Ain Shams Engineering Journal (2015) 6, 1211–1216

Ain Shams University

Ain Shams Engineering Journal

www.elsevier.com/locate/asejwww.sciencedirect.com

ENGINEERING PHYSICS AND MATHEMATICS

Meshless method for the numerical solution

of the Fokker–Planck equation

* Corresponding author. Tel.: +98 21 64542524.E-mail addresses: maysam_askari@yahoo.com (M. Askari), adibih@

aut.ac.ir (H. Adibi).

Peer review under responsibility of Ain Shams University.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.asej.2015.04.0122090-4479 � 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Maysam Askari, Hojatollah Adibi *

Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran

Received 23 January 2015; revised 12 April 2015; accepted 29 April 2015

Available online 13 June 2015

KEYWORDS

Fokker–Planck equation;

Meshless method;

Multiquadric;

Collocation

Abstract In this paper numerical meshless method for solving Fokker–Planck equation is consid-

ered. This meshless method is based on multiquadric radial basis function and collocation method

to approximate the solution. Here we apply h-weighted finite difference method. The stability anal-

ysis of the method is dealt with, using a linearized stability method. Numerical examples illustrate

the validity and applicability of the method.� 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an

open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The Fokker–Planck equation was first applied to the Brownianmotion by Fokker and Planck. This equation is used in a num-

ber of different fields of physics, chemistry and biology such assolid-state physics, quantum optics, chemical physics, theoret-ical biology and circuit theory [1]. The general Fokker–Planckequation for the variable x has the form

@u

@t¼ � @

@xAðxÞ þ @2

@x2BðxÞ

� �u ð1Þ

with the initial condition

uðx; 0Þ ¼ fðxÞ; x 2 R

where AðxÞ > 0;BðxÞ > 0 are the drift and diffusion coeffi-

cients respectively. The drift and diffusion coefficients mayalso depend on time, i.e.

@u

@t¼ � @

@xAðx; tÞ þ @2

@x2Bðx; tÞ

� �u ð2Þ

Eq. (2) is the equation of motion for the distribution func-

tion uðx; tÞ. Mathematically, this equation is a parabolic linearsecond order partial differential equation. Roughly speaking,it is a diffusion equation with an additional first-order deriva-

tive with respect to x. In mathematical literature, Eq. (2) iscalled the forward Kolmogorov equation. The similar partialdifferential equation

@u

@t¼ �Aðx; tÞ @

@xþ Bðx; tÞ @

2

@x2

� �u ð3Þ

is called backward Kolmogorov equation. A generalization of(1) to N variables X ¼ ðx1; . . . ; xNÞ has the form

@u

@t¼ �

XNi¼1

@

@xi

AiðXÞ þXNi;j¼1

@2

@xi@xj

Bi;jðXÞ" #

u ð4Þ

uðX; 0Þ ¼ fðXÞ; X 2 RN

1212 M. Askari, H. Adibi

Notice that there is a more general form of Fokker–Planckequation, which is a nonlinear equation and has importantapplications in various areas including plasma physics, surface

physics, population dynamics, biophysics, neurosciences, non-linear hydrodynamics, polymer physics, laser physics, patternformation, psychology and marketing (see [2] and references

therein). In one variable this equation has the form

@u

@t¼ � @

@xAðx; t; uÞ þ @2

@x2Bðx; t; uÞ

� �u: ð5Þ

and for N variables x1; . . . ; xN this equation can be written as

@u

@t¼ �

XNi¼1

@

@xi

AiðX; t; uÞ þXNi;j¼1

@2

@xi@xj

Bi;jðX; t; uÞ" #

u ð6Þ

where X ¼ ðx1; . . . ; xNÞ.For the solution of the Fokker–Planck equation various

methods are proposed. In [3] a fast and accurate algorithmfor numerical solution of Fokker–Planck equation was pre-

sented. Reif and Barakat [4] employed Chebyschev approxima-tions to solve the one-dimensional, time-dependent Fokker–Planck equation in presence of two barriers a finite ‘‘distance’’

apart. Harrison applied a variation of the moving finite ele-ment method for numerical solution of Fokker–Planck equa-tion [5]. In [6] differential transform method (DTM) was

applied to devise a simple scheme for solving Fokker–Planckequation and some similar equations. Salehian et al. [7] usedthe flatlet oblique multiwavelets for the solution of Fokker–Planck equation. Lakestani and Dehghan applied cubic B-

spline scaling functions for solving Fokker–Planck equation[8]. Authors of [2] investigated the application of Adomiandecomposition method for solving Fokker–Planck equation.

In [9] two numerical meshless methods based on radial basisfunction are presented. In [10] homotopy perturbation method(HPM) was considered to solve the linear and nonlinear

Fokker–Planck equation. Authors of [11] implementedthe He’s variational iteration method (VIM) for solvingFokker–Planck equation. In [12] the combined Hermite

spectral-upwinding difference methods were applied to theFokker–Planck equation and showed that the Hermite basedspectral methods were convergent with spectral accuracy inweighted Sobolev space.

Radial basis function (RBF) methods were first studied byHardy for the approximation of two-dimensional geographi-cal surfaces [13]. These methods can be easily used for scat-

tered data approximation. The existence, uniqueness andconvergence of the RBF methods were studied by manyresearchers. In 1986 Micchelli [14] showed that for distinct

interpolation points the generated matrix from multiquadric(MQ) method is invertible. In [15], Madych and Nelsonshowed that MQ method has spectral convergence rate. We

know by Schoenberg theorem [16] for distinct interpolationpoints the system of matrices for GA, IMQ and IQ are pos-itive definite. In RBF methods the shape parameter hasimportant rule in accuracy of the solution. Although many

researchers work on the optimal value of the shape parame-ter, but the optimal choice of the shape parameter is still anopen problem [17–19]. Radial basis function methods for

solving partial differential equations in probability have alsobeen developed by Ballestra and Pacelli. Authors of [20]proposed a numerical method to compute the survival

(first-passage) probability density function in jump-diffusion

models. This function is obtained by numerical approxima-tion of the associated Fokker–Planck partial integro-differential equation, with suitable boundary conditions and

delta initial condition. In [21] the Fokker–Planck partialintegro-differential equation associated to the problem ofcomputing the survival (first-passage) probability density

function of jump-diffusion models with two stochastic factorsis solved using a meshless collocation approach based onradial basis functions.

In this paper we present a numerical meshless methodbased on multiquadric radial basis function to approximatethe solution of the Fokker–Planck equation by using colloca-tion method. For this aim we apply h-weighted finite difference

method. The stability analysis of the method is investigated byusing a linearized stability method.

The outline of this paper is as follows: In Section 2 we pro-

pose the meshless method for solution of the Fokker–Planckequation. In Section 3 the stability analysis of the method isdealt with using a linearized stability method. In Section 4

numerical results are presented for some examples and wecompare these results with exact solutions and at the end inSection 5 we a have brief conclusion.

2. Meshless method for the solution of Fokker–Planck equation

Consider the Fokker–Planck equation

@u

@t¼ � @

@xAðx; tÞ þ @2

@x2Bðx; tÞ

� �u; x 2 ½a; b�; t P 0 ð7Þ

with the following initial and boundary conditions

uðx; 0Þ ¼ fðxÞ

uða; tÞ ¼ gaðtÞ; uðb; tÞ ¼ gbðtÞ

For solving the Fokker–Planck equation we discretize Eq.

(7) using h-weighted (0 6 h 6 1) finite difference method as

unþ1 � un

Dtþ h

@

@xðAuÞ � @2

@x2ðBuÞ

� �nþ1

þ ð1� hÞ @

@xðAuÞ � @2

@x2ðBuÞ

� �n¼ 0 ð8Þ

where un ¼ uðx; tnÞ and Dt is a time step size. This equation can

be rewritten as

unþ1 þ hDt@

@xðAuÞ � @2

@x2ðBuÞ

� �nþ1

¼ un þ ðh� 1ÞDt @

@xðAuÞ � @2

@x2ðBuÞ

� �nð9Þ

or

unþ1þhDt@An

@xunþ1þ@u

nþ1

@xAn�@

2Bn

@x2unþ1�2

@Bn

@x

@unþ1

@x�Bn @

2unþ1

@x2

� �

¼ unþðh�1ÞDt @

@xðAuÞ� @2

@x2ðBuÞ

� �nð10Þ

where An ¼ Aðx; tnÞ and Bn ¼ Bðx; tnÞ.Now we choose the nodes xi; i ¼ 1; . . . ;N over interval

½a; b� such that xi; i ¼ 2; . . . ;N� 1 are interior and x1; xN are

boundary points and approximate unþ1 using combination of

RBFs as

Table 1 Some well-known radial basis functions.

Gaussian (GA) /ðrÞ ¼ e�c2r2

Inverse quadric (IQ) /ðrÞ ¼ 1r2þc2

Inverse multiquadric (IMQ) /ðrÞ ¼ 1ffiffiffiffiffiffiffiffiffir2þc2p

Multiquadric (MQ) /ðrÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ c2p

Thin plate spline (TPS) /ðrÞ ¼ r2mlogðrÞ

Meshless method for the numerical solution of the Fokker–Planck equation 1213

uðx; tnþ1Þ ’XNj¼1

knþ1j /jðrÞ ð11Þ

where /jðrÞ ¼ /ðkx� xjkÞ and k � k denotes the Euclidean

norm. Some of the commonly used RBFs are presented inTable 1.

By substituting Eq. (11) into Eq. (10) and collocating at

each node xi; i ¼ 2; . . . ;N� 1 we have

XNj¼1

knþ1j /jðxiÞþhDt

@An

@xðxiÞXNj¼1

knþ1j /jðxiÞþAnðxiÞ

XNj¼1

knþ1j /0jðxiÞ

"

�@2Bn

@x2ðxiÞXNj¼1

knþ1j /jðxiÞ�2

@Bn

@xðxiÞXNj¼1

knþ1j /0jðxiÞ

�BnðxiÞXNj¼1

knþ1j /00j ðxiÞ

#¼ unþðh�1ÞDt

� @

@xðAuÞ� @2

@x2ðBuÞ

� �nðxiÞ ð12Þ

Using simple manipulation the above equation becomes

XNj¼1

knþ1j

�1þhDt

@An

@xðxiÞ�hDt

@2Bn

@x2ðxiÞ

�/jðxiÞþhDt

�AnðxiÞ�2

@Bn

@xðxiÞ

�u0ðxiÞ

�

�hDtBnðxiÞ/00j ðxiÞi¼ unþðh�1ÞDt @

@xðAuÞ� @2

@x2ðBuÞ

� �nðxiÞ ð13Þ

Also, by using boundary conditions at boundary nodes

x1; xN we have

XNj¼1

knþ1j /jðx1Þ ¼ gaðtnþ1Þ ð14Þ

XNj¼1

knþ1j /jðxNÞ ¼ gbðtnþ1Þ ð15Þ

So, the following system of linear equations obtains

MKnþ1 ¼ b ð16Þ

where

Mij ¼Di/jðxiþ1ÞþEi/

0jðxiþ1Þ�hDtBnðxiþ1Þ/00j ðxiþ1Þ; 16 i6N�2

/jðx1Þ; i¼N�1

/jðxNÞ; i¼N

8><>:

ð17Þ

Table 2 L2 and L1 errors of multiquadric for Example 4.1

Time t = 1 t = 2 t = 3

L1 1.900e�12 2.900e�12 3.900e�12L2 1.444e�12 2.384e�12 3.329e�12

bi¼

unðxiþ1Þþðh�1ÞDt @@xðAuÞ� @2

@x2ðBuÞ

h inðxiþ1Þ; 16 i6N�2

gaðtnþ1Þ; i¼N�1

gbðtnþ1Þ; i¼N

8>>>>><>>>>>:

ð18Þ

Knþ1 ¼ ½knþ11 ; . . . ; knþ1

N �

Di ¼ 1þ hDt@An

@xðxiþ1Þ � hDt

@2Bn

@x2ðxiþ1Þ;

Ei ¼ hDt An � 2@Bn

@x

� �ðxiþ1Þ

For the nonlinear Fokker–Planck Eq. (7) we can use thediscretization

unþ1 � un

Dtþ h

@

@xðAnunþ1Þ � @2

@x2ðBnunþ1Þ

� �

þ ð1� hÞ @

@xðAuÞ � @2

@x2ðBuÞ

� �n¼ 0 ð19Þ

where An ¼ Aðx; tn; unÞ, Bn ¼ Bðx; tn; unÞ.We can summarize the proposed method in the following

algorithm:Algorithm of the method

� Step 1: Choose N collocation points in ½a; b�.� Step 2: Choose Dt and 0 6 h 6 1.

� Step 3: Calculate

U0 ¼ ½u0ðx1Þ; . . . ; u0ðxN Þ�;U0x ¼ ½u0

xðx1Þ; . . . ; u0xðxN Þ� and

U0xx ¼ ½u0

xxðx1Þ; . . . ; u0xxðxN Þ� using initial condition.

� Step 4: Set n :¼ 0� Step 5: Construct the matrix M and vector b.

� Step 6: Solve the system of linear equation MKnþ1 ¼ b.� Step 7: Set n :¼ nþ 1.� Step 8: If nDt < T (T is final time), go to step 5 else stop.

Remark 2.1. Note that the presented method is valid forany value of h 2 ½0; 1�, but we use h ¼ 1=2 (the famous

Crank-Nicholson scheme).

3. Stability analysis

Here we apply linear stability analysis method for investigatingthe stability of the presented method. Although, the

application of the linear stability analysis to nonlinearequations cannot be rigorously justified, it provides, however,the necessary conditions for stability, and it is found to be

effective in practice [22,23].

t = 5 t = 7 t= 10

5.899e�12 7.899e�12 1.090e�115.224e�12 7.119e�12 9.964e�12

Figure 1 Maximum error of multiquadric for Example 4.1.

Figure 2 Maximum error of multiquadric for Example 4.2.

1214 M. Askari, H. Adibi

At first we freeze locally one variable in nonlinear terms. Byusing the above method for the locally constant Fokker–Planck equation

@u

@t¼ � @

@xAþ @2

@x2B

� �u ð20Þ

and set knj /jðxkÞ ¼ nn

j eibxk , by using the Fourier analysis and

presented method in previous section, for each k ¼ 1; . . . ;Nwe have

nnþ1j 1þ hDt

@A

@x� hDt

@2B

@x2

� �eibxk þ hDt A� 2

@B

@x

� �ibnnþ1

j eiBxk

þ hDtBb2nnþ1j eibxk ¼ nn

j eibxk þ ðh� 1ÞDtnn

j eibxk

@A

@xþ iAb� @

2B

@x2� 2

@B

@xibþ Bb2

� �: ð21Þ

Then

nj 1þ hDt@A

@x� hDt

@2B

@x2þ hDt A� 2

@B

@x

� �ibþ hDtBb2

� �

¼ 1þ ðh� 1ÞDt @A@xþ iAb� @

2B

@x2� 2

@B

@xibþ Bb2

� �:

ð22Þ

So

nj ¼1þ ðh� 1ÞDt @A

@x� @2B

@x2þ Bb2

� �þ i Ab� 2 @B

@xb

h i1þ hDt @A

@x� @2B

@x2þ Bb2

� �þ i Ab� 2 @B

@xb

h i ð23Þ

For h ¼ 12this relation becomes

Table 3 L2 and L1 errors of multiquadric for Example 4.2.

Time t = 1 t = 2 t = 3

L1 4.008e�10 5.037e�10 4.004e�L2 2.097e�10 2.913e�10 1.999e�

nj ¼1� 1

2Dt @A

@x� @2B

@x2þ Bb2

� �þ iðAb� 2 @B

@xbÞ

h i1þ 1

2Dt @A

@x� @2B

@x2þ Bb2

� �þ i Ab� 2 @B

@xb

h i ð24Þ

if we set K1 ¼ @A@x� @2B

@x2þ Bb2 and K2 ¼ Ab� 2 @B

@xb we have

nj ¼1� 1

2Dt½K1 þ iK2�

1þ 12Dt½K1 þ iK2�

ð25Þ

Then

jnjj2 ¼1� 1

2DtK1

2 þ 14ðDtÞ2K2

2

1þ 12DtK1

2 þ 14ðDtÞ2K2

2

¼1þ 1

4ðDtÞ2K2

1 þ 14ðDtÞ2K2

2

h i� DtK1

1þ 14ðDtÞ2K2

1 þ 14ðDtÞ2K2

2

h iþ DtK1

: ð26Þ

Notice that if K1 P 0 then jnjj2 6 1.

From the above discussion we proved that for h ¼ 12if

K1 P 0 then jnjj 6 1. therefore, the necessary condition for

the stability of the proposed method holds.

4. Numerical results

In this section we present some examples to illustrate the

numerical results of the previous section. Here we use twonorms

L2 ¼ ku� ~uk2 ¼ DxXNj¼1juj � ~ujj2

!12

t = 5 t = 7 t = 10

9 3.004e�8 9.869e�10 5.011e�69 1.185e�8 5.853e�10 2.345e�6

Table 4 L2 and L1 errors of multiquadric for Example 4.3.

Time t = 1 t = 2 t = 3 t = 5 t = 7 t = 10

L1 4.421e�10 5.511e�10 4.032e�9 4.984e�8 4.019e�7 4.967e�6L2 2.825e�10 3.109e�10 2.830e�9 2.912e�8 2.829e�7 2.913e�6

Figure 3 Maximum error of multiquadric for Example 4.3. Figure 4 Maximum error of multiquadric for Example 4.4.

Meshless method for the numerical solution of the Fokker–Planck equation 1215

L1 ¼ ku� ~uk1 ¼ max16j6N

juj � ~ujj

where u and ~u are the exact and approximate solutions respec-

tively and we consider fxi ¼ i � Dx; i ¼ 0; . . . ;Ng. In all exam-ples we use Dt ¼ 0:01;Dx ¼ 0:1; a ¼ 0 and b ¼ 1, also, theapproximate solutions are computed using multiquadric radial

basis function. The computations have been performed inMAPLE 16, with hardware configuration: desktop 32-bitIntel core 2 Duo CPU, 2G of RAM.

Example 4.1. Consider [8,9] the Eq. (1) withAðxÞ ¼ �1;BðxÞ ¼ 1 and fðxÞ ¼ x. The exact solution of this

problem is uðx; tÞ ¼ xþ t. Here we choose c ¼ 10�12 and by

using the presented meshless method, compute the approxi-mate solution. Table 2 shows the L1 and L2 errors of thepresented method at t= 1, 2, 3, 5, 7, 10, using multiquadric.

Also, Fig. 1 displays the maximum error of multiquadric int 2 ½0; 10�.

Example 4.2. Consider [8,9] the Eq. (1) with

AðxÞ ¼ x;BðxÞ ¼ x2

2and fðxÞ ¼ x. The exact solution of this

problem is uðx; tÞ ¼ xet. Here we choose c ¼ 10�12 and by

using the presented meshless method, compute the

Table 5 L2 and L1 errors of multiquadric for Example 4.4.

Time t = 1 t = 2 t = 3

L1 4.500e�13 6.007e�11 2.250e�13L2 2.669e�13 2.852e�11 1.335e�13

approximate solution. Table 3 shows the L1 and L2 errorsof the presented method at t = 1, 2, 3, 5, 7, 10, using multi-

quadric. Also, Fig. 2 displays the maximum error of multi-quadric in t 2 ½0; 10�.

Example 4.3. Consider [8,9] the backward Kolmogorov Eq.

(3) with Aðx; tÞ ¼ �ðxþ 1Þ;Bðx; tÞ ¼ x2et and fðxÞ ¼ xþ 1.The exact solution of this problem is uðx; tÞ ¼ ðxþ 1Þet.Here we choose c ¼ 10�12 and by using the presented meshless

method, compute the approximate solution. Table 4 shows theL1 and L2 errors of the presented method at t = 1, 2, 3, 5, 7,10, using multiquadric. Also, Fig. 3 displays the maximum

error of multiquadric in t 2 ½0; 10�.

Example 4.4. Consider [8,9] the Eq. (5) with

Aðx; t; uÞ ¼ 72u;Bðx; t; uÞ ¼ xu and fðxÞ ¼ x. The exact solution

of this problem is uðx; tÞ ¼ xtþ1. Here we choose c ¼ 10�12 and

by using the presented meshless method, compute the approx-

imate solution. Table 5 shows the L1 and L2 errors of the pre-sented method at t= 1, 2, 3, 5, 7, 10 using multiquadric. Also,Fig. 4 displays the maximum error of multiquadric int 2 ½0; 10�.

t = 5 t = 7 t= 10

5.997e�11 1.125e�13 4.955e�122.384e�11 6.674e�14 2.881e�12

1216 M. Askari, H. Adibi

5. Conclusion

We applied the meshless method using multiquadric radialbasis functions to solve the Fokker–Planck equation. Here

we used h-weighted finite difference method and then approx-imated the solution of the equation using the combination ofradial basis functions. Then by using the collocation method,

we gained the system of linear equations. Also, we presentedthe stability analysis of the method using linearized stabilityanalysis.

Acknowledgements

This paper has been extracted from the first author’s Ph.D.

thesis, which has been supported by Islamic Azad University,Central Tehran Branch.The authors would like to thank anonymous reviewers for

their careful reading and constructive comments which havehelped improved the quality of the paper.

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equation. Comput Phys Comm 2012;183:1259–68.

Maysam Askari He has received his B.Sc. in

applied mathematics from Amirkabir

University of Technology, Iran 2001. He has

received his M.Sc. in applied mathematics

from the same university in 2003. Currently he

is a Ph.D. student in applied mathematics at

the department of mathematics, Islamic Azad

university, Central Tehran Branch, Iran. His

fields of interest are numerical methods for the

solution of ordinary and partial differential

equations, Integral equations.

Hojatollah Adibi He was born in Iran in1948,

completed High School in 1966. He obtained

his Bachelor’s in mathematics in 1970 from the

Teacher’s Training University in Tehran. He

continued his studies and gained his M.Sc.

from the Sharif University of Technology in

1974. After two years teaching at the School of

Planing and Computer Applications, he

became employed as a lecturer of mathematics

at the Amirkabir University of Technology in

1976. From October 1984 he started his post-

graduate studies at the City University in London and obtained his

Ph.D. in applied mathematics under the supervision of Professor M.A.

Jaswon in 1989. Then he returned back to Iran and continued his

teaching and research at the Amirkabir University of Technology. So

far, he has supervised three PhD students and Currently has about forty

published paper mainly in ISI journals.