Numerical Solutions of Fractional Fokker-Planck Equations Using Iterative Laplace Transform Method

Abstract

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: 𝜕𝑢=−𝜕𝜕𝑡𝐴(𝑥)+𝜕𝜕𝑥𝜕𝑥²²𝐵(𝑥)𝑢(𝑥,𝑡) (1) with the initial condition 𝑢(𝑥,0)=𝑓(𝑥), (2) 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]: 𝜕𝑢=−𝜕𝜕𝑡𝐴()+𝜕𝜕𝑥𝑥,𝑡,𝑢²𝜕𝑥²𝐵(𝑥,𝑡,𝑢)𝑢(𝑥,𝑡). (3)

The space-time fractional Fokker-Planck is as follows [26]: 𝐷^𝛼_𝑡𝑢=−𝐷^𝛽_𝑥𝐴(𝑥,𝑡,𝑢)+𝐷_𝑥^2𝛽𝐵(𝑥,𝑡,𝑢)𝑢(𝑥,𝑡) (4) which is the generalized fractional form of (3). Here 𝐷^𝛼_𝑡(⋅),𝐷^𝛽_𝑥(⋅),𝐷_𝑥^2𝛽(⋅) 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. Preliminaries

Definition 1. The Caputo fractional derivative [27, 28] of function 𝑢(𝑥,𝑡) is defined as 𝐷^𝛼_𝑡𝑢()=1𝑥,𝑡Γ(𝑚−𝛼)^𝑡₀𝑡−𝜂^{𝑚−𝛼−1}𝑢^(𝑚)𝑥,𝜂𝑑𝜂,𝑚−1<𝛼≤𝑚,𝑚∈𝑁, (5) where Γ(⋅) denotes the gamma function.

Definition 2. The Laplace transform of 𝑓(𝑡) is defined as [27, 28] 𝐹(𝑠)=ℒ𝑓(𝑡)=^∞₀𝑒^−𝑠𝑡𝑓(𝑡)𝑑𝑡. (6)

Definition 3. Laplace transform of 𝐷^𝛼_𝑡𝑢(𝑥,𝑡) is given as [27, 28] ℒ𝐷^𝛼_𝑡𝑢(𝑥,𝑡)=𝑠^𝛼ℒ[𝑢(𝑥,𝑡)]−^𝑚−1_𝑘=0𝑢^(𝑘)(𝑥,0)𝑠^{𝛼−1−𝑘},𝑚−1<𝛼≤𝑚, (7) 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 𝐷^𝛼_𝑡𝑢=𝒜𝑢,𝐷^𝛽_𝑥𝑢,𝐷_𝑥^2𝛽𝑢,…,𝑚−1<𝛼≤𝑚,𝑛−1<𝛽≤𝑛,𝑚,𝑛∈𝑁, (8) with initial value conditions 𝑢^(𝑘)(𝑥,0)=ℎ_𝑘(𝑥),𝑘=0,1,…,𝑚−1, (9) where 𝒜(𝑢,𝐷^𝛽_𝑥𝑢,𝐷_𝑥^2𝛽𝑢,…) is a linear or nonlinear operator of 𝑢,𝐷^𝛽_𝑥𝑢,𝐷_𝑥^2𝛽𝑢,…, and 𝑢=𝑢(𝑥,𝑡) is the unknown function that will be determined later.

Taking Laplace transfer of both sides of (8) results in 𝑠^𝛼ℒ[𝑢(𝑥,𝑡)]−^𝑚−1_𝑘=0𝑠^{𝛼−1−𝑘}𝑢^(𝑘)()𝒜𝑥,0=ℒ𝑢,𝐷^𝛽_𝑥𝑢,𝐷_𝑥^2𝛽𝑢,…. (10) Equivalently, ℒ[𝑢(𝑥,𝑡)]=^𝑚−1_𝑘=0𝑠^−1−𝑘𝑢^(𝑘)(𝑥,0)+𝑠^−𝛼ℒ𝒜𝑢,𝐷^𝛽_𝑥𝑢,𝐷_𝑥^2𝛽𝑢,…. (11)

Operating with Laplace inverse (denoted by ℒ⁻¹ throughout the present paper) on both sides of (11) gives 𝑢(𝑥,𝑡)=ℒ^−1𝑚−1_𝑘=0𝑠^−1−𝑘𝑢^(𝑘)(𝑥,0)+ℒ⁻¹𝑠^−𝛼ℒ𝒜𝑢,𝐷^𝛽_𝑥𝑢,𝐷_𝑥^2𝛽𝑢,…, (12) which can be rewritten as the form 𝑢(𝑥,𝑡)=ℒ^−1𝑚−1_𝑘=0𝑠^−1−𝑘𝑢^(𝑘)(𝑥,0)+ℬ𝑢,𝐷^𝛽_𝑥𝑢,𝐷_𝑥^2𝛽𝑢,…, (13) where ℬ(𝑢,𝐷^𝛽_𝑥𝑢,𝐷_𝑥^2𝛽𝑢,…)=ℒ⁻¹[𝑠^−𝛼ℒ[𝒜(𝑢,𝐷^𝛽_𝑥𝑢,𝐷_𝑥^2𝛽𝑢,…)]].

The iterative Laplace transform method represents the solution as an infinite series: 𝑢(𝑥,𝑡)=^∞_𝑛=0𝑢_𝑛, (14) where the terms 𝑢_𝑛 are to be recursively computed. The linear or nonlinear operator ℬ(𝑢,𝐷^𝛽_𝑥𝑢,𝐷_𝑥^2𝛽𝑢,…) can be decomposed as follows: ℬ^∞_𝑛=0𝑢_𝑛,𝐷^𝛽_𝑥^∞_𝑛=0𝑢_𝑛,𝐷_𝑥^∞2𝛽_𝑛=0𝑢_𝑛𝑢,…=ℬ₀,𝐷^𝛽_𝑥𝑢₀,𝐷_𝑥^2𝛽𝑢₀+,…_∞𝑗=1ℬ^𝑗_𝑘=0𝑢_𝑘,𝐷^𝛽_𝑥^𝑗_𝑘=0𝑢_𝑘,𝐷_𝑥^𝑗2𝛽_𝑘=0𝑢_𝑘−,…_∞𝑗=1ℬ^𝑗−1_𝑘=0𝑢_𝑘,𝐷^𝛽_𝑥^𝑗−1_𝑘=0𝑢_𝑘,𝐷_𝑥^{2𝛽𝑗−1}_𝑘=0𝑢_𝑘,…. (15) Substituting (14) and (15) into (13) yields ^∞_𝑛=0𝑢_𝑛=ℒ^−1𝑚−1_𝑘=0𝑠^−1−𝑘𝑢^(𝑘)(𝑢𝑥,0)+ℬ₀,𝐷^𝛽_𝑥𝑢₀,𝐷_𝑥^2𝛽𝑢₀+,…_∞𝑗=1ℬ^𝑗_𝑘=0𝑢_𝑘,𝐷^𝛽_𝑥^𝑗_𝑘=0𝑢_𝑘,𝐷_𝑥^𝑗2𝛽_𝑘=0𝑢_𝑘,…−ℬ^𝑗−1_𝑘=0𝑢_𝑘,𝐷^𝛽_𝑥^𝑗−1_𝑘=0𝑢_𝑘,𝐷_𝑥^{2𝛽𝑗−1}_𝑘=0𝑢_𝑘,…. (16)

We set 𝑢₀=ℒ^−1𝑚−1_𝑘=0𝑠^−1−𝑘𝑢^(𝑘)(,𝑢𝑥,0)₁=ℬ𝑢₀,𝐷^𝛽_𝑥𝑢₀,𝐷_𝑥^2𝛽𝑢₀,𝑢,…_𝑚+1=ℬ^𝑚_𝑘=0𝑢_𝑘,𝐷^𝛽_𝑥^𝑚_𝑘=0𝑢_𝑘,𝐷_𝑥^𝑚2𝛽_𝑘=0𝑢_𝑘,…−ℬ^𝑚−1_𝑘=0𝑢_𝑘,𝐷^𝛽_𝑥^𝑚−1_𝑘=0𝑢_𝑘,𝐷_𝑥^{2𝛽𝑚−1}_𝑘=0𝑢_𝑘,…,𝑚≥1. (17)

Therefore the 𝑚-term numerical solution of (8)-(9) is given by 𝑢(𝑥,𝑡)≅𝑢₀(𝑥,𝑡)+𝑢₁(𝑥,𝑡)+⋯+𝑢_𝑚(𝑥,𝑡),𝑚=1,2,…. (18)

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] 𝐷^𝛼_𝑡𝑢=𝜕𝑢+𝜕𝜕𝑥²𝑢𝜕𝑥²,0<𝛼≤1, (19) subject to 𝑢(𝑥,0)=𝑥. (20)
Taking Laplace transform on both sides of (19) gives 𝑠^𝛼ℒ[𝑢(𝑥,𝑡)]−𝑠^𝛼−1𝑢(𝑥,0)=ℒ𝜕𝑢+𝜕𝜕𝑥²𝑢𝜕𝑥²,ℒ[𝑢(=𝑥𝑥,𝑡)]𝑠+1𝑠^𝛼ℒ𝜕𝑢+𝜕𝜕𝑥²𝑢𝜕𝑥². (21)(22)
Operating with Laplace inverse on both sides of (22) results in 𝑢(𝑥,𝑡)=𝑥+ℒ⁻¹1𝑠^𝛼ℒ𝜕𝑢+𝜕𝜕𝑥²𝑢𝜕𝑥². (23)
Substituting (14) and (15) into (23) and applying (17), we obtain the components of the solution as follows: 𝑢₀()()𝑢𝑥,𝑡=𝑢𝑥,0=𝑥,₁(𝑥,𝑡)=ℒ⁻¹1𝑠^𝛼ℒ𝜕𝑢₀+𝜕𝜕𝑥²𝑢₀𝜕𝑥²=𝑡^𝛼Γ(),𝑢1+𝛼₂()𝑢𝑥,𝑡=0,₃(𝑥,𝑡)=⋯=𝑢_𝑛(𝑥,𝑡)=⋯=0. (24)
Therefore, the solution of (19)-(20) in a closed form can be obtained as follows: 𝑢()𝑡𝑥,𝑡=𝑥+^𝛼Γ().1+𝛼 (25)
If we take 𝛼=1, then (25) can be reduced to 𝑢(𝑥,𝑡)=𝑥+𝑡, (26) 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]: 𝐷^𝛼_𝑡𝑢=−𝐷^𝛽_𝑥𝑢𝑥6+𝐷_𝑥^2𝛽𝑢𝑥²,12𝑡>0,𝑥>0,0<𝛼,𝛽≤1, (27) subject to 𝑢(𝑥,0)=𝑥². (28)
Taking Laplace transform on both sides of (27) gives 𝑠^𝛼ℒ[𝑢(𝑥,𝑡)]−𝑠^𝛼−1𝑢(𝑥,0)=ℒ−𝐷^𝛽_𝑥𝑢𝑥6+𝐷_𝑥^2𝛽𝑢𝑥²,ℒ[𝑢(=𝑢()12𝑥,𝑡)]𝑥,0𝑠+1𝑠^𝛼ℒ−𝐷^𝛽_𝑥𝑢𝑥6+𝐷_𝑥^2𝛽𝑢𝑥².12 (29)(30)
Operating with Laplace inverse on both sides of (30), we obtain the following Laplace equation: 𝑢()()𝑥,𝑡=𝑢𝑥,0+ℒ⁻¹1𝑠^𝛼ℒ−𝐷^𝛽_𝑥𝑢𝑥6+𝐷_𝑥^2𝛽𝑢𝑥².12 (31) Following the algorithm given in (17), the first three components of the solution are as follows: 𝑢₀(𝑥,𝑡)=𝑢(𝑥,0)=𝑥²,𝑢₁(𝑥,𝑡)=ℒ⁻¹1𝑠^𝛼ℒ−𝐷^𝛽_𝑥𝑢₀𝑥6+𝐷_𝑥^2𝛽𝑢₀𝑥²=−112Γ𝑥4−𝛽^3−𝛽+2Γ𝑥5−2𝛽^4−2𝛽⋅𝑡^𝛼Γ(),𝑢1+𝛼₂()=𝑥,𝑡4−𝛽𝑥6Γ5−2𝛽^4−2𝛽−5−3𝛽𝑥3Γ6−3𝛽^5−3𝛽−5−𝛽4−𝛽𝑥12Γ6−3𝛽^5−3𝛽+6−2𝛽5−2𝛽6Γ7−4𝛽×𝑥^6−4𝛽⋅𝑡^2𝛼Γ().1+2𝛼 (32)
The solution in series form is then given by 𝑢(𝑥,𝑡)=𝑥²+−1Γ𝑥4−𝛽^3−𝛽+2Γ𝑥5−2𝛽^4−2𝛽⋅𝑡^𝛼Γ()+1+𝛼4−𝛽𝑥6Γ5−2𝛽^4−2𝛽−5−3𝛽𝑥3Γ6−3𝛽^5−3𝛽−5−𝛽4−𝛽𝑥12Γ6−3𝛽^5−3𝛽+6−2𝛽5−2𝛽𝑥6Γ7−4𝛽^6−4𝛽⋅𝑡^2𝛼Γ(1+2𝛼)+⋯. (33)
Setting 𝛼=𝛽=1 in (33), we get the solution of the problem by 𝑢(𝑥,𝑡)=𝑥²𝑡1+2+(𝑡/2)²2!+⋯ (34) and in a closed form by 𝑢(𝑥,𝑡)=𝑥²𝑒^𝑡/2 (35) 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]: 𝐷^𝛼_𝑡𝑢=−𝐷^𝛽_𝑥4𝑢²𝑥−𝑥𝑢3+𝐷_𝑥^2𝛽𝑢²,0<𝛼,𝛽≤1, (36) subject to 𝑢(𝑥,0)=𝑥². (37)
Taking Laplace transform on both sides of (35) gives 𝑠^𝛼ℒ[𝑢(𝑥,𝑡)]−𝑠^𝛼−1𝑢(𝑥,0)=ℒ−𝐷^𝛽_𝑥4𝑢²𝑥−𝑥𝑢3+𝐷_𝑥^2𝛽𝑢²,ℒ[𝑢(=𝑥𝑥,𝑡)]²𝑠+1𝑠^𝛼ℒ−𝐷^𝛽_𝑥4𝑢²𝑥−𝑥𝑢3+𝐷_𝑥^2𝛽𝑢². (38)(39)
Operating with Laplace inverse on both sides of (38), we obtain the following Laplace equation: 𝑢(𝑥,𝑡)=𝑥²+ℒ⁻¹1𝑠^𝛼ℒ−𝐷^𝛽_𝑥4𝑢²𝑥−𝑥𝑢3+𝐷_𝑥^2𝛽𝑢². (40)
Following the algorithm given in (17), the first few components of the solution are as follows: 𝑢₀(𝑥,𝑡)=𝑥²,𝑢₁(𝑥,𝑡)=ℒ⁻¹1𝑠^𝛼ℒ−𝐷^𝛽_𝑥4𝑢²₀𝑥−𝑥𝑢₀3+𝐷_𝑥^2𝛽𝑢²₀=−22Γ𝑥4−𝛽^3−𝛽+24Γ𝑥5−2𝛽^4−2𝛽⋅𝑡^𝛼Γ(),𝑢1+𝛼₂(𝑥,𝑡)=ℒ⁻¹1𝑠^𝛼ℒ−𝐷^𝛽_𝑥4𝑢₀+𝑢₁²𝑥−𝑥𝑢₀+𝑢₁3+𝐷_𝑥^2𝛽𝑢₀+𝑢₁²−ℒ⁻¹1𝑠^𝛼ℒ−𝐷^𝛽_𝑥4𝑢²₀𝑥−𝑥𝑢₀3+𝐷_𝑥^2𝛽𝑢²₀Γ()Γ=−1+2𝛼8−4𝛽Γ²()Γ()ΓΓ⋅1+𝛼1+3𝛼8−5𝛽5−2𝛽2304Γ+5−2𝛽1056Γ𝑡4−𝛽^3𝛼𝑥^7−5𝛽−1Γ(1+2𝛼)Γ6−3𝛽184Γ6−2𝛽Γ+5−2𝛽44Γ6−𝛽Γ4−𝛽⋅𝑡^2𝛼𝑥^5−3𝛽+Γ(1+2𝛼)Γ²()Γ()ΓΓ×1+𝛼1+3𝛼7−4𝛽4−𝛽484Γ7−2𝛽Γ+4−𝛽4224Γ7−3𝛽Γ5−2𝛽⋅𝑡^3𝛼𝑥^6−4𝛽+506Γ5−𝛽3Γ(1+2𝛼)Γ5−2𝛽Γ4−𝛽⋅𝑡^2𝛼𝑥^4−2𝛽−1936Γ(1+2𝛼)Γ6−2𝛽Γ²(1+𝛼)Γ(1+3𝛼)Γ6−3𝛽Γ²4−𝛽⋅𝑡^3𝛼𝑥^5−3𝛽+576Γ(1+2𝛼)Γ9−4𝛽Γ²(1+𝛼)Γ(1+3𝛼)Γ8−5𝛽Γ²5−2𝛽⋅𝑡^3𝛼𝑥^8−6𝛽+48Γ7−2𝛽Γ(1+2𝛼)Γ7−4𝛽Γ5−2𝛽⋅𝑡^2𝛼𝑥^6−4𝛽+22Γ𝑥4−𝛽^3−𝛽−24Γ𝑥5−2𝛽^4−2𝛽⋅𝑡^𝛼Γ(),⋮1+𝛼 (41) The solution in series form is then given by 𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+𝑢₁(𝑥,𝑡)+𝑢₂(𝑥,𝑡)+⋯. (42) If we take 𝛼=𝛽=1, the first few components of the solution are as follows: 𝑢₀(𝑥,𝑡)=𝑥²,𝑢₁(𝑥,𝑡)=𝑥²𝑢𝑡,₂()=𝑥𝑥,𝑡²𝑡²2,⋮ (43)
For this special case, the exact solution of (36) and (37) is therefore given by 𝑢(𝑥,𝑡)=𝑥²𝑒^𝑡 (44) 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.

Table 1 

Several approximate values and exact solutions for (36) and (37) when 𝛼=𝛽=1.

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.