CHAPTER – 3

SOLUTION OF FRACTIONAL TELEGRAPH

AND FRACTIONAL FOKKER-PLANCK

EQUATIONS WITH CAPUTO FRACTIONAL

DERIVATIVE BY ADM

______________________________________________

The main results established in this chapter have been presented/ published as

detailed below:

1. Solution of space-time fractional telegraph equation by Adomian

decomposition method, Journal of Inequalities and Special Functions 2(1).

1-7 (2011). (Also presented in “22nd

National Conference of Rajasthan Ganita

Parishad”, held during March 05-06, 2011 at Government Engineering

College, Jhalawar)

2. Exact Solution of space-time fractional Fokker-Planck equations by

Adomian decomposition method, Journal of International Academy of

Physical Sciences 15. 1-12 (2012). (Also presented in “13th

International

Conference of International Academy of Physical Sciences”, held during

June 14-16, 2011 at University of Petroleum and Energy Studies, Dehradun)

3.1 INTRODUCTION

In the present chapter, first we obtain solutions of two space-time

fractional telegraph equations using Adomian decomposition method (ADM),

discussed in Chapter 2. The space and time fractional derivatives are

considered in Caputo sense and the solutions are obtained in terms of Mittag-

Leffler type functions. The first equation considered here, is a homogeneous

space-time fractional telegraph equation. On specializing the parameters of this

equation, we get solution of classical telegraph equation, solved earlier by

Kaya (2000) and space fractional telegraph equations, solved by Momani

(2005), Odibat & Momani (2008) and Yildirim (2010a). The second equation

solved here, is a nonhomogeneous space-time fractional telegraph equation.

We also obtain solutions of corresponding space fractional, time fractional and

ordinary telegraph equations as its special cases. Next, we derive closed form

solutions of two fractional Fokker-Planck equations (FFPE) by means of the

ADM extended for nonlinear fractional partial differential equations. The first

FFPE is a nonlinear time fractional FPE, which in the special case yields the

solution of a nonlinear FPE studied recently by Yildrim (2010b). The second

FFPE is a linear space-time fractional FPE. On specializing the parameters, we

obtain solutions of corresponding space fractional, time fractional and ordinary

FPE. Two linear Fokker-Planck equations studied recently by Yildirim (2010b)

also follow as its special cases.

Chapter 3 95

3.2 FRACTIONAL TELEGRAPH EQUATIONS

The telegraph equation is used in modeling reaction-diffusion and signal

analysis for propagation of electrical signals in a cable of transmission line

[Debnath (1997) and Metaxas & Meredith (1993)]. Both current I and voltage

V satisfy an equation of the form (3.2.1). This equation also arises in the

propagation of pressure waves in the study of pulsatile blood flow in arteries

and in one-dimensional random motion of bugs along a hedge. Compared to

the heat equation, the telegraph equation is found to be a superior model for

describing certain fluid flow problems involving suspensions [Eckstein et al.

(1999)]. The classical telegraph equation is a partial differential equation with

constant coefficients given by [Debnath (1997)]

2 0,tt xx tu c u au bu (3.2.1)

where ,a b and c are constants.

The classical and space/time fractional telegraph equations have been

studied by a number of researchers namely Biazar et al. (2009), Cascaval et al.

(2002), Kaya (2000), Momani (2005), Odibat & Momani (2008), Sevimlican

(2010) and Yildirim (2010a). Orsingher & Zhao (2003) have shown that the

law of the iterated Brownian motion and the telegraph processes with

Brownian time are governed by time fractional telegraph equations given by

2

2 * 2 *

0 02, 2 , , 0 1.t t

uf D u x t D u x t

x

(3.2.2)

Chapter 3 96

Orsingher & Beghin (2004) presented that the transition function of a

symmetric process with discontinuous trajectories satisfies the space fractional

telegraph equation given by

22

22 , 1 2.

u u uf

t tx

(3.2.3)

Several techniques such as transform method, Adomian decomposition

method, juxtaposition of transforms, generalized differential transform method,

variational iteration method, and homotopy perturbation method have been

used to solve space or time fractional telegraph equations. In the next section,

we obtain closed form solutions of homogeneous and nonhomogeneous space-

time fractional telegraph equations using Adomian decomposition method.

3.3 SOLUTION OF FRACTIONAL TELEGRAPH EQUATIONS

USING ADM

In this section, we consider space-time fractional telegraph equations in

the following form

2 * * *

0 0 0, , , , , ,p r

x t tc D u x t D u x t a D u x t bu x t f x t 0 1, 0,x t

(3.3.1)

where 1/ ,q , , ,p q r 1 2, 1 2,p 0 1,r

* * * *

0 0 0 0...p

t t t tD D D D ( p times), * * * *

0 0 0 0...r

t t t tD D D D ( r times), *

0 xD &

Chapter 3 97

*

0 tD are Caputo fractional derivatives defined by equation (0.5.4), ,a b and c

are constants and ,f x t is given function.

The above space-time fractional telegraph equation is obtained by

replacing the integer order space and time derivatives in telegraph equation

(3.2.1) by Caputo fractional derivatives (0.5.4). Thus for

2, 1,q 2,p 1,r 0f , space-time fractional telegraph equation

(3.3.1) reduces to classical telegraph equation (3.2.1).

PROBLEM 3.1

Consider the following homogeneous space-time fractional telegraph

equation

* * *

0 0 0, , , , ,0 1, 0,p r

x t tD u x t D u x t D u x t u x t x t (3.3.2)

where 1/ ,q , , ,p q r 1 2, 1 2,p 0 1,r

* * * *

0 0 0 0...p

t t t tD D D D ( p times), * * * *

0 0 0 0...r

t t t tD D D D ( r times), *

0 xD ,

*

0 tD are Caputo fractional derivatives defined by equation (0.5.4), p r is odd

and initial conditions are given by

0,u t E t and 0, ,xu t E t (3.3.3)

where E t denotes the Mittag-Leffler function defined by (0.2.1).

Chapter 3 98

To apply Adomian decomposition method as explained in Chapter 2, we

take *

0 xL D and 1

0 xL I defined by equations (0.5.4) and (0.5.1)

respectively.

Now, operating 1L on equation (3.3.2) and making use of result (0.5.6),

we get from equation (2.3.2)

2 1

* *

0 0 000

, 1 , .!

kk p r

x x t txk

xu x t D u I D D u x t

k

(3.3.4)

This gives the following recursive relations using equation (2.3.6)

1

0 00

, ,!

kk

x xk

xu x t D u

k

(3.3.5)

* *

1 0 0 0, 1 , , 0,1,2,... .p r

k x t t ku x t I D D u x t k

(3.3.6)

Which using result (0.5.5), definition (0.5.1) and initial conditions (3.3.3)

gives

0 , 1u x t x E t , (3.3.7)

1

1 ,1 2

x xu x t E t

, (3.3.8)

2 2 1

2 ,1 2 2 2

x xu x t E t

(3.3.9)

and so on for other components.

Chapter 3 99

Substituting (3.3.7)-(3.3.9) in equation (2.3.3) and making use of

definitions (0.2.1) and (0.2.2), we get the solution of problem (3.3.2) as

,2, .u x t E x xE x E t

(3.3.10)

REMARK 3.2

Setting 2, 1p q r , the space-time fractional telegraph equation

(3.3.2) reduces to the space fractional telegraph equation and the solution is

same as obtained by Momani (2005), Odibat & Momani (2008) and Yildirim

(2010a) using Adomian decomposition method, generalized differential

transform method and homotopy perturbation method respectively.

REMARK 3.3

Setting 2, equation (3.3.2) reduces to time fractional telegraph

equation, with the meaning of various symbols and parameters as given with

equation (3.3.2), as follows

2 * *

0 0, , , , ,0 1, 0,p r

x t tD u x t D u x t D u x t u x t x t (3.3.11)

with initial condition given by (3.3.3) and solution

, .xu x t e E t (3.3.12)

Chapter 3 100

REMARK 3.4

Setting 2, 2, 1p q r , equation (3.3.2) reduces to the classical

telegraph equation and the solution is same as obtained by Kaya (2000) using

Adomian decomposition method.

PROBLEM 3.5

Consider the following nonhomogeneous space-time fractional telegraph

equation

* * *

0 0 0, , , , 2 ,p r

x t tD u x t D u x t D u x t u x t E x E t

0 1, 0,x t (3.3.13)

where 1 ,q

, , ,p q r 1 2, 1 2,p 0 1,r ...p

t t t tD D D D

( p times), * * * *

0 0 0 0...r

t t t tD D D D ( r times), *

0 xD , *

0 tD are Caputo fractional

derivatives defined by equation (0.5.4), E t denotes the Mittag-Leffler

function defined by (0.2.1), p and r are even and initial conditions are

0, , 0, 0xu t E t u t

. (3.3.14)

To apply Adomian decomposition method, we take *

0 xL D and

1

0 xL I defined by equation (0.5.4) and (0.5.1) respectively.

Now, operating 1L on equation (3.3.13) and making use of relation

(0.5.6), we get from equation (2.3.2)

Chapter 3 101

2 1

* *

0 0 000

, 1 ,!

kk p r

x x t txk

xu x t D u I D D u x t

k

02 ,xI E x E t

(3.3.15)

this gives the following recursive relations using equation (2.3.6)

1

0 000

, 2 ,!

kk

x xxk

xu x t D u I E x E t

k

(3.3.16)

* *

1 0 0 0, 1 , , 0,1,2,... .p r

k x t t ku x t I D D u x t k

(3.3.17)

Which using result (0.5.5), definition (0.5.1) and initial conditions

(3.3.14) gives

1

0

0

, 31 1

k

k

xu x t E x E t E t

k

, (3.3.18)

1 2

1

0 0

, 3 31 1 2 1

k k

k k

x xu x t E t E t

k k

,

(3.3.19)

2 3

2

2

0 0

, 3 32 1 3 1

k k

k k

x xu x t E t E t

k k

(3.3.20)

and so on for other components.

Substituting (3.3.18)-(3.3.20) in equation (2.3.3), the solution ,u x t of

the problem (3.3.13) is given by

,u x t E x E t

. (3.3.21)

Chapter 3 102

REMARK 3.6

Setting 2, 4, 2q p r , equation (3.3.13) reduces to nonhomogeneous

space fractional telegraph equation, with the meaning of various symbols and

parameters as given with the equation (3.3.13), as follows

* 2

0 , , , , 2 ,0 1, 0,t

x t tD u x t D u x t D u x t u x t E x e x t

(3.3.22)

with initial conditions

1/2

1/20, , 0, 0xu t E t u t (3.3.23)

and solution

, .tu x t E x e

(3.3.24)

REMARK 3.7

Setting 2, equation (3.3.13) reduces to nonhomogeneous time

fractional telegraph equation, with the meaning of various symbols and

parameters as given with the equation (3.3.13), as follows

2 * *

0 0, , , , 2 ,0 1, 0,p r x

x t tD u x t D u x t D u x t u x t e E t x t

(3.3.25)

with initial conditions given by (3.3.14) and solution

, .xu x t e E t (3.3.26)

Chapter 3 103

REMARK 3.8

Setting 2, 2,q 4,p 2r , equation (3.3.13) reduces to

nonhomogeneous telegraph equation, with the meaning of various symbols and

parameters as given with the equation (3.3.13), as follows

2 2 1/2

1/2, , , , 2 ,0 1, 0,x

x t tD u x t D u x t D u x t u x t e E t x t

(3.3.27)

with initial conditions given by (3.3.23) and solution

1/2

1/2, .xu x t e E t (3.3.28)

3.4 FRACTIONAL FOKKER-PLANCK EQUATIONS

The Fokker-Planck equation (FPE) arises in kinetic theory [Fokker

(1914)], where it describes the evolution of the one-particle distribution

function of a dilute gas with long-range collisions, such as a Coulomb gas. The

FPE, first applied to investigate Brownian motion [Chandrashekhar (1943)] and

the diffusion mode of chemical reactions [Kramers (1940)], is now largely

employed, in various generalized forms, in physics, chemistry, engineering and

biology [Risken (1989)]. For some applications of this equation one can refer

to the works of He & Wu (2006), Jumarie (2004), Kamitani & Matsuba (2004),

Xu et al. (2004), and Zak (2006). The general FPE for the motion of a

Chapter 3 104

concentration field ,u x t of one space variable x at time t has the form

[Risken (1989)]

2

2, ,

uA x B x u x t

t x x

(3.4.1)

with the initial condition given by

,0 ,u x f x x , (3.4.2)

where 0B x is called the diffusion coefficient and A x is the drift

coefficient. The drift and diffusion coefficients may also depend on time.

Mathematically, this equation is a linear second-order partial differential

equation of parabolic type. Roughly speaking, it is a diffusion equation with an

additional first-order derivative with respect to x.

There is a more general form of FPE, which is called the nonlinear FPE.

The nonlinear FPE has important applications in various areas such as plasma

physics, surface physics, population dynamics, biophysics, engineering,

neurosciences, nonlinear hydrodynamics, polymer physics, laser physics,

pattern formation, psychology and marketing [see, Frank (2004) and references

therein]. In one variable case, the nonlinear FPE is written in the form

2

2, , , , , ,

uA x t u B x t u u x t

t x x

(3.4.3)

with the initial condition given by

,0 ,u x f x x . (3.4.4)

Chapter 3 105

Due to the vast range of applications of the FPE, lots of work has been

done to find the numerical solution of this equation. In this context, the works

of Harrison (1988), Palleschi et al. (1990), Vanaja (1992), Zorzano et al.

(1999) and Buet et al. (2001) are worth mentioning.

It has been observed that diffusion processes where the diffusion takes

place in a highly nonhomogeneous medium, the traditional FPE may not be

adequate [Benson et al. (2000a, 2000b)]. The nonhomogeneities of the medium

may alter the laws of Markov diffusion in a fundamental way. In particular, the

corresponding probability density of the concentration field may have a heavier

tail than the Gaussian density, and its correlation function may decay to zero at

a much slower rate than the usual exponential rate of Markov diffusion,

resulting in long-range dependence. This phenomenon is known as anomalous

diffusion [Bouchaud & Georges (1990)]. Fractional derivatives play key role in

modeling particle transport in anomalous diffusion including the space

fractional Fokker-Planck (advection-dispersion) equation describing Levy

flights, the time fractional FPE depicting traps, and the space-time fractional

equation characterizing the competition between Levy flights and traps

[Metzler & Klafter (2000) and Zaslavasky (2002)]. Different assumptions on

this probability density function lead to a variety of space-time fractional FPEs.

The nonlinear space-time fractional FPE can be written in the following

general form

Chapter 3 106

* * * 2

0 0 0, , , , , ,t x xD u D A x t u D B x t u u x t (3.4.5)

where 0, 0, 0 1, 1 2 2t x . It can be obtained from the general

Fokker-Planck equation by replacing the space and time derivatives by Caputo

fractional derivatives *

0 tD and *

0 xD defined by (0.5.4). The function ,u x t is

assumed to be a causal function of time and space, i.e., vanishing for 0t and

0x . Particularly for 1 , the fractional FPE (3.4.5) reduces to the

ordinary FPE given by (3.4.3) in the case 0x .

Recently, several numerical methods have been proposed for solutions of

space and/or time fractional FPEs [Zhuang et al. (2006), Yang et al. (2009,

2010) and Yildirim (2010b)]. In Section 3.6, we obtain closed form solutions of

a nonlinear time fractional and a linear space-time fractional FPE using

Adomian decomposition method.

3.5 ADM FOR NONLINEAR FRACTIONAL PARTIAL

DIFFERENTIAL EQUATIONS [Odibat & Momani (2008)]:

We consider the nonlinear fractional partial differential equation written

in an operator form as

*

0 , , , , , 0,tD u x t Lu x t Nu x t g x t x (3.5.1)

where *

0 tD is Caputo fractional derivative of order , 1 ,m m m ,

defined by equation (0.5.4), L is a linear operator which might include other

Chapter 3 107

fractional derivatives of order less than , N is a nonlinear operator which

also might include other fractional derivatives of order less than and ,g x t

is source term.

We apply the Riemann-Liouville fractional integral 0 tI defined by (0.5.1)

to both sides of equation (3.5.1), use relation (0.5.6) to obtain

1

0 000

, , , , .!

kmk

t t ttk

tu x t D u I g x t I Lu x t Nu x t

k

(3.5.2)

Next, we decompose the unknown function u into sum of an infinite

number of components given by the decomposition series

0

n

n

u u

(3.5.3)

and the nonlinear term is decomposed as follows

0

,n

n

Nu A

(3.5.4)

where nA are Adomian polynomials given by

0 0

1, 0,1,2,... .

!

n ni

n ini

dA N u n

n d

(3.5.5)

The components 0 1 2, , ,....u u u are determined recursively by substituting

(3.5.3), (3.5.4) in (3.5.2) that leads to

1

0 000 0 0 0

, ,!

kmk

n t t t n ntn k n n

tu D u I g x t I L u A

k

(3.5.6)

Chapter 3 108

this can be written as

0 1 2 ...u u u

1

0 0 0 1 2 0 1 200

, ... ... .!

kmk

t t ttk

tD u I g x t I L u u u A A A

k

(3.5.7)

Adomian method uses the formal recursive relations as

1

0 000

, ,!

kmk

t ttk

tu D u I g x t

k

1 0 , 0.n t n nu I L u A n

(3.5.8)

3.6 SOLUTIONS OF FRACTIONAL FOKKER-PLANCK EQUATIONS

PROBLEM 3.9

We consider the nonlinear time fractional FPE

* 2

0

4, , , 0, 0,0 1,

3t x x

u xD u x t D D u u x t t x

x

(3.6.1)

where *

0 tD is the Caputo fractional derivative defined by (0.5.4), initial

condition is

2,0 .u x x (3.6.2)

Chapter 3 109

Applying Riemann-Liouville fractional integral 0 tI to both sides of

equation (3.6.1), using relation (0.5.6), we have

1 12 2

0 000

4.

! 3

kk

t t x t x xtk

t xuu D u I D I D D u

k x

(3.6.3)

This gives the following recursive relations using equation (3.5.8),

0

0 00

,!

kk

t tk

tu D u

k

(3.6.4)

1 , 0,1,2,... ,3

nn t x t n

xuu I D I A n

(3.6.5)

where

2

2

00

1 4, 0,1,2,... .

!

n ni

n x x ini

dA D D u n

n x d

(3.6.6)

Which using result (0.5.5), definition (0.5.1) and initial condition (3.6.2)

gives

2

0 ,u x (3.6.7)

0 0,A (3.6.8)

2

1 ,1

tu x

(3.6.9)

1 0,A (3.6.10)

22

2 ,1 2

tu x

(3.6.11)

Chapter 3 110

2 0,A (3.6.12)

32

31 3

tu x

(3.6.13)

and so on for the other components.

Substituting 0 1 2, , ,.....u u u in equation (3.5.3) and making use of definition

(0.2.1), the solution of problem (3.6.1) is obtained as

2, .u x t x E t (3.6.14)

REMARK 3.10

Setting 1 , in the Problem 3.9, it reduces to the nonlinear FPE, which

is studied recently by Yildirim (2010b).

PROBLEM 3.11

Consider the linear space-time fractional FPE [Garg & Manohor (2011)]

* * * 2 2

0 0 0, , ,t x xD u x t D px D qx u x t

0, 0,0 1,1 2 2, , ,t x p q (3.6.15)

where *

0 ,tD

*

0 xD are Caputo fractional derivatives defined by (0.5.4), initial

condition is

1,0 .au x x (3.6.16)

Chapter 3 111

Applying Riemann-Liouville fractional integral 0 tI to both sides of

equation (3.6.15), using relation (0.5.6), we have

1 1

* * 2 2

0 0 000

, , .!

kk

t t x xtk

tu x t D u I D px D qx u x t

k

(3.6.17)

This gives the following recursive relations using equation (3.5.8),

0

0 00

,!

kk

t tk

tu D u

k

(3.6.18)

* * 2 2

1 0 0 0 , 0,1,2,... .n t x x nu I D px D qx u n

(3.6.19)

Which using relation (0.5.6), definition (0.5.1) and initial condition

(3.6.16) gives

1

0 ,au x (3.6.20)

1

1 ,1

atu bx

(3.6.21)

22 1

21 2

atu b x

(3.6.22)

and so on for the other components. Where 2

b q a p a

.

Substituting (3.6.20)-(3.6.22) in equation (3.5.3) and making use of

definition (0.2.1), the solution of problem (3.6.15) is given by

1

2, , .au x t x E bt b q a p a

(3.6.23)

Chapter 3 112

REMARK 3.12

Setting 1, in the equation (3.6.15), it reduces to the following linear

space fractional FPE, with the meaning of various symbols and parameters as

given with equation (3.6.15)

* * 2 2

0 0 , , 0, 0,1 2 2,x x xD u D px D px u x t t x

(3.6.24)

with initial condition given by (3.6.16) and solution is

1

2, , .a btu x t x e b q a p a

(3.6.25)

REMARK 3.13

Setting 1 , in the equation (3.6.15), it reduces to the following linear

time fractional FPE with the meaning of various symbols and parameters as

given with equation (3.6.15)

* 2 2

0 , , , 0, 0,0 1, , ,t x xD u x t D px D qx u x t t x p q

(3.6.26)

with initial condition given by (3.6.16) and solution is

1 2, , .au x t x E bt b qa a q p

(3.6.27)

Chapter 3 113

REMARK 3.14

Setting 1, 1 , in the equation (3.6.15), it reduces to the following

ordinary linear FPE

2 2 , , 0, 0, , ,x x

uD px D qx u x t t x p q

t

(3.6.28)

with initial condition given by (3.6.16) and solution is

1 2, , .a btu x t x e b qa a q p (3.6.29)

REMARK 3.15

Setting 1, 1, 2, 1, 1/ 2a p q , in the Problem 3.11, it reduces to

the ordinary linear FPE, which is studied recently by Yildirim (2010b).

REMARK 3.16

Setting 1, 1, 3, 1/ 6, 1/12a p q , in the Problem 3.11, it

reduces to the ordinary linear FPE, which is studied recently by Yildirim

(2010b).

Chapter 3 114

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