An approximate analytical solution description of time-fractional order Fokker-Planck equation by using FRDTM

$Page 1: An approximate analytical solution description of time-fractional order Fokker-Planck equation by using FRDTM$
Asia Pacific Journal of Engineering Science and Technology 1 (1) (2015) 34-47

*E-mail address: [email protected]

© 2015 Author(s)

Asia Pacific Journal of Engineering Science and Technology

journal homepage: www.apjest.com

Full length article

An approximate analytical solution description of time-fractional order

Fokker-Planck equation by using FRDTM

Neeraj Dhimana*

, Anand Chauhanb

aDepartment of Mathematics, Graphic Era Hill University, Dehradun-248001, India

bDepartment of Mathematics, Graphic Era University, Dehradun-248001, India

(Received: 25 December 2015; accepted 29 December 2015; published online 30 December 2015)

ABSTRACT

The fractional Fokker-Plank equation paid much more attention by researchers due to its

wide range of applications in several areas of sciences and engineering. In this article, our main

motto is to present a new approximate solution of time-fractional FPE by means of a new semi-

analytical approach called fractional reduced differential transform method (FRDTM), with

appropriate initial condition. In FRDTM, fractional derivative is considered in the Caputo sense.

The validity and efficiency of FRDTM is illustrated by considering three numeric experiments.

The solutions behavior and the effects of different values of fractional order are depicted

graphically.

Keywords: Fokker-Planck equation; FRDTM; Caputo time derivative; exact solution

1. Introduction

In the recent years, fractional calculus theory gained a great attention in both sciences and

engineering [1-10] due to its application to model many real world problems and fractional

dynamical systems, e.g., in earthquake modeling, measurement of viscoelastic material

properties, the traffic flow model, fluid flow model with fractional derivatives etc. The fractional

order system equation being converges to the integer order equation, have also paid more

attention to the researchers. Risken [11] have first time described the Brownian motion of

particles by mean of Fokker-Planck equations (FPE) as one of dynamical equation. The simplest

FPE equation for the motion of a small particle of mass m immersed in a fluid of temperature T

is given as

35 N. Dhiman, A. Chauhan Asia Pacific J. Eng. Sci. Tech. 1 (1) (2015) 34-47

1 1 2( ) ,t v vD u D vu KT m D u

(1)

where2

1 2

2, ;t v

u uD u D u and

t v

,u x t and v are the distribution function and the velocity for

the Brownian motion of the particle, respectively, K denotes the Boltzmann’s constant and

the fraction constant. The forward Kolmogorov equation, a more general form of FPE (1),

representing the motion for the distribution function ,u x t , can be expressed as

1 1 2 , , 0t x xD u D A x D B x u x I t

(2)

subject to the initial condition

,0 , ,u x x x I where ( , )I a b and is known function and A x

denotes the drift

coefficient while 0B x the diffusion coefficient. These coefficients may be function of time t

, that is,

1 1 2, , .t x xD u D A x t D B x t u

(3)

It is clear that Eq. (1) is a special case of (3), when ,A x t is linear in x and ,B x t is

constant. Risken [11] has introduced a similar partial differential equation, a more general form

of FPE, called Backward Kolmogorov equation is given by

1 1 2, , ,t x xD u A x t D B x t D u

(4)

The nonlinear time-fractional Fokker-Planck equations of order (0 1) is expressed as

1 2, , , , .t x xD u D A x t u D B x t u u

(5)

Eq. (5) is reduced to the most general form classical nonlinear FPE when 1 . The FPE have its

wide range of applications in several areas such as solid-state physics, theoretical biology, circuit

theory, quantum theory, chemical physics, neurosciences, nonlinear hydrodynamics, polymer

physics, laser physics, engineering, pattern formation, psychology and marketing and so on (see

[12] and the references therein). The most important role of FPE is in describing the system

dynamics [13–26], e.g. the dynamics of energy cascade in turbulence [13], the population

dynamics [14], the fission dynamics [15] the chaotic universe dynamics [16], the fatigue crack

growth dynamics [17], the Langevin approach for microscopic dynamics [18], the dynamics of

distributions of heavy quarks [19], financial returns [20], the electron dynamics of plasmas and

semiconductors [21], the spin relaxation dynamics [22], the critical dynamics [23], and the fiber

dynamics [24] had been investigated by using the Fokker-Planck equation, see [25, 26] and the

references therein. The fractional diffusion equations got their attention among the researchers

due to its arising in the modeling of anomalous diffusive and sub-diffusive systems, continuous

time random walks, unification of diffusion and wave propagation phenomenon and so on [27].

The fractional FPE have been intensively studied by several researchers in [28-42] due to their

broad applications.


The numeric and analytical solutions of FPE have been intensively studied since the work

of Risken [11], and the various approaches have been developed for getting the analytic [43-47]

and numeric [47-50] solution of FPE. Some of these schemes are moving finite element method

[43], differential transform method (DTM) [44], variational iteration method (VIM) and

homotopy-perturbation methods (HPM) [45], Adomian decomposition method (ADM) [46],

cubic B-spline method [47] and so on.

The fractional order FPE has been solved analytically by means of iterative Laplace

Transform method [27], homotopy perturbation method (HPM) [28], finite element method [39]

and homotopy perturbation transform method (HPTM) [41], for more schemes, see [28-42] and

references therein. The major drawback of these approaches is that they require a very

complicated and huge calculation. The fractional reduced differential transform method

(FRDTM) is proposed by Keskin and Oturanc [51] to overcome from such type of drawbacks.

This FRDTM is very easy to implement as it takes small size of computation contrary to the

other numerical methods, in the literature. The method is very reliable, effective and efficient

powerful approach applicable to a wide range of problems arising in sciences and engineering,

see [52-54]. In this paper, our main motto is to develop a new approximate analytical solutions

approach for nonlinear time-fractional Fokker-Planck equations of order (0 1) by means

of FRDTM, in the series form converges to the exact solution rapidly.

The rest of the paper is organized as follows: in Section 2, we revisit some basic

preliminaries and notations based on fractional calculus theory while Section 3 is the basics of

FRDTM which are used for further study. In Section 4, three test problems of time-fractional

homogenous and non-homogeneous fractional Fokker Planck equations are presented. The

comparison is made with the exact solutions available, in the literature. The concluding remarks

are presented in Section 5.

2. Fractional Calculus Theory

The basic preliminaries are revisited in this section which we use in this paper. Among

several results based on fractional integrals and derivatives proposed, in the literature, we

consider the first major contribution to give a proper and most meaningful result proposed by

Liouville [3].

Definition 2.1 A real valued function x , x>0f is said to be in the space , C if there

exists a real number q > such that qx =x g xf , where g x C[0, ) , and is said to be in

the spacemC

if

, mm

f C .

Definition 2.2 Let f . Riemann-Liouville fractional integral operator [2] of order 0 is

defined by


x1

0

0

1= x-t dt, 0, 0,

= .

x

x

J f x f t x

J f x f x

(6)

In his work, Caputo and Mainardi [2] proposed a modified fractional differentiation operatorxD

on the theory of visco-elasticity to overcome the discrepancy of Riemann-Liouville derivative [3]

while modeling the real world problems using the fractional differential equations. They further,

demonstrated that their proposed Caputo fractional derivative allow the utilization of initial and

boundary conditions involving integer order derivatives, is a straightforward physical

interpretations.

Definition 2.3 The fractional derivative of f x , in the Caputo sense [2] is defined as

x

1

0

1= x-t dt,

m mm m

x x xD f x J D f x f tm

(7)

for 11 , , 0, mm m m x f C . The basic properties of the Caputo fractional derivative

can be given by the following

Lemma 2.1 If 1 , m m m and , -1mf C , then we have

0

= , x>0,

= 0 , x>0,!

x x

kmk

x x

k

D J f x f x

xJ D f x f x f

k

(8)

In the present work, the Caputo fractional derivative is considered because it allows the

traditional initial and boundary conditions to be included in the formulation of the physical

problems, for further important characteristics of fractional derivatives, see [1-10].

3. The Basic Idea of FRDTM

In this section, the basic properties of the fractional reduced differential transform

method are described. Let ,x t be a function of two variables, which can be represented as a

product of two single-variable functions, that is ,x t f x g t . Using the properties of the

one-dimensional differential transform (DT) method, ,x t

can be written as

0 0 0 0

, , ,i j i j

i j i j

x t f i x g j t i j x t

(9)


where ,i j f i g j is referred to as the spectrum of ,x t .

Let DR and 1

DR

denotes operators for fractional reduced differential transform (FRDT)

and inverse FRDT, respectively. The basic definition and properties of the FRDTM is described

below.

Lemma 3.1 If ,x t is analytic and continuously differentiable with respect to space variable x

and time variable t in the domain of interest, then the t -dimensional spectrum is given by

0

1( ) , ,

1

k

k tt t

x D x tk

(10)

is referred as the FRDT of , where the parameter describes the order of time-fractional

derivative. Throughout the paper, lowercase ( , )x t is used for the original function while its

fractional reduced transformed function is represented by the uppercase x .

The inverse FRDT of k x is defined as

0

0

, : .k

k

k

x t x t t

(11)

Using Eq. (10) and (11), the following results can be obtained

0

0

0

1, , .

1

kk

tt t

k

x t D x t t tk

(12)

In particular, for 0 0t , Eq. (12) reduces to

0

0

1, , .

1

k k

tt

k

x t D x t tk

(13)

This shows that FRDTM is a special case of the power series expansion of a function.

Let us suppose ,u x t and ,v x t be any functions such that 1, D ku x t R U x ,

1, D kv x t R V x and the convolution denotes the fractional reduced differential

transform version of the multiplication, then the fundamental properties of the FRDT have been

presented in Table 1, where denotes the gamma function defined by 1

0

: ,t zz e t dt z

,

is the continuous extension to the factorial function, and the function is defined by

1 if 0( ) : .

0 otherwise

kk

Table 1: Basic properties of the FRDTM

Original Function

,f x t

Fractional Reduced Differential Transformed

Function

( , )D kR f x t F x


, ,u x t v x t 0

k

k k r k r

r

U x V x U x V x

1 2, ,a u x t a v x t 1 2k kaU x a V x

,m nx t u x t , if

0, .

m

k nx U x k n

else

,N

tD u x t

1 ( )

1k N

k NU x

k

,l

xD u x t l

x kD U x

mx

( )mx k

te

!

k

k

Definition 3.2 The Mittag-Leffler function ( )E z with 0 is defined by the following series

representation, is valid in the whole complex plane

0

E :1

k

k

zz

k

(14)

which is an advanced form of exp z , in particular, 1

exp limE .z z

4. Numerical Computations

In this section, we describe the method explained in the Section 2 using the following

three examples of the time fractional Fokker-Planck equation (FPE) to validate the efficiency

and reliability of the FRDTM.

Example 4.1: Consider the following time fractional order Fokker-Planck equation [29, 42]

2

2 , , 0,0 12

t x x

x uD u D xu D x t

(15)


,0 .u x x

(16)

Applying the RDTM to Eq. (15), we obtain the following recurrence relation

22

1

1.

1 2k D x D x

k x uU x R D xu R D

k

(17)

Using the RDTM to the initial condition (16), we obtain the expression

0 .U x x (18)


Using Eq. (18) into Eq. (17), we get the following kU x values successively

1 2, ,..., ,... 1 1 2 1

k

x x xU x U x U x

k

(19)

Using the differential inverse reduced transform of xkU , we have

0

1 0 0

1, x x = = E ,

1 1

k

k k

k

k k k

tu x t U U t x t x x t

k k

(20)

where E t is the well known Mittag-leffler function. It is noticed that solution (20) is in full

agreement with the solutions obtained by HPM [29] and HPTM [42]. In particular, for 1 in

Eq. (15), we get

, ,tu x t xe

(21)

which is the exact solution for the Fokker-Planck Eq. (15) with 1 .

Fig. 1. Comparison between FRDTM solution and exact solution of Example 4.1 for 1, 1t .

Fig. 2. Comparison between (a) exact solution and (b) approx. FRDTM solution of Example 4.1 for 1, 1t .


Fig. 3. Physical behavior of approximate FRDTM solution ( , )u x t of Example 4.1 vs. at 1x t (left) and vs.

at 1t x (right) for different values of .

Fig. 1 shows the comparison between the exact solution and approx. FRDTM solution

while Fig. 2 shows the comparison between the surface (a) exact solution and (b) approx.

FRDTM solution of Example 4.1. It is clearly seen from Fig. 1 and Fig. 2 that the solution

obtained by FRDTM is identical with the exact solution for classical Foker-Plank Eq. (14).

Fig. 3 shows the physical behavior of the approximate FRDTM solution for different fraction

Brownian motion 0.6, 0.7, 0.8, 0.9 vs. at 1x t and vs. at 1t x respectively of Example 4.1.

Example 4.2: Consider the following time fractional order Fokker-Planck equation [29, 42]:

22 , , 0,0 1,

6 12t x x

xu x uD u D D x t

(22)

under the initial condition

2,0 .u x x

(23)


22

1

1.

1 6 12k D x D x

k xu x uU x R D R D

k

(24)

Using the RDTM in Eq. (23), we obtain the expression

2

0 .U x x (25)


2 2 2

1 2 2, ,..., ,...

2 1 2 1 2 2 1k k

x x xU x U x U x

k

(26)



2

0

0 1 0

2 2

0

1, x x x

1 2

= = E .2 1 2

k

k k

k k

k k k

k

kk

tu x t U t U U t x

k

t tx x

k

(27)

The same exact solution was obtained by HPM [29] and HPTM [42]. In particular, for 1 in

Eq. (22), we get

2 2, ,t

u x t x e

(28)

which is the exact solution for the classical Fokker-Planck Eq. (22) with 1 .

Fig. 4. Comparison between exact solutions and FRDTM solutions of Example 4.2 for 1, 1t .

Fig. 5. Comparison between Exact solution (a) and approx. FRDTM solution (b) for 1, 1t , Example 4.2.


Fig. 6 Physical characteristics of approximate FRDTM solution ( , )u x t of Example 4.2 vs. at 1x t (left) and vs.


Fig. 4 depicts the comparison between the exact solution and approx. FRDTM solution

for while Fig. 5 shows the comparison between the surface of (a) exact solution and (b) approx.

FRDTM solution of Example 4.2. Fig. 4 and Fig. 5 show that FRDTM solution is identical with

the exact solution of (21) when 1 . Fig. 6 depicts the physical characteristics of the

approximate FRDTM solution for different fraction Brownian motion 0.6, 0.7,0.8,0.9 vs.

at 1x t and vs. at 1t x , respectively of Example 4.2.

Example 4.3: Consider the following nonlinear time fractional Fokker-Planck equation [29, 42]

2

2 24, , 0,0 1

3t x x

u xuD u D D u x t

x

(29)


2,0 .u x x

(30)


22 2

1

1 4.

1 3k D x D x

k u xuU x R D R D u

k x

(31)

Using the RDTM to the initial condition (30), we obtain the expression

2

0 .U x x (32)


2 2 2

1 2, ,..., ,... 1 1 2 1

k

x x xU x U x U x

k

(33)



2

0

0 1 0

2 2

0

1, x x x

1

= = E .1

kk k

k k

k k k

k

k

u x t U t U U t x tk

tx x t

k

(34)

which is exactly the same as obtained in [29] using HPM, and in [42] using HPTM. In particular,

for 1 in Eq. (29), we get

2, ,tu x t x e

(35)

which is the exact solution for the classical Fokker-Planck Eq. (29) with 1 .

Fig. 7. Comparison between exact solutions and FRDTM solutions of Example 4.3 for 1, 1t .

Fig. 8. Comparison between Exact solution (a) and approx. FRDTM solution (b) for 1, 1t , Example 4.3.


Fig. 9. Physical signature of approximate FRDTM solution ( , )u x t of Example 4.3 vs. at 1x t (left) and vs.


Fig. 7 shows the comparison between the exact solution and approx. FRDTM solution

whereas Fig. 8 depicts the comparison between the surface of (a) exact solution and (b) approx.

FRDTM solution of Example 4.3. It evident from Fig. 7 and Fig. 8 that FRDTM solution is

identical with the exact solution for classical FPE (28). Fig. 9 shows the physical signature of the

approximate FRDTM solution for different fraction Brownian motion 0.6, 0.7, 0.8, 0.9 vs.

at 1x t and vs. at 1t x respectively of Example 4.3.

5. Conclusions

In this article, a new approximate solution of time-fractional FPE arising in several

physical dynamical systems has been obtained by means of FRDTM with appropriate initial

condition. The proposed approximate solutions of time-fractional FPE are obtained in terms of a

power series, without using any kind of discretization, perturbation, or any other restrictive

condition, etc. The validity and efficiency of FRDTM is illustrated by considering three numeric

experiments. This method is very powerful analytical approach which can be easily applicable to

broad classes of real world problems arising in physical dynamical systems and engineering.

References

[1] J. Klafter, S. C. Lim, and R. Metzler, Fractional Dynamics: Recent Advances, World Scientific, 2012.

[2] A. Carpinteri, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag,

Wien, New York, 1997.

[3] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations,

Wiley, New York, 1993.

[4] D. Baleanu, J. A. T. Machado, and A. C. Luo, Fractional Dynamics and Control, Springer, 2012.

[5] V. E. Tarasov, Fractional Dynamics, Springer, 2010.

[6] G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, UK,

2005.


[7] E. Goldfain, Fractional dynamics, Cantorian space-time and the gauge hierarchy problem, Chaos, Solitons

and Fractals 22(3) (2004) 513–520.

[8] R. Hilfer, Applications of fractional calculus in physics, World scientific, Singapore, 2000.

[9] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.

[10] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

[11] H. Risken, The Fokker–Planck Equation Method of Solution and Applications, Springer Verlag, Berlin,

Heidelberg, 1989.

[12] T.D. Frank, Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker–Planck

equations, Phys. Lett. A 331 (2004) 391–408.

[13] A. Naert, R. Friedrich and J. Peinke, Fokker-Planck equation for the energy cascade in turbulence, Physical

Review E-Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics 56(6) (1997) 6719-

6722.

[14] M. Saad and T. Goudon, On a Fokker-Planck equation arising in population dynamics, Revista

Matematica Complutense 11(2) (1998) 353–372.

[15] J. R. Nix, A. J. Sierk, H. Hofmann, F. Scheuter, and D. Vautherin, Stationary Fokker-Planck equation

applied to fission dynamics, Nuclear Physics, Section A 424(2) (1984) 239–261.

[16] D. H. Coule and K. O. Olynyk, Chaotic universe dynamics using a Fokker-Planck equation, Modern

Physics Letters A 2(11) (1987) 811–817.

[17] A. Tsurui and H. Ishikawa, Application of the Fokker-Planck equation to a stochastic fatigue crack

growth model, Structural Safety 4(1) (1986) 15–29.

[18] T. D. Frank, A Langevin approach for the microscopic dynamics of nonlinear Fokker-Planck equations,

Physica A: Statistical Mechanics and its Applications 301(1–4) (2001) 52–62.

[19] D. B. Walton and J. Rafelski, Equilibrium distribution of heavy quarks in Fokker-Planck dynamics,

Physical Review Letters 84(1) (2000) 31–34.

[20] D. Sornette, Fokker-Planck equation of distributions of financial returns and power laws, Physica A:

Statistical Mechanics and its Applications 290(1-2) (2001) 211–217.

[21] V. I. Kolobov, Fokker-Planck modeling of electron kinetics in plasmas and semiconductors,

Computational Materials Science 28 (2) (2003) 302–320.

[22] A. Yoshimori and J. Korringa, A Fokker-Planck equation for spin relaxation, Physical Review 128 (3)

(1962) 1054–1060.

[23] C. P. Enz, Fokker-Planck description of classical systems with application to critical dynamics, Physica

A: Statistical Mechanics and its Applications 89(1) (1977) 1–36.

[24] A. Klar, P. Reutersw ard, and M. Sea¨ıd, A semi-lagrangian method for a Fokker-Planck equation

describing fiber dynamics, Journal of Scientific Computing 38(3) (2009) 349–367.

[25] L. P. Kadanoff, Statistical Physics: Statics, Dynamics and Renormalization, World Scientific, 2000.

[26] T. D. Frank and A. Daffertshofer, Multivariate nonlinear Fokker-Planck equations and generalized

thermostatistics, Physica A: Statistical Mechanics and its Applications 292(1–4) (2001) 392–410.

[27] O. P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear

Dynamics 29 (2002) 145.

[28] L. Yan, Numerical solutions of fractional Fokker-Planck equations using iterative Laplace Transform

method, Abstract and applied analysis, vol. 2013 (2013), Article ID 465160.

[29] A. Yildrim, Analytical approach to Fokker-Planck equation with space- and time- fractional derivatives

by means of the homotopy perturbation method, J. King Saud University (Science) 22 (2010) 257-264.

[30] F. Liu, V. Anh, I. Turner, Numerical solution of the space fractional Fokker–Planck equation, Journal of

Computational and Applied Mathematics 166 (2004) 209–219.

[31] R. Metzler and T. F. Nonnenmacher, Space- and time fractional diffusion and wave equations, fractional

Fokker-Planck equations, and physical motivation, Chemical Physics 284(1-2) (2002) 67–90.

[32] R. Metzler, E. Barkai, and J. Klafter, Deriving fractional fokker-planck equations from a generalised

master equation, Europhysics Letters 46(4) (1999) 431–436.

[33] S. A. El-Wakil and M. A. Zahran, Fractional Fokker-Planck Equation, Chaos, Solitons and Fractals 11(5)

(2000) 791–798.

[34] A.V.Chechkin, J.Klafter, and I. M. Sokolov, Fractional Fokker-Planck equation for ultraslow kinetics,

Europhysics Letters 63(3) (2003) 326–332.

[35] A. A. Stanislavsky, Subordinated Brownian motion and its fractional Fokker-Planck equation, Physica

Scripta 67(4) (2003) 265–268.


[36] K. Kim and Y. S. Kong, Anomalous behaviors in fractional Fokker-Planck equation, Journal of the

Korean Physical Society, 40 (6) (2002) 979–982.

[37] I. M. Sokolov, Thermodynamics and fractional Fokker-Planck equations, Physical Review E, Statistical,

Nonlinear, and Soft Matter Physics 63(5) (2001) 561111–561118.

[38] V. E. Tarasov, Fokker-Planck equation for fractional systems, International Journal of Modern Physics B

21(6) (2007) 955–967.

[39] F. Liu, V. Anh, and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, Journal

of Computational and Applied Mathematics 166(1) (2004) 209–219.

[40] W. Deng, Finite element method for the space and time fractional fokker-planck equation, SIAM Journal

on Numerical Analysis, 47(1) (2004) 204–226.

[41] Z. Odibat and S. Momani, Numerical solution of Fokker- Planck equation with space- and time-fractional

derivatives, Physics Letters A: General, Atomic and Solid State Physics 369(5-6) (2007) 349–358.

[42] S. Kumar, Numerical computation of time fractional Fokker-Planck equation arising in solid state physics

and circuit theory, 2013.

[43] G. Harrison, Numerical solution of the Fokker–Planck equation using moving finite elements’’, Numer.

Methods Partial Differential Equations 4 (1988) 219–232.

[44] S. Hesama, A.R. Nazemia, A. Haghbinb, Analytical solution for the Fokker–Planck equation by

differential transform method, Scientia Iranica B 19 (4) (2012) 1140–1145.

[45] A. Sadighi, D.D. Ganji, Y. Sabzehmeidani, A study on Fokker–Planck equation by variational iteration

and homotopy-perturbation methods, Int. J. Nonlinear Sci. 4 (2007) 92–102.

[46] M. Tatari, M. Dehghan, M. Razzaghi, Application of the Adomian decomposition method for the

Fokker–Planck equation, Math. Comput. Modelling 45 (2007) 639–650.

[47] M. Lakestani, M. Dehghan, Numerical solution of Fokker–Planck equation using the cubic B-spline

scaling functions, Numer. Methods Partial Differential Equations 25 (2009) 418–429.

[48] C. Buet, S. Dellacherie, R. Sentis, Numerical solution of an ionic Fokker–Planck equation with electronic

temperature, SIAM J. Numer. Anal. 39 (2001) 1219-1253.

[49] V. Palleschi, F. Sarri, G. Marcozzi, M.R. Torquati, Numerical solution of the Fokker–Planck equation: a

fast and accurate algorithm, Phys. Lett. A 146 (1990) 378-386.

[50] V. Vanaja, Numerical solution of simple Fokker–Planck equation, Appl. Numer. Math. 9 (1992) 533-540.

[51] Y. Keskin, G. Oturanc, Reduced differential transform method: a new approach to factional partial

differential equations, Nonlinear Sci. Lett. A 1 (2010) 61-72.

[52] V. K. Srivastava, M. K. Awasthi, M. Tamsir, RDTM solution of Caputo time fractional-order hyperbolic

Telegraph equation, AIP Advances 3 (2013), 032142.

[53] V.K. Srivastava, M.K. Awasthi, S. Kumar. Analytical approximations of two and three dimensional time-

fractional telegraphic equation by reduced differential transform method. Egyptian Journal of Basic and

Applied Sciences 1(2014) 60-66.

[54] V.K. Srivastava, S. Kumar, M.K. Awasthi, B. K. Singh. Two-dimensional time fractional-order biological

population model and its analytical solution. Egyptian Journal of Basic and Applied Sciences 1 (2014)

71-76.

Documents

An approximate analytical solution description of time-fractional order Fokker-Planck equation by using FRDTM