Fractional Integral Representation of Master Equation

\

Chaos\ Solitons + Fractals Vol[ 09\ No[ 8\ pp[ 0434Ð0437\ 0888 Þ 0888 Elsevier Science Ltd[ All rights reserved 9859Ð9668:88:, ! see front matter

Pergamon

PII] S9859!9668"87#99065!2

Fractional Integral Representation of Master Equation S[ A[ ELWAKIL and M[ A[ ZAHRAN Department of Physics\ Faculty of Science\ Mansoura University\ Mansoura\ Egypt "Acceptedd 18 June 0887# Abstract*Using the de_nition of LiouvilleÐRiemann "LÐR# fractional integral operator\ master equation can be represented in the domain of fractal time evolution with a critical exponent a"9³a¾0#[ The relation between the continuous time random walks "CTRW# and fractional master equation "FME# has been achieved by obtaining the corresponding waiting time density "WTD# c"t#[ The latter is obtained in a closed form in terms of the generalized MittagÐLe/er "MÐL# function[ The asymptotic expansion of the "MÐL# function show the same behavior considered in the theory of random walk[ Applying the Fourier and LaplaceÐMellin transforms to "FME#\ one obtains the solution\ in closed form\ in terms of the Fox function[ Þ 0888 Elsevier Science Ltd[ All rights reserved[

0[ INTRODUCTION

Large classes of dynamical process in disordered media display di}usion!like behavior[ In Eucli! dean system\ the mean!square displacement\ Rðr1 "t#Ł½t\ is proportional to time for any number of spatial dimension\ d\ "Ficks law#[ However\ in disordered systems\ this law is not valid in general\ rather the di}usion becomes anomalous ð0Ł\ ðr1 "t#Ł×½ta \

"0[0#

with a1:dw "dw×1# in a system with spatial complexity "geometric constraints# ð1Ł[ Hilfer and Anton ð2Ł introduced the fractional master equation in which the time derivative is replaced with a derivative of fractional order\ a[ The fractional order\ a\ plays the role of the dynamical critical exponent ð2Ł[ They showed that there exists a precise and rigorous relation between the fractional equation and the continuous time random walk "CTRW#[ So the corresponding waiting time density "WTD#\ c"t#\ is obtained\ in closed form\ in terms of the generalized MittageÐLe/er function[ In this work\ the "WTD#\ c"t#\ is obtained directly from the fractional master equation only[ Moreover\ by using the de_nition "CF#\ one can obtain the fractional integral in the Fourier domain[ By applying the LaplaceÐMellin technique\ the "CF# is obtained\ in closed form\ in terms of the Fox function[ 1[ RELATION BETWEEN THE FRACTIONAL MASTER EQUATION AND FRACTAL WALKS

Consider the fractional master equation ð2Ł\ 9

D at p"r\t#s W"r−r?#p"r?\t#\ 9³a³0\

"1[0#

r?

where p"r\t# is the probability density of _nding a di}using entity at the position r$R d at time t

 Author to whom correspondence should be addressed[ 0434

0435

S[ A[ ELWAKIL and M[ A[ ZAHRAN

if it was at the origin\ r9\ at time t9[ W"r# is the fractional transition rate which measures the propensity for a displacement\ r\ in units of "0:time# 0:a[ 9 D at is the LiouvilleÐRiemann "LÐR# fractional di}erential operator\ de_ned by ð3Ł 9

0 d D at p"r\t# G"a# dt

g

t

"t−t?# −a p"r\t#dt?[

"1[1#

9

Assuming a decoupling between r and t in the distribution function\ i[e[\ p"r\t#c"t#8"r#\ then the fractional di}erential master equation becomes 9

D at c"t#f"r#sW"r−r?#c"t#f"r#[

"1[2#

r?

The time part of the above equation will give the fractal time evolution of fractional Brownian motion\ 9

D at c"t# −lc"t#\

"1[3#

where l is the constant of separation[ Using the LaplaceÐMillen technique ð4Ł to solve eqn "1[3# by de_ning

g



g



c"u#L"c"t#\u#

e−ut c"t#dt\

"1[4#

ts−0 c"t#dt\

"1[5#

9

as the Laplace transform of c"t# and c"s#M"c"t#\s#

9

as the Mellin transform of c"t#\ the connection between the Laplace and Mellin techniques is de_ned by] 0 M"L"c"t#\u#\0−s#\ M"c"t#\s# G"0−s#

"1[6#

where G"s# is the gamma function[ Therefore\ eqn "1[3# in the Laplace domain yields c"u#

c9 l¦ua

\

"1[7#

where c9 is a constant[ Using eqn "1[6# in "1[7# gives

0

0

s

10

0

s

F G − G 0− ¦ a a a a 0 j −s:a "l# c"s#"c9 :l# Ja"l−0:a # G"0−s# f

1 Jf

F[ j

"1[8#

Inverting eqn "1[8# into the time domain leads to the solution\ "9\0:a# c9 0:a c"t# ta−0 H 00 \ 01 l a "9\0:a# "0−a\0#

0 b

1

"1[09#

where H 00 01 is de_ned as the Fox function ð5Ł[ Using the series expansion for the Fox function\ one can obtain

Fractional integral representation of master equation 

c"t#lta−0 c9 s n9

0 "−lta # n \ G"na¦a#

0436

"1[00#

showing that c"t# behaves as c"t#½t0−0¦a

"1[01#

for small t:9\ while for t:\ it will give the asymptotic expansion of c"t# as c"t#½t−0−a [

"1[02#

This result shows that the waiting time distribution has a form of the kind usually considered in the theory of random walk ð2Ł[

2[ THE FRACTIONAL MASTER EQUATION AND ITS SOLUTION

In view of the possible general validity of eqn "0[0#\ one can use the integral form of the fractional master equation as p"r\t#dr9 ¦9 D −a sW"r−r?#p"r\t#\ t

"2[0#

R?

is the fractional "LÐR# integral operator\ which has the form ð3Ł\ Where 9 D −a t 9

g

0 D −a t p"r\t# G"a# 9

t

"t−t?# 0−a p"r\t#dt?[

"2[1#

9

dr\9 denotes the initial condition at t9[ From the de_nition of the characteristic function as the Fourier transform of the probability density\

g



dreikr p"r\t#\

p"k\t#

"2[2#

−

where k−9[ p"k\t# is known as the intermediate scattering function in the theory of liquids ð6Ł and represents the correlation of densityÐdensity ~uctuations of a wavevector in the ~uid[ Equation "2[0# in the Fourier domain becomes p"k\t#0¦9 D −a t W"k#p"k\t#\

"2[3#

where W"k# is the Fourier transform of the transition rate probability W"r#[ Equation "2[3# is an integral equation with di}erence kernel\ "t−t?# a−0[ A Laplace transform of eqn "2[3# yields p"k\u#

ua−0 "ua −W"k##

\

"2[4#

taking W"k# −ck1\ where c is a constant related to the di}usion coe.cient[ Therefore\ the above expression becomes p"k\u#

ua−0 "ua ¦ck1 #

[

"2[5#

From the properties of the characteristic function\ the mean!square displacement is given by

0437

S[ A[ ELWAKIL and M[ A[ ZAHRAN

R½ðr1 "t#ŁÐp"r\t#r1 dr −

d1 dk1

p"k\t#

b

[

"2[6#

k9

Using eqn "2[5# in eqn "2[6#\ one obtains 0 ðr1 "t#Ł½ a¦0 \ u

"2[7#

which\ in the time domain\ will give the anomalous behavior of eqn "0[0# as ðr1 "t#Ł½ta [

"2[8#

To obtain the inverse Laplace transformation for eqn "2[5#\ one can use the connection between the Laplace and Mellin transforms "eqns "1[5# and "1[6## for eqn "2[5#\ which gives 0 p"k\s# G"0−s#

g



9

ua−s−0 "ck1 ¦ua−0 #

du\

"2[09#

which can be integrated to obtain ð7Ł p"k\s#

0 "ck1 # −s:a b""s:a#\"0−s:a##\ aG"0−s#

"2[00#

where b is the special beta function[ Now the inversion of eqn "2[00# into the time domain yields "9\0:a# 0 1 0:a p"k\t# H 00 t [ 01 "ck # a "9\0:a# "9\0#

0

b

1

"2[01#

The Fox function\ H 00 01 \ can be written in a series expansion such that "−0# n "ck1 t# 1an [ n9 G"1na¦0# 

p"k\t#"0:a# s

"2[02#

The above solution is a monotonic decreasing function with p"k\9#dr\9\ and the asymptotic behavior at t: is given by p"k\t#½t−a \

"2[03#

which is the same result as that obtained in the study of the electrochemical response of rough electrodes ð8Ł[ REFERENCES 0[ 1[ 2[ 3[ 4[ 5[

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