Meshless method for the numerical solution of the ...

Ain Shams Engineering Journal (2015) xxx, xxx–xxx

Ain Shams University

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ENGINEERING PHYSICS AND MATHEMATICS

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

KEYWORDS Fokker–Planck equation; Meshless method; Multiquadric; Collocation

Abstract In this paper numerical meshless method for solving Fokker–Planck equation is considered. 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 analysis of the method is dealt with, using a linearized stability method. Numerical examples illustrate the validity and applicability of the method. Ó 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 Brownian motion by Fokker and Planck. This equation is used in a number of different fields of physics, chemistry and biology such as solid-state physics, quantum optics, chemical physics, theoretical biology and circuit theory [1]. The general Fokker–Planck equation for the variable x has the form @u @ @2 ð1Þ ¼ AðxÞ þ 2 BðxÞ u @t @x @x with the initial condition uðx; 0Þ ¼ fðxÞ;

x2R

* Corresponding author. Tel.: +98 21 64542524. E-mail addresses: [email protected] (M. Askari), adibih@ aut.ac.ir (H. Adibi). Peer review under responsibility of Ain Shams University.

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where AðxÞ > 0; BðxÞ > 0 are the drift and diffusion coefficients respectively. The drift and diffusion coefficients may also depend on time, i.e. @u @ @2 ¼ Aðx; tÞ þ 2 Bðx; tÞ u ð2Þ @x @t @x Eq. (2) is the equation of motion for the distribution function uðx; tÞ. Mathematically, this equation is a parabolic linear second order partial differential equation. Roughly speaking, it is a diffusion equation with an additional first-order derivative with respect to x. In mathematical literature, Eq. (2) is called the forward Kolmogorov equation. The similar partial differential equation @u @ @2 ð3Þ ¼ Aðx; tÞ þ Bðx; tÞ 2 u @x @t @x is called backward Kolmogorov equation. A generalization of (1) to N variables X ¼ ðx1 ; . . . ; xN Þ has the form " # N N X X @u @ @2 ¼ Ai ðXÞ þ Bi;j ðXÞ u ð4Þ @xi @xj @t @xi i¼1 i;j¼1 uðX; 0Þ ¼ fðXÞ;

X 2 RN

http://dx.doi.org/10.1016/j.asej.2015.04.012 2090-4479 Ó 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/). Please cite this article in press as: Askari M, Adibi H, Meshless method for the numerical solution of the Fokker–Planck equation, Ain Shams Eng J (2015), http:// dx.doi.org/10.1016/j.asej.2015.04.012

2 Notice that there is a more general form of Fokker–Planck equation, which is a nonlinear equation and has important applications in various areas including plasma physics, surface physics, population dynamics, biophysics, neurosciences, nonlinear hydrodynamics, polymer physics, laser physics, pattern formation, psychology and marketing (see [2] and references therein). In one variable this equation has the form @u @ @2 ð5Þ ¼ Aðx; t; uÞ þ 2 Bðx; t; uÞ u: @t @x @x and for N variables x1 ; . . . ; xN this equation can be written as " # N N X X @u @ @2 Bi;j ðX; t; uÞ u ¼ Ai ðX; t; uÞ þ ð6Þ @t @xi @xi @xj i¼1 i;j¼1 where X ¼ ðx1 ; . . . ; xN Þ. For the solution of the Fokker–Planck equation various methods are proposed. In [3] a fast and accurate algorithm for numerical solution of Fokker–Planck equation was presented. Reif and Barakat [4] employed Chebyschev approximations 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 element method for numerical solution of Fokker–Planck equation [5]. In [6] differential transform method (DTM) was applied to devise a simple scheme for solving Fokker–Planck equation and some similar equations. Salehian et al. [7] used the flatlet oblique multiwavelets for the solution of Fokker– Planck equation. Lakestani and Dehghan applied cubic Bspline scaling functions for solving Fokker–Planck equation [8]. Authors of [2] investigated the application of Adomian decomposition method for solving Fokker–Planck equation. In [9] two numerical meshless methods based on radial basis function are presented. In [10] homotopy perturbation method (HPM) was considered to solve the linear and nonlinear Fokker–Planck equation. Authors of [11] implemented the He’s variational iteration method (VIM) for solving Fokker–Planck equation. In [12] the combined Hermite spectral-upwinding difference methods were applied to the Fokker–Planck equation and showed that the Hermite based spectral methods were convergent with spectral accuracy in weighted Sobolev space. Radial basis function (RBF) methods were first studied by Hardy for the approximation of two-dimensional geographical surfaces [13]. These methods can be easily used for scattered data approximation. The existence, uniqueness and convergence of the RBF methods were studied by many researchers. In 1986 Micchelli [14] showed that for distinct interpolation points the generated matrix from multiquadric (MQ) method is invertible. In [15], Madych and Nelson showed that MQ method has spectral convergence rate. We know by Schoenberg theorem [16] for distinct interpolation points the system of matrices for GA, IMQ and IQ are positive definite. In RBF methods the shape parameter has important rule in accuracy of the solution. Although many researchers work on the optimal value of the shape parameter, but the optimal choice of the shape parameter is still an open problem [17–19]. Radial basis function methods for solving partial differential equations in probability have also been developed by Ballestra and Pacelli. Authors of [20] proposed a numerical method to compute the survival (first-passage) probability density function in jump-diffusion

M. Askari, H. Adibi models. This function is obtained by numerical approximation of the associated Fokker–Planck partial integrodifferential equation, with suitable boundary conditions and delta initial condition. In [21] the Fokker–Planck partial integro-differential equation associated to the problem of computing the survival (first-passage) probability density function of jump-diffusion models with two stochastic factors is solved using a meshless collocation approach based on radial basis functions. In this paper we present a numerical meshless method based on multiquadric radial basis function to approximate the solution of the Fokker–Planck equation by using collocation method. For this aim we apply h-weighted finite difference method. The stability analysis of the method is investigated by using a linearized stability method. The outline of this paper is as follows: In Section 2 we propose the meshless method for solution of the Fokker–Planck equation. In Section 3 the stability analysis of the method is dealt with using a linearized stability method. In Section 4 numerical results are presented for some examples and we compare these results with exact solutions and at the end in Section 5 we a have brief conclusion. 2. Meshless method for the solution of Fokker–Planck equation Consider the Fokker–Planck equation @u @ @2 ¼ Aðx; tÞ þ 2 Bðx; tÞ u; x 2 ½a; b; t P 0 @t @x @x

ð7Þ

with the following initial and boundary conditions uðx; 0Þ ¼ fðxÞ uða; tÞ ¼ ga ðtÞ;

uðb; tÞ ¼ gb ðtÞ

For solving the Fokker–Planck equation we discretize Eq. (7) using h-weighted (0 6 h 6 1) finite difference method as nþ1 unþ1 un @ @2 þh ðAuÞ 2 ðBuÞ @x Dt @x n @ @2 ð8Þ ðAuÞ 2 ðBuÞ ¼ 0 þ ð1 hÞ @x @x where un ¼ uðx; tn Þ and Dt is a time step size. This equation can be rewritten as nþ1 @ @2 u þ hDt ðAuÞ 2 ðBuÞ @x @x n @ @2 ðAuÞ 2 ðBuÞ ¼ un þ ðh 1ÞDt @x @x nþ1

ð9Þ

or n @A nþ1 @unþ1 n @ 2 Bn nþ1 @Bn @unþ1 @ 2 unþ1 u þ A u 2 Bn 2 2 @x @x @x @x @x @x n 2 @ @ ð10Þ ðAuÞ 2 ðBuÞ ¼ un þ ðh 1ÞDt @x @x

unþ1 þ hDt

where An ¼ Aðx; tn Þ and Bn ¼ Bðx; tn Þ. 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

Please cite this article in press as: Askari M, Adibi H, Meshless method for the numerical solution of the Fokker–Planck equation, Ain Shams Eng J (2015), http:// dx.doi.org/10.1016/j.asej.2015.04.012

Meshless method for the numerical solution of the Fokker–Planck equation Table 1

Some well-known radial basis functions. 2 2

Gaussian (GA) Inverse quadric (IQ)

/ðrÞ ¼ ec r 1 /ðrÞ ¼ r2 þc 2 1 ffi /ðrÞ ¼ pffiffiffiffiffiffiffiffi 2 þc2 prffiffiffiffiffiffiffiffiffiffiffiffiffiffi /ðrÞ ¼ r2 þ c2 /ðrÞ ¼ r2m logðrÞ

Inverse multiquadric (IMQ) Multiquadric (MQ) Thin plate spline (TPS)

N X uðx; tnþ1 Þ ’ knþ1 /j ðrÞ j

ð11Þ

j¼1

where /j ðrÞ ¼ /ðkx xj kÞ and k k denotes the Euclidean norm. Some of the commonly used RBFs are presented in Table 1. By substituting Eq. (11) into Eq. (10) and collocating at each node xi ; i ¼ 2; . . . ; N 1 we have " N N N X X X @An nþ1 kj /j ðxi Þ þ hDt /0j ðxi Þ /j ðxi Þ þ An ðxi Þ knþ1 ðxi Þ knþ1 j j @x j¼1 j¼1 j¼1 N N X X @ 2 Bn @Bn nþ1 knþ1 /0j ðxi Þ / k ðx Þ 2 ðx Þ ðx Þ i i i j j j @x2 @x j¼1 j¼1 # N X 00 / ðx Þ Bn ðxi Þ knþ1 ¼ un þ ðh 1ÞDt i j j

bi ¼

3

h in 8 @ @2 > un ðxiþ1 Þ þ ðh 1ÞDt @x ðAuÞ @x ðxiþ1 Þ; 1 6 i 6 N 2 2 ðBuÞ > > > > < nþ1

ga ðt Þ; > > > > > : gb ðtnþ1 Þ;

i¼N1 i¼N

ð18Þ

nþ1 Knþ1 ¼ ½knþ1 1 ; . . . ; kN

@An @ 2 Bn Di ¼ 1 þ hDt ðxiþ1 Þ hDt ðxiþ1 Þ; @x @x2 n @B Ei ¼ hDt An 2 ðxiþ1 Þ @x For the nonlinear Fokker–Planck Eq. (7) we can use the discretization unþ1 un @ @2 þh ðAn unþ1 Þ 2 ðBn unþ1 Þ Dt @x @x n 2 @ @ ðAuÞ 2 ðBuÞ ¼ 0 þ ð1 hÞ ð19Þ @x @x where An ¼ Aðx; tn ; un Þ, Bn ¼ Bðx; tn ; un Þ. We can summarize the proposed method in the following algorithm: Algorithm of the method

j¼1

n @ @2 ðAuÞ 2 ðBuÞ ðxi Þ @x @x

ð12Þ

Using simple manipulation the above equation becomes N X

@An @ 2 Bn @Bn ðxi Þ u0 ðxi Þ ðxi Þ hDt 2 ðxi Þ /j ðxi Þ þ hDt An ðxi Þ 2 @x @x @x j¼1 n 2 i @ @ ð13Þ ðAuÞ 2 ðBuÞ ðxi Þ hDtBn ðxi Þ/00j ðxi Þ ¼ un þ ðh 1ÞDt @x @x knþ1 j

1 þ hDt

Also, by using boundary conditions at boundary nodes x1 ; xN we have N X knþ1 /j ðx1 Þ ¼ ga ðtnþ1 Þ j

ð14Þ

N X

ð15Þ

j¼1

j¼1

knþ1 /j ðxN Þ ¼ gb ðtnþ1 Þ j

So, the following system of linear equations obtains MKnþ1 ¼ b

Remark 2.1. Note that the presented method is valid for any value of h 2 ½0; 1, but we use h ¼ 1=2 (the famous Crank-Nicholson scheme). 3. Stability analysis

ð16Þ

where 8 00 0 n > < Di /j ðxiþ1 Þ þ Ei /j ðxiþ1 Þ hDtB ðxiþ1 Þ/j ðxiþ1 Þ; 1 6 i 6 N 2 i¼N1 Mij ¼ /j ðx1 Þ; > : i¼N /j ðxN Þ;

ð17Þ

Table 2

Step 1: Choose N collocation points in ½a; b. Step 2: Choose Dt and 0 6 h 6 1. Step 3: Calculate U0 ¼ ½u0 ðx1 Þ; . . . ; u0 ðxN Þ; U0x ¼ ½u0x ðx1 Þ; . . . ; u0x ðxN Þ and U0xx ¼ ½u0xx ðx1 Þ; . . . ; u0xx ðxN 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 n þ 1. Step 8: If nDt < T (T is final time), go to step 5 else stop.

Here we apply linear stability analysis method for investigating the stability of the presented method. Although, the application of the linear stability analysis to nonlinear equations cannot be rigorously justified, it provides, however, the necessary conditions for stability, and it is found to be effective in practice [22,23].

L2 and L1 errors of multiquadric for Example 4.1

Time

t=1

t=2

t=3

t=5

t=7

t = 10

L1 L2

1.900e12 1.444e12

2.900e12 2.384e12

3.900e12 3.329e12

5.899e12 5.224e12

7.899e12 7.119e12

1.090e11 9.964e12


4

M. Askari, H. Adibi h

i 2 1 12 Dt @A @@xB2 þ Bb2 þ iðAb 2 @B bÞ @x @x h nj ¼

i 2 @@xB2 þ Bb2 þ i Ab 2 @B b 1 þ 12 Dt @A @x @x

ð24Þ

2

@@xB2 þ Bb2 and K2 ¼ Ab 2 @B b we have if we set K1 ¼ @A @x @x

nj ¼

1 12 Dt½K1 þ iK2 1 þ 12 Dt½K1 þ iK2

ð25Þ

Then

jnj j2 ¼

1 12 DtK1

2

2

þ 14 ðDtÞ2 K22

1 þ 12 DtK1 þ 14 ðDtÞ2 K22 h i 1 þ 14 ðDtÞ2 K21 þ 14 ðDtÞ2 K22 DtK1 i : ¼h 1 þ 14 ðDtÞ2 K21 þ 14 ðDtÞ2 K22 þ DtK1

Figure 1

Maximum error of multiquadric for Example 4.1.

At first we freeze locally one variable in nonlinear terms. By using the above method for the locally constant Fokker– Planck equation @u @ @2 ð20Þ ¼ Aþ 2B u @t @x @x and set knj /j ðxk Þ ¼ nnj eibxk , by using the Fourier analysis and presented method in previous section, for each k ¼ 1; . . . ; N we have @A @ 2 B ibxk @B þ hDt A 2 eiBxk hDt e ibnnþ1 nnþ1 1 þ hDt j j @x @x2 @x þ hDtBb2 nnþ1 eibxk ¼ nnj eibxk þ ðh 1ÞDtnnj eibxk j @A @2B @B ib þ Bb2 : þ iAb 2 2 @x @x @x

Then @A @2B @B nj 1 þ hDt hDt 2 þ hDt A 2 ib þ hDtBb2 @x @x @x 2 @A @ B @B 2 þ iAb 2 2 ib þ Bb : ¼ 1 þ ðh 1ÞDt @x @x @x

ð26Þ

Notice that if K1 P 0 then jnj j2 6 1. From the above discussion we proved that for h ¼ 12 if K1 P 0 then jnj j 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 two norms !12 N X 2 L2 ¼ ku u~k2 ¼ Dx juj u~j j j¼1

ð21Þ

ð22Þ

So h

i 2 @@xB2 þ Bb2 þ i Ab 2 @B b 1 þ ðh 1ÞDt @A @x @x h nj ¼

i 2 @2 B @B þ Bb þ i Ab 2 b 1 þ hDt @A 2 @x @x @x

ð23Þ Figure 2

For h ¼ 12 this relation becomes

Table 3


L2 and L1 errors of multiquadric for Example 4.2.

Time

t=1

t=2

t=3

t=5

t=7

t = 10

L1 L2

4.008e10 2.097e10

5.037e10 2.913e10

4.004e9 1.999e9

3.004e8 1.185e8

9.869e10 5.853e10

5.011e6 2.345e6


Meshless method for the numerical solution of the Fokker–Planck equation Table 4

5


Time

t=1

t=2

t=3

t=5

t=7

t = 10

L1 L2

4.421e10 2.825e10

5.511e10 3.109e10

4.032e9 2.830e9

4.984e8 2.912e8

4.019e7 2.829e7

4.967e6 2.913e6

Figure 3


L1 ¼ ku u~k1 ¼ max juj u~j j 16j6N

where u and u~ are the exact and approximate solutions respectively and we consider fxi ¼ i Dx; i ¼ 0; . . . ; Ng. In all examples we use Dt ¼ 0:01; Dx ¼ 0:1; a ¼ 0 and b ¼ 1, also, the approximate solutions are computed using multiquadric radial basis function. The computations have been performed in MAPLE 16, with hardware configuration: desktop 32-bit Intel core 2 Duo CPU, 2G of RAM. Example 4.1. Consider [8,9] the Eq. (1) with AðxÞ ¼ 1; BðxÞ ¼ 1 and fðxÞ ¼ x. The exact solution of this problem is uðx; tÞ ¼ x þ t. Here we choose c ¼ 1012 and by using the presented meshless method, compute the approximate solution. Table 2 shows the L1 and L2 errors of the presented method at t = 1, 2, 3, 5, 7, 10, using multiquadric. Also, Fig. 1 displays the maximum error of multiquadric in t 2 ½0; 10. [8,9] the Eq. (1) with Example 4.2. Consider 2 AðxÞ ¼ x; BðxÞ ¼ x2 and fðxÞ ¼ x. The exact solution of this problem is uðx; tÞ ¼ xet . Here we choose c ¼ 1012 and by using the presented meshless method, compute the

Table 5

Figure 4


approximate solution. Table 3 shows the L1 and L2 errors of the presented method at t = 1, 2, 3, 5, 7, 10, using multiquadric. Also, Fig. 2 displays the maximum error of multiquadric in t 2 ½0; 10. Example 4.3. Consider [8,9] the backward Kolmogorov Eq. (3) with Aðx; tÞ ¼ ðx þ 1Þ; Bðx; tÞ ¼ x2 et and fðxÞ ¼ x þ 1. The exact solution of this problem is uðx; tÞ ¼ ðx þ 1Þet . Here we choose c ¼ 1012 and by using the presented meshless method, compute the approximate solution. Table 4 shows the L1 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 72 u; Bðx; t; uÞ ¼ xu and fðxÞ ¼ x. The exact solution x of this problem is uðx; tÞ ¼ tþ1 . Here we choose c ¼ 1012 and by using the presented meshless method, compute the approximate solution. Table 5 shows the L1 and L2 errors of the presented method at t = 1, 2, 3, 5, 7, 10 using multiquadric. Also, Fig. 4 displays the maximum error of multiquadric in t 2 ½0; 10.


Time

t=1

t=2

t=3

t=5

t=7

t = 10

L1 L2

4.500e13 2.669e13

6.007e11 2.852e11

2.250e13 1.335e13

5.997e11 2.384e11

1.125e13 6.674e14

4.955e12 2.881e12


6 5. Conclusion We applied the meshless method using multiquadric radial basis functions to solve the Fokker–Planck equation. Here we used h-weighted finite difference method and then approximated the solution of the equation using the combination of radial basis functions. Then by using the collocation method, we gained the system of linear equations. Also, we presented the stability analysis of the method using linearized stability analysis. 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 have helped improved the quality of the paper. References [1] Risken H. The Fokker–Planck equation: method of solution and applications. Berlin, Heidelberg: Springer Verlag; 1989. [2] Tatari M, Dehghan M, Razzaghi M. Application of the Adomian decomposition method for the Fokker–Planck equation. Math Comput Model 2007;45:639–50. [3] Palleschi V, Sarri F, Marcozzi G, Torquati MR. Numerical solution of the Fokker–Planck equation: a fast and accurate algorithm. Phys Lett A 1990;146:378–86. [4] Reif J, Barakat R. Numerical solution of the Fokker–Planck equation via Chebyschev polynomial approximations with reference to first passage time probability density functions. J Comput Phys 1977;23:425–45. [5] Harrison GW. Numerical solution of the Fokker–Planck equation using moving finite elements. Numer Methods Partial Differential Eqs. 1988;4:219–32. [6] Hesam S, Nazemi AR, Haghbin A. Analytical solution for the Fokker–Planck equation by differential transform method. Scientia Iranica B 2012;19:1140–5. [7] Salehian MV, Abdi-mazraeh S, Irandoust-pakchin S, Rafati N. Numerical solution of Fokker–Planck equation using the flatlet oblique multiwavelets. Int J Nonlinear Sci 2012;13:387–95. [8] Lakestani M, Dehghan M. Numerical solution of Fokker–Planck equation using the cubic B-spline scaling functions. Numer Methods Partial Differential Eqs 2009;25:418–29. [9] Kazem S, Rad JA, Parand K. Radial basis functions methods for solving Fokker–Planck equation. Eng Anal Bound Elem 2012;36:181–9. [10] Biazar J, Hosseini K, Gholamin P. Homotopy perturbation method Fokker–Planck equation. Int Math Forum 2008;3:945–54. [11] Dehghan M, Tatari M. The use of He’s variational iteration method for solving a Fokker–Planck equation. Phys Scr 2006;74:310–6. [12] Fok JCM, Guo B, Tang T. Combined Hermite spectral-finite difference method for the Fokker–Planck equation. Math Comput 2001;71:1497–528.

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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 postgraduate 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.
