,0)]+ℒ-1[s-αℒ[𝒜(u,Dxβu,Dx2βu,…)]],
which can be rewritten as the form
(13)u(x,t)=ℒ-1[∑k=0m-1s-1-ku(k)(x,0)]+ℬ(u,Dxβu,Dx2βu,…),
where ℬ(u,Dxβu,Dx2βu,…)=ℒ-1[s-αℒ[𝒜(u,Dxβu,Dx2βu,…)]].
The iterative Laplace transform method represents the solution as an infinite series:
(14)u(x,t)=∑n=0∞un,
where the terms un are to be recursively computed. The linear or nonlinear operator ℬ(u,Dxβu,Dx2βu,…) can be decomposed as follows:
(15)ℬ(∑n=0∞un,Dxβ∑n=0∞un,Dx2β∑n=0∞un,…)=ℬ(u0,Dxβu0,Dx2βu0,…)+∑j=1∞ℬ(∑k=0juk,Dxβ∑k=0juk,Dx2β∑k=0juk,…)-∑j=1∞ℬ(∑k=0j-1uk,Dxβ∑k=0j-1uk,Dx2β∑k=0j-1uk,…).
Substituting (14) and (15) into (13) yields
(16)∑n=0∞un=ℒ-1[∑k=0m-1s-1-ku(k)(x,0)]+ℬ(u0,Dxβu0,Dx2βu0,…)+∑j=1∞[ℬ(∑k=0juk,Dxβ∑k=0juk,Dx2β∑k=0juk,…)-ℬ(∑k=0j-1uk,Dxβ∑k=0j-1uk,Dx2β∑k=0j-1uk,…)].
We set
(17)u0=ℒ-1[∑k=0m-1s-1-ku(k)(x,0)],u1=ℬ(u0,Dxβu0,Dx2βu0,…),um+1=ℬ(∑k=0muk,Dxβ∑k=0muk,Dx2β∑k=0muk,…)-ℬ(∑k=0m-1uk,Dxβ∑k=0m-1uk,Dx2β∑k=0m-1uk,…),m≥1.
Therefore the m-term numerical solution of (8)-(9) is given by
(18)u(x,t)≅u0(x,t)+u1(x,t)+⋯+um(x,t),m=1,2,….
4. Numerical Solutions of Fractional Fokker-Planck Equations
The iterative Laplace transform method, described in Section 3, will be applied to solve three special cases of space-time fractional Fokker-Planck equations with initial conditions.
Example 1.
Consider the Fokker-Planck equation in the case that [30]
(19)Dtαu=∂u∂x+∂2u∂x2,0<α≤1,
subject to
(20)u(x,0)=x.
Taking Laplace transform on both sides of (19) gives
(21)sαℒ[u(x,t)]-sα-1u(x,0)=ℒ[∂u∂x+∂2u∂x2],(22)ℒ[u(x,t)]=xs+1sαℒ[∂u∂x+∂2u∂x2].
Operating with Laplace inverse on both sides of (22) results in
(23)u(x,t)=x+ℒ-1[1sαℒ[∂u∂x+∂2u∂x2]].
Substituting (14) and (15) into (23) and applying (17), we obtain the components of the solution as follows:
(24)u0(x,t)=u(x,0)=x,u1(x,t)=ℒ-1[1sαℒ[∂u0∂x+∂2u0∂x2]]=tαΓ(1+α),u2(x,t)=0,u3(x,t)=⋯=un(x,t)=⋯=0.
Therefore, the solution of (19)-(20) in a closed form can be obtained as follows:
(25)u(x,t)=x+tαΓ(1+α).
If we take α=1, then (25) can be reduced to
(26)u(x,t)=x+t,
which is exactly the same as that obtained by homotopy perturbation method in [30].
It should be pointed out that the iterative Laplace transform method is the generalization algorithm of iterative method proposed by Daftardar-Gejji and Jafari [18]. When these two methods are used to solve differential equations with integer order derivatives, especially for linear cases, they are not different from each other.
Example 2.
Consider the following space-time fractional Fokker-Planck equation with initial value conditions [26]:
(27)Dtαu=-Dxβ(ux6)+Dx2β(ux212),t>0,x>0,0<α,β≤1,
subject to
(28)u(x,0)=x2.
Taking Laplace transform on both sides of (27) gives
(29)sαℒ[u(x,t)]-sα-1u(x,0)=ℒ[-Dxβ(ux6)+Dx2β(ux212)],(30)ℒ[u(x,t)]=u(x,0)s+1sαℒ[-Dxβ(ux6)+Dx2β(ux212)].
Operating with Laplace inverse on both sides of (30), we obtain the following Laplace equation:
(31)u(x,t)=u(x,0)+ℒ-1[1sαℒ[-Dxβ(ux6)+Dx2β(ux212)]].
Following the algorithm given in (17), the first three components of the solution are as follows:
(32)u0(x,t)=u(x,0)=x2,u1(x,t)=ℒ-1[1sαℒ[-Dxβ(u0x6)+Dx2β(u0x212)]]=(-1Γ(4-β)x3-β+2Γ(5-2β)x4-2β)·tαΓ(1+α),u2(x,t)=[4-β6Γ(5-2β)x4-2β-5-3β3Γ(6-3β)x5-3β-(5-β)(4-β)12Γ(6-3β)x5-3β+(6-2β)(5-2β)6Γ(7-4β)×x6-4β4-β6Γ(5-2β)]·t2αΓ(1+2α).
The solution in series form is then given by
(33)u(x,t)=x2+(-1Γ(4-β)x3-β+2Γ(5-2β)x4-2β)·tαΓ(1+α)+[4-β6Γ(5-2β)x4-2β-5-3β3Γ(6-3β)x5-3β-(5-β)(4-β)12Γ(6-3β)x5-3β+(6-2β)(5-2β)6Γ(7-4β)x6-4β]·t2αΓ(1+2α)+⋯.
Setting α=β=1 in (33), we get the solution of the problem by
(34)u(x,t)=x2(1+t2+(t/2)22!+⋯)
and in a closed form by
(35)u(x,t)=x2et/2
which is in full agreement with the results by homotopy perturbation method in [26].
Example 3.
Consider the following space-time fractional nonlinear initial value problem that describes Fokker-Planck equation [12]:
(36)Dtαu=-Dxβ(4u2x-xu3)+Dx2βu2,0<α,β≤1,
subject to
(37)u(x,0)=x2.
Taking Laplace transform on both sides of (35) gives
(38)sαℒ[u(x,t)]-sα-1u(x,0)=ℒ[-Dxβ(4u2x-xu3)+Dx2βu2],(39)ℒ[u(x,t)]=x2s+1sαℒ[-Dxβ(4u2x-xu3)+Dx2βu2].
Operating with Laplace inverse on both sides of (38), we obtain the following Laplace equation:
(40)u(x,t)=x2+ℒ-1[1sαℒ[-Dxβ(4u2x-xu3)+Dx2βu2]].
Following the algorithm given in (17), the first few components of the solution are as follows:
(41)u0(x,t)=x2,u1(x,t)=ℒ-1[1sαℒ[-Dxβ(4u02x-xu03)+Dx2βu02]]=(-22Γ(4-β)x3-β+24Γ(5-2β)x4-2β)·tαΓ(1+α),u2(x,t)=ℒ-1[(4(u0+u1)2x-x(u0+u1)3)1sαℒ[-Dxβ(4(u0+u1)2x-x(u0+u1)3)+Dx2β(u0+u1)2(4(u0+u1)2x-x(u0+u1)3)]]-ℒ-1[1sαℒ[-Dxβ(4u02x-xu03)+Dx2βu02]]=-Γ(1+2α)Γ(8-4β)Γ2(1+α)Γ(1+3α)Γ(8-5β)Γ(5-2β)·(2304Γ(5-2β)+1056Γ(4-β))t3αx7-5β-1Γ(1+2α)Γ(6-3β)(184Γ(6-2β)Γ(5-2β)+44Γ(6-β)Γ(4-β))·t2αx5-3β+Γ(1+2α)Γ2(1+α)Γ(1+3α)Γ(7-4β)Γ(4-β)×(484Γ(7-2β)Γ(4-β)+4224Γ(7-3β)Γ(5-2β))·t3αx6-4β+506Γ(5-β)3Γ(1+2α)Γ(5-2β)Γ(4-β)·t2αx4-2β-1936Γ(1+2α)Γ(6-2β)Γ2(1+α)Γ(1+3α)Γ(6-3β)Γ2(4-β)·t3αx5-3β+576Γ(1+2α)Γ(9-4β)Γ2(1+α)Γ(1+3α)Γ(8-5β)Γ2(5-2β)·t3αx8-6β+48Γ(7-2β)Γ(1+2α)Γ(7-4β)Γ(5-2β)·t2αx6-4β+(22Γ(4-β)x3-β-24Γ(5-2β)x4-2β)·tαΓ(1+α),⋮
The solution in series form is then given by
(42)u(x,t)=u0(x,t)+u1(x,t)+u2(x,t)+⋯.
If we take α=β=1, the first few components of the solution are as follows:
(43)u0(x,t)=x2,u1(x,t)=x2t,u2(x,t)=x2t22,⋮
For this special case, the exact solution of (36) and (37) is therefore given by
(44)u(x,t)=x2et
which is exactly the result obtained by homotopy perturbation transformation method in [12].
Table 1 shows the numerical solutions for (36) and (37) by using iterative Laplace transform method, homotopy perturbation transform method, Adomian decomposition method, and the exact solution as α=β=1. It should be pointed out that only three terms of these methods are used to evaluate the numerical solutions in Table 1. It is obvious that the iterative Laplace transform method used in the present paper has the same convergence as the convergence of homotopy perturbation transform method and Adomian decomposition method for solving this fractional nonlinear Fokker-Planck equation. Therefore, iterative Laplace transform method is an effective method for solving fractional partial differential equations just as homotopy perturbation transform method and Adomian decomposition method.
Several approximate values and exact solutions for (36) and (37) when α=β=1.
x
t
SolutionILTM
SolutionHPM
SolutionADM
Exact solution
0.25
0.1
0.0690729
0.0690729
0.0690729
0.0690732
0.2
0.0763333
0.0763333
0.0763333
0.0763377
0.4
0.0931667
0.0931667
0.0931667
0.093239
0.8
0.137833
0.137833
0.137833
0.139096
0.5
0.1
0.276292
0.276292
0.276292
0.276293
0.2
0.305333
0.305333
0.305333
0.305351
0.4
0.372667
0.372667
0.372667
0.372956
0.8
0.551333
0.551333
0.551333
0.556385
0.75
0.1
0.621656
0.621656
0.621656
0.621659
0.2
0.687
0.687
0.687
0.687039
0.4
8385
0.8385
0.8385
0.839151
0.8
1.2405
1.2405
1.2405
1.25187
5. Conclusions
With the aid of the symbolic computation system Mathematica, the iterative Laplace transform method is first successfully applied to solve fractional Fokker-Planck equations. The results obtained by the iterative Laplace transform method are the same as those obtained by homotopy perturbation transform method and Adomian decomposition method. The method finds the solutions without unnecessary linearization, small perturbation and other restrictive assumptions. Therefore, the method considerably reduces the computational work to a great extent. It is worth mentioning that the method can also be applied to solve other nonlinear fractional differential equations with initial value conditions.
Acknowledgments
The author would like to express her sincere thanks to the referees for their valuable suggestions which lead to an improved version. This research is supported by Excellent Young Scientist Foundation of Shandong Province under Grant no. BS2013HZ026.
SabatierJ.AgrawalO. P.Tenreiro MachadoJ. A.BaleanuD.DiethelmK.ScalasE.TrujilloJ. J.LiuY.XinB.Numerical solutions of a fractional predator-prey systemMaJ.LiuY.Exact solutions for a generalized nonlinear fractional Fokker-Planck equationRehmanM.Ali KhanR.The Legendre wavelet method for solving fractional differential equationsLiY.Solving a nonlinear fractional differential equation using Chebyshev waveletsWuJ. L.A wavelet operational method for solving fractional partial differential equations numericallyLepikÜ.Solving fractional integral equations by the Haar wavelet methodErtürkV. S.MomaniS.Solving systems of fractional differential equations using differential transform methodLiY.SunN.Numerical solution of fractional differential equations using the generalized block pulse operational matrixSaadatmandiA.DehghanM.A new operational matrix for solving fractional-order differential equationsLiuY.Approximate solutions of fractional nonlinear equations using homotopy perturbation transformation methodLiuY.Variational homotopy perturbation method for solving fractional initial boundary value problemsSweilamN. H.KhaderM. M.Al-BarR. F.Numerical studies for a multi-order fractional differential equationDasS.Analytical solution of a fractional diffusion equation by variational iteration methodJafariH.KhaliqueC. M.NazariM.Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion-wave equationsOngunM. Y.The Laplace adomian decomposition method for solving a model for HIV infection of CD4^{+} cellsDaftardar-GejjiV.JafariH.An iterative method for solving nonlinear functional equationsJafariH.BhalekarS.Daftardar-GejjiV.Solving evolution equations using a new iterative methodDaftardar-GejjiV.BhalekarS.Solving fractional boundary value problems with Dirichlet boundary conditions using a new iterative methodJafariH.NazariM.BaleanuD.KhaliqueC. M.A new approach for solving a system of fractional partial differential equationsJafariH.SeifiS.Solving a system of nonlinear fractional partial differential equations using homotopy analysis methodRiskenH.FrankT. D.Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equationsYiidirimA.Analytical approach to Fokker-Planck equation with space- and time fractional derivatives by means of the homotopy perturbation methodMillerK. S.RossB.PodlubnyI.HilferR.BiazarJ.HosseiniK.GholaminP.Homotopy perturbation method Fokker-Planck equation