Transform Methods & Special Functions, Sofia'94

Transform Methods

& Special

Functions,

Sofia'94

by a fractional derivati ve of order o: with O < o: < 2. It reads aO:u

Proceedings

of International

Workshop,

12

- 17

August

1994

atO:

where x, t denote the space-time

a2u

= D ax2 '

D >0,

(1.1)

variables, and u = u(x, t) is the response field

va.riable. The formal derivative of order o: is to be intended as a fractional derivative in the Riemann-Liouville sense, whose definition and properties are found in texts on Fractional Calculus (see e.g. [3], [7], [9], [23-25]). Later we will provide its explicit expression in terms of a convolution integraI. ON A SPECIAL IN TRE

TIME

FRACTIONAL

FUNCTION

ARISING

DIFFUSION-WAVE

EQUATION

F. Mainardi*, M. Tomirotti**

Abstract The ti me fractional

diffusion-wave equation is obtained from the classical dif-

fusion or wave equation by replacing the first- or second-order time derivative by a fractional deriyative of order o: with O < o: < 2. Using the method of the Laplace transform, it is shown that the fundamental solution of the basic Cauchy problem can be expressed in terms of an auxiliary function M(z; (3), where z is the similarity variable and (3 = 0:/2. This function, which reduces to the Gaussian function in the case of classical diffusion ((3 = 1/2), is shown to be a Wright-type function, and therefore it turns out to be an entire function of the complex variable z for any value of the parameter (3 in its range O < (3 < 1. For it we provide series and integral representations and the differenti al equation of fractional order to be satisfied in the complex pIane; the above properties allow us to consider M(z; (3) as a sort of generalized hyper-Airy function. The probIem of its asymptotic representation for Iarge Izj is preliminarily discussed using the saddIe point method. Mathematics

Subject Classification: 26A33, 33C, 44A

Key Words and Phrases: Laplace transforms, functions, Wright functions.

fractional

calcuIus, hyper-Airy

1. Introduction Recently, in a series of papers [11-15], Mainardi has investigated the so-called time fractional diffusion-wave equation. This equation is obtained from the classical diffusion or wave equation by replacing the first- or second-prder time derivative

.

* The first author was partially supported by the National Research Council (CNR-FISBAT) and by the Ministry of University, Italy (60 % grants)

The equation (1.1) has been introduced in physics, with O < o: < 1, by Nigmatullin [19-20] to describe the diffusion process in media with fractal geometry, while, with 1 < o: < 2, by Mainardi [11] to describe the propagation of mechanical diffusive waves in viscoelastic media which exhibit a power-Iaw creep. We expect that this equaton can be connected with the so-called fractional Brownian motion introduced by Mandelbrot (see e.g. [5], [16]) like the ordinary diffusion equation is related to the ordinary Brownian motion (see e.g. [IO)). Mathematical aspects of (1.1) have been treated in two relevant papers by Wyss [30] and Schneider & Wyss [26], who have used Mellin transforms and Fox H functions; however, because of the great level of generality, the results appear of not easy application. In sect. 2 we resume the recent results by Mainardi, based on the Laplace transforms, which allows us to obtain for (1.1) the fundamental solution of the initial value problem, the so-called Green function, in terms of an auxiliary function M(z;(3), where z = Ixl/(VDt,B) (O < (3 = 0:/2 < 1) is the similarity variable. The treatment turns out to be much more accessible than that proposed in [26], [30] and can be easiIy applied to specific initial-value probIems to yield plots of the solutions ([11-13)). Our auxiliary function turns out to a particular case of a special function known as Wright function [4], entire in the complex pIane. The main properties of the function (series and integraI representations) will be presented in sect. 3, where we also provide the differential equation (of fractional order) to be satisfied in the complex pIane. In particular, in the Appendix A, we shown that for (3 = 1/2, 1/3 M(z; (3) reduce respectively to the well known Gaussian and Airy functions. In view of its properties our function can be considered as a generalized hyper-Airy function. In sect. 4 we provide for M(z; (3) its asymptotic representation as z -+ 00 on the positive real axis using the saddle-point method; for a detailed asymptotic analysis in the complex pIane we defer to our recent report [15].

..

2. The Laplace transform

approach

sO that, inverting this image using standard

to the Green function

( )'

8t

=D

where z is the similarity variable.

1,

For'the

fPu

is stated imposing the following conditions

u(x,O-)=O, { u(:f:oo,t) = O,

cases

Q

=F

1 the above analysis can be generalized using the causaI

(generalized) junction, introduced by Ge'lfand & Shilov [7], tÀ-l >.E.) ,

(2.1)

8X2 ,

(2.8)

.Jfj t1{2'

Ix\9c(x,t) = "2,fii exp -4

The typical initial value problem (IV P or Cauchy problem) for the ordinary

8u

(2.4) is recovered.

From (2.4) we note the following relevant property Z2 z 1 Z=~

Let us first recall the statement of the initial value problem for the ordinary diffusion equation and the Laplace transform method to obtain the corresponding fundamental solution, in order to better understand the extension to the generalized case with fractional derivative. diffusion equation, recovered from (1.1) for Q =

techniques,

where r(>.) is the Gamma function.

(2.2)

>. = -n

(n = 0,1, . . . ) , reduces

1£O
. and, for

to the causai Dirac distributions

8~n)(t)

.

~ 1 the IV p is stated as in (2.2), but the correspondinginhomoge-

neous equation reads (+oo

u(x,t) = l-oo 9c(ç,t)f(x - ode

(2.3)

where the function 9c(x, t), referred to as the Greenfunction for the Cauchy problem, turns out to be 9c(x,t)=

~r1/2e-x2/(4Dt); 2v 7rD

~l-a(t)

where * denotes integraI time-convolution "j., '~1

+00 -st e 9c(x, t) dt,

10-

I ~

1

.

2,jD Sl/2 e-(lxl/VD)sl/2

(2.1Gb)

82u

+ g(x)

=

(2.5)

Then, applying G.LT to (2.10a) and (2.10b) with f(x)

s E CC,

(2.6)

,

,

~2-0(t)

where g(x) provides the value of Ut(x, t) at t = 0+. Since we are interested in a particular solution depending on a unique source function f( x) and continuous in the transition from Q = 1- to Q 1+ , we agree to consider g(x) ==Oin (2.1Gb).

a

-

d2Qc

8 9c-D :I ,I ~j ,

and henceforth,

extending

=

= 8(x)

provides

(2.11)

0-1

dx2 =8(x)s

the procedure

Qc,(X S )

where the lower limit of integration is just written as 0- to account for the possibility of causaI (generalized) functions. After some simple manipulations we obtain Qc(x,s)=

from O to t.

~2-a(t) * 8t2 - D 8X2 = f(x) ~l-a(t)

where o+(t) is the causaI Dirac delta. distribution. The G-LT is based on the following definition 9c(x, s):=

~l-a(t),

1£ 1 < Q :S 2 the IV P must be equipped with an additional function g( x) , since in this case we expect two linearly independent solutions; as a consequence, the corresponding inhomogeneous equation reads 82u

it represents the solution for f(x) = o(x), where o is denoting the Dirac delta diJtribution. According to Kevorkian [8], 9c(x, t) can be obtained by applying a "generalized" technique of Laplace transform (G-LT) to the inhomogeneous diffusion equation which incorporates the initial condition at t = 0+, i.e.

-

(2.1Ga)

= f(x)

.~'.

(2.4)

8u 82u 8t - D 8X2 = f(x) o+(t),

8u 82u * 8t - D 8X2

,

described in [8] for the case Q =

1 2,jD 81-0/2

1,

'

(2.12)

e-(\x\/JD)sO/2 '

that generalizes the expression (2.7). Even if the explicit inversion of (2.12) is not available in the ordinary tables of Laplace transforms (except for Q = 1), we are

(2.7)

.1 .~

~!....

.

'I interested to generalize the relation (2.8) introducing actual aimilarity variable

an auxiliary function of the

Q z=-EL Vl5 t(3, (3="2 (O< (3< 1).

whose radius of convergence turns out to infinite for O < (3 < 1. An alternative series representation can be obtained from (3.2) using the well-known reflection formula for the Gamma function

(2.13)

This function is expected to play the same role as the Gaussian function in (2.8) for the ordinary diffusion. In fact, if we write, according to the inversion formula,

~ r est - (lxIfVD) s(3sl-(3 ds 2Vl5 21ri JBr ,

Ixl9c(x,t) = -EL

l

(2.14)

.~

where Br denotes the Bromwich path [a straight line from c - ioo to c + ioo with

.i

an arbitrary c> O],we obtain

(2.15)

where M(z; (3), the auxiliary function of our initial value problem, is given by

M(z; (3) :=

r é' -

~

JBr

21rZ

3. The properties

~ ul-(3

zu(3

,

of the auxiliary

l

i§

.~

z . Ixl9c(x, t) = '2 M(z; (3),

z > O.

JI ~t

(2.16)

J'I

function

-

zu(3

=~

~

eU

21rZ J},H a

~ =~

n=o

(-l)nzn I

n.

(-l)nzn ~ n! n=o 1

l

~

U eu

(3n+(3-1

[ 21rz Ha

..

-

W(Zj.À,IL):=

r

00 zn 1 u n=On!r(>.n + IL) := 21ri JHae

L

+ zu->.

du

u/J'

du

(3)J'

M(z; (3) = W( -z; -(3, 1 - (3).

(3.4)

(3.5)

]

(3.2)

=

z2/4) ,

(3.6)

M(z; 1/3) = 32/3Ai (z/31/3) .

(3.7)

M(z; 1/2) = vf1rexp (-

We recognize from (3.6) that for /3 = 1/2 the Gauaaian auxiliary function in (2.8) is recovered.

(-l)nzn

M(z; (3) :='L n=on! r[-(3n + (l

(3.3)

We thus point out that equations (3.1) and (3,2-3) provide respectively the integral and series representations of our function M(z; (3), valid on ali of O [28] and oniy iater -1 < ). < O [29]. The Wright function has been used in the framework of Laplace transforms by Mikusinski [17] and Stankovic [27], [6].

Writing M(z;(3)

1

M(z; (3):=- 11'

.;j 'I

lies along the negative real axis, and ends up at u = -00 + ib (b > O)],which is equivalent to the originaI path (at least) for z > O. We therefore obtain eU

it reads

where ). > -1, IL> O. We thus recognize that

of O), encircles the branch cut that

~ r 21rZ)Ha

~

,I

The above definition of M(z;(3) (Bromwich repreaentation) can be analytically continued 'Vz E CC,adopting suitable integral and series representations valid in alI

M(z;(3):=

r(z) = :;;:r(l- z) sin lrZ;

.~ a:)

,~

1

r

Now, introducing the notion of the derivative of order 1/ > O of an anaiytic function j(z) (see e.g. [2], [21], [25]), we are going to show that, for any (3 (O
l),

di/ P-l (3.8)

dzI//3-l M(z; (3) + éi7r/P /3Z M(z; (3) = O. For this purpose let us consider the integraI representation the following change of integration variabie U --+ T

M(h)(Oj (3)

(3.1) and Iet us make

= u/3 .

j

where Cp is the path (in the complex T-pIane) obtained from the originai Hankel path (in the complex u-plane) by the transformation (3.9). Taking in (3.10) the derivati ve of order v = 1//3- 1, we obtain

-

Z

j

e-ZT + TI/P T 1//3-1dT.

21T"i/3zM(z;/3) = -

=-

j

(3.11)

C(3

Cf!eT

TI/(3

e

dT

e

(

e-ZT

-ZT

[

+ ] Cf!

)

j

4. The asymptotic

d

Cf!dT

Since at the extremes of the path C/3 the contribution

21T"i /3Z M(z; (3) = ~ k(3e -ZT By comparing

+ TI/P

TI//3

( ) e

e -ZT d T.

is proved to vanish, we get

M(z/ /3;(3) :=

'-

For /3 = 1/ q (q ~ 2) , the followingordinary differential equation of order (q-1) (-l)q

ZM(z; l/q)

.

function

~

]

[

{ exp (/3TI/(3 - ZT)/ /3

21T"~/3 JCf!

dT,

(4.1)

where C/3 is the path (in the complex T-pIane), obtained £rom the originai Hankel path, which extends from 00 exp - i1T" /3 to 00 exp + i1T" /3. Introducing the change of variable (scale transformation)

.,'"

T= Izl/3/(l-/3)s

is recovered

dq-I

of the auxiliary

(3.12) TI/ P-l dT.

(3.11) and (3.12), (3.8) turns out to be proved.

dzq-l M(z; l/q) + -q

representation

Let us preliminarily discuss the problem of the asymptotic representation of our function as Izi --+ 00 in the complex pIane, applying the saddle-point method to its integrai representation (3.10), which is the most suitabie for this purpose. For the sake of convenience we shaIl consider as a variable z / /3 rather than z, so re-writing (3.10) in the form

dT

-

(3.14)

We aIso point out that recent1y Podlubny [22], basing on the method of combined Laplace-Fourier transforms, has provided an alternative solution to the probIem of fractionaI diffusion, in terms of a Mittag-Leffier function in two parameters, which can be proved to be equivaIent to ours.

On the other hand, we recognize from (3.10) that

l/P d

h = O,1, . . . k - 1,

For the fractional diffusion-wave equation (1.1), only positive values of Z are of interest. As a matter offact it turns out that for Z E ]R" the function M(z; (3) is monotonic decreasing for O < /3 ::; 1/2, while for 1/2 < /3 < 1 it first increases and then decreases exhibiting a maximum vaIue Mo(/3) at some point zo«(3) [12]. As /3 --+ 1-, Mo(/3) --+ +00 and zo(/3) --+ 1; the limit case (3 = 1, for which (1.1) reduces to the wave equation, is singular since M(z; 1) = li(z - 1).

1T"Z Cf!

(- l )l/P /3

r[/3(h + 1)] sin[1T"/3(h + 1)],

where k = [1//3 -1] (i.e. k is the minimum positive integer greater or equaI to 1//3 -1). In view of the above considerations, our auxiIiary function can be referred to as a generalized hyper-Airy function.

(3.9)

As a consequence, since dT = (/3/ul-(3) du , we get the followingaIternative integrai representation 1 l/P /3M(z;/3):=-2 ' e-ZT+T dT, (3.10)

di//3-1 21T"i M (z' (3) = d I/P-I'

= (-l)h 1T"

= O.

(3.13)

The above equation is, for q ~ 4, akin to the hyper-Airy differentiaI equation order q-l [1].

,

(4.2)

the integrai (4.1) reads

of

M(z / /3;(3) = 21T"li /3 Iz/,r - 1 kf! exp [ 'ir

.

(/3SI/P - ei8s)] ds,

(4.3)

l

I

where 8

= arg z , and l ì=

1-/3

ì-l=

> l,

/3 l-/3=/3ì>O.

(4.4)

The saddle point method is known to provide the asymptotic following integral

I( >',8):=

J

>. >

(s, 8) ds ,

O,

S

representation

In view of the term sin[7r(n + l)/q], which vanishes when (n + l)/q is integer, we are led to write (n + l) = qm + h where m = O, 1,2, ... and h = l, 2, ... , q -l, so that the infinite series in n is split (re-arranged) in q - l infinite series in m. Since the sine-term reduces to

of the

sin (7rm + 7rh/q) = cos (7rm) sin (7rh/q) (A.4) reads

E 3

Sc:r tP~'-LSHIA/6-

P. RUSEV I. DIMOVSKI V. KIRYAKOVA

S{A/HtI~

-'ff(f.s-