This is the author’s version of a work that was submitted/accepted for pub-lication in the following source:

Liu, Fawang, Anh, Vo V., & Turner, Ian W. (2004) Numerical solution of thespace fractional Fokker-Planck equation. Journal of Computational andApplied Mathematics, 166(1), pp. 209-219.

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http://dx.doi.org/10.1016/j.cam.2003.09.028

Numerical Solution of the Space Fractional

Fokker-Planck Equation ⋆

F. Liu a,b,∗, V. Anh b , I. Turner b

aDepartment of Mathematics, Xiamen University, Xiamen 361005, China

bSchool of Mathematical Sciences, Queensland University of Technology, GPO Box

2434, Brisbane, Qld. 4001, Australia

Abstract

The traditional second-order Fokker-Planck equation may not adequately describethe movement of solute in an aquifer because of large deviation from the dynamicsof Brownian motion. Densities of α-stable type have been used to describe theprobability distribution of these motions. The resulting governing equation of thesemotions is similar to the traditional Fokker-Planck equation except that the orderα of the highest derivative is fractional.

In this paper, a space fractional Fokker-Planck equation (SFFPE) with instan-taneous source is considered. A numerical scheme for solving SFFPE is presented.Using the Riemann-Liouville and Grunwald-Letnikov definitions of fractional deriva-tives, the SFFPE is transformed into a system of ordinary differential equations(ODE). Then the ODE system is solved by a method of lines. Numerical results forSFFPE with a constant diffusion coefficient are evaluated for comparison with theknown analytical solution. The numerical approximation of SFFPE with a time-dependent diffusion coefficient is also used to simulate Levy motion with α−stabledensities. We will show that the numerical method of SFFPE is able to more accu-rately model these heavy-tailed motions.

Key words: fractional derivative, Fokker-Planck equation, α-stable densities, Levymotion, heavy-tailed motionsPACS: 45.10.Hj, 02.60.Cb, 02.70.-c, 02.90.+p

⋆ This research has been supported by the ARC SPIRT grant C10024101 and theNational Natural Science Foundation of China grant 10271098.∗ Corresponding author.

Email addresses: fwliu@xmu.edu.cn (F. Liu), v.anh@fsc.qut.edu.au (V.Anh), i.turner@fsc.qut.edu.au (I. Turner).

Preprint submitted to New Astronomy 28 August 2010

1 Introduction

A Fokker-Planck equation (FPE) has commonly been used to describe theBrownian motion of particles [36]. An FPE describes the change of probabilityof a random function in space and time; hence it is naturally used to describesolute transport. The general FPE for the motion of a concentration fieldC(x, t) of one space variable x at time t has the form [36]

∂C

∂t= [−

∂

∂xD(1)(x) +

∂2

∂x2D(2)(x)]C (x, t) (1)

where D(2)(x) > 0 is the diffusion coefficient and D(1)(x) > 0 is the driftcoefficient. Equation (1) is a linear second-order partial differential equationof parabolic type.

In many studies of diffusion processes where the diffusion takes place in ahighly non-homogeneous medium, the traditional second-order Fokker-Planckequation may not be adequate [5,6]. The non-homogeneities of the mediummay alter the laws of Markov diffusion in a fundamental way. In particular, thecorresponding probability density of the concentration field may have a heaviertail than the Gaussian density, and its correlation function may decay to zeroat a much slower rate than the usual exponential rate of Markov diffusion,resulting in long-range dependence (LRD). This phenomenon is known asanomalous diffusion [10,32].

Fractional derivatives play a key role in modelling particle transport in anoma-lous diffusion. The derivation supposes that the instantaneous density can beapproximated by the 1st and the αth orders, just as the traditional FPE usesthe 1st and the 2th orders. Different assumptions on this probability densityfunction lead to a variety of space-time fractional Fokker-Planck equations(STFFPE). Some STFFPE were successfully used for modelling relevant phys-ical processes (see [5,6,12,15,18,29]).

Space-time fractional Fokker-Planck equations are obtained from the stan-dard FPE by replacing the second order space-derivative and the first ordertime-derivative by fractional derivatives in the Riemann-Liouville sense. Theseequations have recently been treated by a number of authors. Wyss [40] con-sidered the time fractional diffusion equation and the solution is given in closedform in terms of Fox functions. Schneider and Wyss [38] considered the timefractional diffusion and wave equations. The corresponding Green functionsare obtained in closed form for arbitrary space dimensions in terms of Fox func-tions and their properties are exhibited. Gorenflo, Luchko and Mainardi [17]used the similarity method and the method of Laplace transform to obtain thescale-invariant solution of the time-fractional diffusion-wave equation in terms

2

of the Wright function. Benson, Wheatcraft and Meerschaert [5,6] consideredspace fractional advection-dispersion equation. They gave analytic solution interms of the α-stable error function. Liu, Anh, Turnar and Zhuang [28] consid-ered time fractional advection-dispersion equation and derived the completesolution. Mainardi, Luchko and Pagnini [31] considered the space-time frac-tional diffusion equation and provided a general representation of the Greenfunctions in terms of Mellin-Barnes integrals in the complex plane. Anh andLeonenko [2,3] proposed scaling laws for fractional diffusion-wave equationswith singular data. Angulo, Ruiz-Medina, Anh and Grecksch [1] introduceda fractional heat equation, where the diffusion operator is the compositionof the Bessel and Riesz potentials. Wyss [41] considered a fractional Black-Scholes equation and gave a complete solution of this equation. We note thatthe representation of the Green functions is mostly expressed as convergentexpansions in negative and positive power series. These special functions arenot suitable for a numerical evaluation when x is sufficiently small or suffi-ciently large. Therefore, a new numerical strategy is important in solving theseequations.

In this paper, an SFFPE with instantaneous source is considered. The analyt-ical solution of the SFFPE is only gained when the diffusion coefficient andthe drift coefficient are constants, and relatively simple initial and boundaryconditions are imposed on the SFFPE. In the real world, parameters such asthe diffusion coefficient and the drift coefficient are spatially or temporallyvariable. In order to accurately predict natural movements under complex orvariable conditions, a numerical model is required. The numerical solution ofSFPDE is difficult to obtain accurately using standard discretization methods.In this paper, we concentrate on describing the numerical methods withoutstudying the convergence and stablility of the methods from the theoreti-cal point of view. In order to derive the numerical solution of SFFPE, theRiemann-Liouville and Grunwald-Letnikov definitions [37] are used for thespatial approximation. The SFFPE is transformed by using these computa-tionally efficient numerical techniques into a system of ordinary differentialequations (ODE). This ODE system is then solved by using the backwarddifferentiation formulas of order one through five [11]. Computed results forSFFPE with a constant diffusion coefficient are numerically evaluated for com-parison with an analytical solution. The numerical approximation of SFFPEwith a time-dependent diffusion coefficient is also used to simulate Levy mo-tion with α−stable densities. We will show that the numerical method ofSFFPE is able to more accurately model these heavy-tailed motions.

3

2 Statement of the problem

For a large number of independent solute particles the probability propagatoris replaced by the expected concentration [8]. A special case of the SFFPEmay be written as [6]

∂C

∂t= −v

∂C

∂x+ (

1

2+

β

2)D(t)Dα

a+C + (1

2−

β

2)D(t)Dα

b−C (2)

where v is the drift of the process, that is, the mean advective velocity,D = D(t) is the time-dependent coefficient of dispersion, α (1 < α < 2) is theorder of fractional differentiation, −1 ≤ β ≤ 1 indicates the relative weight offorward versus backward transition probability. The most frequently encoun-tered definition of fractional derivatives is the Riemann-Liouville derivative.

Definition: For functions f(x) given in the interval [a, b], the expressions [37]

Dαa+f(x) =

1

Γ(n − α)

dn

dxn

x∫

a

f(ξ)dξ

(x − ξ)α−n+1, (3)

Dαb−f(x) =

(−1)n

Γ(n − α)

dn

dxn

b∫

x

f(ξ)dξ

(ξ − x)α−n+1(4)

are called left-handed and right-handed fractional derivatives of order α (0 ≤n − 1 < α < n, n being an integer) respectively. When α is an integer,Dα = d

α

dxα. Γ(z) is the gamma function. The notion of left and right fractional

derivatives can be considered from a physical viewpoint. The left derivativeis an operation performed on the past states of the process f and the rightderivative is an operation performed on the future states of the process f .

3 Numerical techniques in evaluating the space fractional deriva-

tive

In this section, algorithms for implementing integro-differentiation to frac-tional order α (0 ≤ n − 1 < α < n, n = 2) are devised and evaluated. Thereare two main definitions: Riemann-Liouville and Grunwald-Letnikov fractionalderivatives. There exists a link between these two approaches to differentiationof arbitrary real order. Let us suppose that the function f(x) is n−1 continu-ously differentiable in the interval [a, b] and that f (n)(x) is integrable in [a, b].Then for every α (0 ≤ n− 1 < α < n) the Riemann-Liouville derivative exists

4

and coincides with the Grunwald-Letnikov derivative [34,35]. The fractionalGrunwald-Letnikov definitions with n = 2 are

Dαa+f(x) =

(x − a)−αf(a)

Γ(1 − α)+

(x − a)1−αf ′(a)

Γ(2 − α)

+1

Γ(2 − α)

x∫

a

f (2)(ξ)dξ

(x − ξ)α−1, (5)

Dαb−f(x) =

(b − x)−αf(b)

Γ(1 − α)+

(b − x)1−αf ′(b)

Γ(2 − α)

+1

Γ(2 − α)

b∫

x

f (2)(ξ)dξ

(ξ − x)α−1. (6)

The relationship between the Riemann-Liouville and Grunwald-Letnikov defi-nitions also has another consequence which is important for the numerical ap-proximation of fractional-order differential equations, formulation of appliedproblems, manipulation with fractional derivatives and formulation of phys-ically meaningful initial-value and boundary-value problems for fractional-order differential equations. This allows the use of the Riemann-Liouville def-initions during problem formulation, and then the Grunwald-Letnikov defini-tions for obtaining the numerical solution.

Assume that the spatial domain is [a, b] . The mesh is L intervals {[xj, xj+1]}L−1j=0

of h = (b − a)/L and xj = a + jh for 0 ≤ j ≤ L. Let

f0 = f(x − l × h) = f(a), f1 = f(x − (l − 1)h) = f(a + h), · · ·,

fl−j = f(x − jh), · · ·, fl = f(x) = f(a + l × h)

and

fL = f(b), fL−1 = f(b − h), · · ·, fl+j = f(x + jh), · · ·, fl = f(x).

The second term of the right hand side of equation (5) can be approximatedby

(x − a)1−αf ′(a)

Γ(2 − α)=

h−α

Γ(2 − α)lα−1(f1 − f0). (7)

5

The third term of the right hand side of equation (5) can be approximated by

1

Γ(2 − α)

x∫

a

f (2)(ξ)dξ

(x − ξ)α−1=

1

Γ(2 − α)

x−a∫

0

f (2)(x − ξ)dξ

ξα

=1

Γ(2 − α)

l−1∑

j=0

(j+1)h∫

jh

f (2)(x − ξ)dξ

ξα−1

=1

Γ(2 − α)

l−1∑

j=0

f(x − (j − 1)h) − 2f(x − jh) + f(x − (j + 1)h)

h2

(j+1)h∫

jh

dξ

ξα−1

=h−α

Γ(3 − α)

l−1∑

j=0

(fl−j+1 − 2fl−j + fl−j−1)[(j + 1)2−α − j1−α].

We obtain an approximation of the left-handed fractional derivative with 1 <α < 2 as

Dαa+f(x) =

h−α

Γ(3 − α)

{

(1 − α)(2 − α)f0

lα+

(2 − α)(f1 − f0)

lα−1

+l−1∑

j=0

(fl−j+1 − 2fl−j + fl−j−1)[(j + 1)2−α − j2−α]

. (8)

Similarly, we can derive an approximation of the right-handed fractional deriva-tive with 1 < α < 2 as

Dαb−f(x) =

h−α

Γ(3 − α)

{

(1 − α)(2 − α)fL

(L − l)α+

(2 − α)(fL − fL−1)

(L − l)α−1

+L−l−1∑

j=0

(fl+j−1 − 2fl+j + fl+j+1)[(j + 1)2−α − j2−α]

. (9)

The upwind difference scheme may be used to approximate the convectionterm.

Using the Grunwald-Letnikov definitions (5) and (6), together with numericalapproximation (8), (9) and an upwind scheme, the SFFPE (2) can be castinto the following ordinary differential equation (ODE):

dCl

dt=−v

Cl − Cl−1

h+ (

1

2+

β

2)

Dh−α

Γ(3 − α)

{

(1 − α)(2 − α)C0

lα

6

+(2 − α)

lα−1(C1 − C0) +

l−1∑

j=0

(Cl−j+1 − 2Cl−j + Cl−j−1)[(j + 1)2−α − j2−α]

+ (1

2−

β

2)

Dh−α

Γ(3 − α)

{

(1 − α)(2 − α)CL

(L − l)α+

(2 − α)

(L − l)(α−1)(CL − CL−1)

+L−l−1∑

j=0

(Cl+j−1 − 2Cl+j + Cl+j+1)[(j + 1)2−α − j2−α]

(10)

where Cl = C(t, xl).

4 Method of Lines

The method of lines (MOL) is a well-known technique for solving parabolictype partial differential equations [42]. Essentially, the MOL technique pro-ceeds by leaving the derivatives along one chosen axis untouched (usually intime), while the fractional partial derivatives (in space) are discretised usinga method such as that discussed in Section 3. The system is thereby reducedfrom its PDE to a system of ODE, then integrated in time using an ODEcode. The advantage of the MOL technique is that temporal accuracy can bespecified by the users, therefore the error checking, robustness, order selectionand time-step adaptive features available in sophisticated ODE codes can beapplied to the time integration of the PDE.

In the MOL, time integration is accomplished using an ODE or a differential-algebraic equations (DAE) integrator. Some of the mathematical and numer-ical difficulties in working with these approaches will be encountered. In par-ticular, error control, solution-order adjustment, time-step adjustment, andnonlinear and linear algebraic equation resolution are the central issues ofconcern. A lot of efficient techniques have been proposed [11,14,21,39]. Bre-nan et al. [11] developed the differential/algebraic system solver (DASSL),which is based on the backward difference formulas (BDF). DASSL approxi-mates the derivatives using the k-th order BDF, where k ranges from one tofive. At every step it chooses the order k and stepsize based on the behaviourof the solution. Liu [21] proposed an efficient second-order L-stable variablestep predictor-corrector method for stiff problems, which was applied to thespatially descretizes semi-conductor device equations. In this process, the timestep length was controlled according to the preformance of the solution result-ing from the nonlinear equations and the estimate of the local truncation errorat each time step. Tocci and Kelley [39] proposed an accurate economical so-lution of the pressure-head form of Richards equation by the MOL and haveshown that DASSL is the most efficient approach for the problems considered,often significantly more efficient than the other ODE/MOL approaches.

7

In this work, we used DASSL as our ODE solver. This technique has beenused to solve adsorption problems involving step gradients in bidisperse solids[22,25], hyperbolic models of transport in bidisperse solids [23], transport prob-lems involving steep concentration gradients [24], and modelling saltwater in-trusion into coastal aquifers [26,27].

5 Results and discussion

In this section, the following SFFPE with instantaneous source is considered[5,6]:

∂C

∂t= −v

∂C

∂x+ (

1

2+

β

2)D(t)Dα

a+C + (1

2−

β

2)D(t)Dα

b−C + c0x0δ(t, x)(11)

where c0x0δ(t, x) denotes the initial solute concentration, c0 being the spreadover some injection distance x0 which is mathematically concentrated into adelta function.

In this section, the following two numerical results are shown.

(i) Computed results for the SFFPE with a constant diffusion coefficient arenumerically evaluated for comparison with known analytical solution.

Benson et al. [6] considered symmetric transitions, i.e. β = 0, and using amean-removed equation (i.e. shifting coordinates by the mean travel distance= vt) gave the mass balance equation for symmetric dispersion (or diffusion)using a fractional divergence [4]. Benson et al. [6] used the Fourier transformof fractional derivatives and obtained an analytical solution of the simplifiedsymmetric fractional divergence equation with β = 0 and D = 1.

In Figure 1, the analytical solution [6] and the numerical solution for α = 1.7,t = 1 and t = 10 are shown. From Figure 1, it can be seen that the numericalsolution from our algorithm is in good agreement with the analytical solution.

The parameter β describes the skewness of the transport process. For theSFFPE with a constant diffusion coefficient, the effect of β is shown in Figure2.

(ii) Numerical approximation of the SFFPE is used to simulate Levy motionwith α−stable densities.

If the α−stable solution to Eq. (11) provides a fair representation of the plume,one would also expect to see heavy tails. In order to justify this, numericalapproximation of the SFFPE is used for comparison with an experiment data

8

x-vt-x 0

-20 -10 0 10 20

C

0.00

0.05

0.10

0.15

0.20

0.25

0.30

t=1, Ex. Sol. t=10,Ex. Sol. t=1, Nu. Sol. t=10,Nu. Sol.

Fig. 1. Comparison of the analytical solution (symbols) and numerical solution(lines)

x-vt-x 0

-20 -10 0 10 20

C

0.00

0.02

0.04

0.06

0.08

0.10

t=10, β =1 t=10, β =0 t=10, β =-1

Fig. 2. Showing the effect of β. Here α = 1.7, D = 1, t = 10.

9

Distance from source (m)

0 10 20 30 40 50 60 70 80 90 100

Co

nce

ntr

atio

n C

0

200

400

600

Exp., t=13 Exp., t=55 Exp., t=111 NS., t=13, α=2.0 NS., t=55, α=2.0 NS., t=111, α =2.0 NS., t=13, α=1.8 NS., t=55, α=1.8 NS., t=111, α=1.8

Fig. 3. Comparison of the experiment data (symbols), numerical solution of sec-ond-order equation (dashed lines) and numerical solution of fractional-order equa-tion with α = 1.8 (solid lines) at early time

set. This test was described in more details in Benson’s thesis [4], where heused the data collected by the USGS during a 511-day long tracer test within asand and gravel aquifer on Cape Cod, Massachusetts, and the value 640mg/L×400cm = 0.256g/cm2 for c0 in the x direction estimated by LeBlanc et al. [20]and Garabedian et al. [13].

In numerical approximation, we use β = 1, v = 0.43. Figures 3 and 4 showthe experiment data, second-order FPE with D = 0.42, α = 2 and numericalsolution of SFFPE with α = 1.8, D(t) = D1t

1/α, D1 = 0.02 for early time andlate time, respectively. From Figures 3 and 4, it is seen that both models fitthe experiment data reasonably well.

The SFFPE with time-dependent dispersivity is able to more accurately modelthese heavy-tailed motions. Notable on the plot of Figure 5 is the concave-upward shape of both the data and the α−stable solution tails.

6 Conclusions

In this paper, a numerical method for solving the SFFPE has been describedand demonstrated. The method is based on the Riemann-Liouville and Grunwald-

10

Distance from source (m)

0 100 200 300 400 500 600

Co

nce

ntr

atio

n C

0

20

40

60

80

100

120

140 Exp., t=203 Exp., t=349 Exp., t=511 NS., t=203, α=2.0 NS., t=349, α=2.0 NS., t=511, α=2.0 NS., t=203, α=1.8 NS., t=349, α=1.8 NS., t=511, α=1.8

Fig. 4. Comparison of the experiment data (symbols), numerical solution of sec-ond-order equation (dashed lines) and numerical solution of fractional-order equa-tion with α = 1.8 (solid lines) at late time.

Letnikov definitions of fractional derivatives. This method has been testedagainst the SFFPE with a constant diffusion coefficient. Good agreement be-tween the numerical solution and analytic solution has been noted.

The numerical approximation of SFFPE with a time-dependent diffusion co-efficient is also used to simulate Levy motion with α−stable densities. Themethod is able to describe these heavy-tailed motions more accurately.

Space and time fractional partial differential equations are difficult to solveaccurately using standard discretisation methods. Further research includingnumerical analysis needs to be carried out.

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