Two-dimensional axisymmetric stresses exerted by ... · PDF fileThe numerical results for the...

International Journal of Solids and Structures 44 (2007) 2324–2348

www.elsevier.com/locate/ijsolstr

Two-dimensional axisymmetric stresses exertedby instantaneous pulses and sources of diffusionin an infinite space in a case of time-fractional

diffusion equation

Y.Z. Povstenko

Institute of Mathematics and Computer Science, Jan Długosz University of Cze�stochowa, al. Armii Krajowej 13/15,

42–200 Cze�stochowa, Poland

Received 9 April 2006; received in revised form 5 July 2006Available online 15 July 2006

Abstract

The theory of diffusive stresses based on the time-fractional diffusion equation is formulated. The source problem isdiscussed as well as the Cauchy problem. The stresses are found in axially symmetric cases (for plane deformation).The numerical results for the concentration and stress distributions are presented graphically for various values of orderof fractional derivative.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Non-Fickean diffusion; Diffusive stresses; Fractional calculus; Mittag–Leffler functions

1. Introduction

The classical theory of heat conduction is based on the Fourier law

0020-7

doi:10

E-m

q ¼ �k grad T ð1Þ
relating the heat flux vector q to the temperature gradient, where k is the thermal conductivity of a solid. Incombination with the law of conservation of energy, this equation leads to the parabolic heat conductionequation
oTot¼ aTDT ; ð2Þ

where aT is the thermal diffusivity coefficient, t is time, D is the Laplace operator. Eq. (2) is a constituent partof the classical theory of thermoelasticity (we restrict our consideration to the uncoupled theory).

683/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

.1016/j.ijsolstr.2006.07.008

ail address: [email protected]

mailto:[email protected]

Y.Z. Povstenko / International Journal of Solids and Structures 44 (2007) 2324–2348 2325

The classical theory of diffusion is based on the Fick law

J ¼ �,grad c ð3Þ
relating the matter flux vector J to the concentration gradient, where , is the diffusion conductivity. In com-bination with the balance equation for mass the Fick law leads to the classical diffusion equation
ocot¼ aDc; ð4Þ

where a is the diffusivity coefficient. Eq. (4) makes an integral part of the classical theory of diffusive stresses.During the past three decades, non-classical theories, in which the Fourier law and Fick law as well as the

heat conduction equation and the diffusion equation were replaced by more general equations, have been pro-posed. Some of these theories were formulated in terms of the theory of heat conduction, other in terms of thediffusion theory. For an extensive bibliography on this subject and further discussion see Podstrigach andKolyano (1976), Chandrasekharaiah (1986), Chandrasekharaiah (1998), Joseph and Preziosi (1989a,b),Hetnarski and Ignaczak (1999), Metzler and Klafter (2000a) and Zaslavsky (2002), among others. The resultsof this paper are formulated in terms of diffusion and diffusive stresses, but they also concern heat conductionand thermal stresses.

In the theory of heat conduction proposed by Gurtin and Pipkin (1968) the Fourier law was generalized totime-non-local dependence between the heat flux vector and the temperature gradient resulting in integro-differential heat conduction equation. The thermoelasticity theory based on this equation was formulatedby Chen and Gurtin (1970). Subsequently, Norwood (1972) and Moodi and Tait (1983) proposed the follow-ing time-non-local equation:

qðtÞ ¼ �kZ t

0

Kðt � sÞgrad T ðsÞds ð5Þ

or in terms of diffusion

JðtÞ ¼ �,Z t

0

Kðt � sÞgradcðsÞds: ð6Þ

Green and Naghdi (1993) set

Kðt � sÞ ¼ 1 ð7Þ
which yields the wave equation for the temperature and thermoelasticty without energy dissipation.
It was noted by Chandrasekharaiah (1986) that the constitutive equation for the heat flux proposed by Cat-taneo (1958) and Vernotte (1958) can also be rewritten in a non-local form with the ‘‘short-tale’’ exponentialtime-non-local kernel

Kðt � sÞ � exp � t � sf

� �; ð8Þ

where f is a non-negative constant.Kaliski (1965) and Lord and Shulman (1967), based on results of Cattaneo (1958) and Vernotte (1958),

introduced the theory of generalized thermoelasticity.The time-non-local dependence between the flux vectors and corresponding gradients with ‘‘long-tale’’

power kernel can be interpreted in terms of fractional integrals and derivatives and yields the time-fractionaldiffusion (or heat conduction) equation

oacota¼ aDc; 0 < a < 2: ð9Þ

For example, the constitutive equation with the kernel

Kðt � sÞ ¼ 1

Cða� 1Þ ðt � sÞa�2; 1 < a < 2 ð10Þ

2326 Y.Z. Povstenko / International Journal of Solids and Structures 44 (2007) 2324–2348

can be written as (see Eq. (15) below)

JðtÞ ¼ �,Ia�1 gradcðtÞ: ð11Þ

Substitution of Eq. (11) into the balance equation for mass and subsequent differentiation gives Eq. (9) with1 < a < 2.

Eq. (9) is usually referred to ‘‘anomalous diffusion’’. Other terms used in this context are: ‘‘anomaloustransport’’ (Zaslavsky, 1992, 2002), ‘‘fractional diffusion’’ (Schneider, 1990), and ‘‘strange kinetics’’ (Shle-singer et al., 1993).

Various types of anomalous transport can be distinguished. The limiting case a = 0 corresponding to theHelmholtz equation is associated with localized diffusion (Kimmich, 2002). The subdiffusion regime is char-acterized by the value 0 < a < 1. The ordinary diffusion corresponds to a = 1. The superdiffusion regime isassociated with 1 < a < 2. The limiting case a = 2 corresponding to the wave equation is known as ballisticdiffusion (Metzler and Klafter, 2000b; Kimmich, 2002).

At the level of individual particle motions the classical diffusion corresponds to the Brownian motion whichis characterized by a mean-squared displacement increasing linearly with time

hx2i � at: ð12Þ

Anomalous diffusion which is exemplified by a mean-squared displacement with the power-law timedependence

hx2i � ata; a 6¼ 1; ð13Þ

at the level of individual particle motion has been modeled in numerous ways. The continuous time randomwalk (CTRW) theory (Montroll and Shlesinger, 1984; Metzler et al., 1998; Metzler and Klafter, 2000b andreferences therein) is in most common use and allows one to extend classical Brownian random walks to var-iable jump lengths and waiting times between successive jumps. The velocity model in a CTRW scheme(Zumofen and Klafter, 1993; Metzler and Compte, 1999) assumes that the particle moves at the constantvelocity to the new site. The power-law tails make it possible to have very long waiting times, and in the sub-diffusion regime (0 < a < 1) particles on the average move slower than in the ordinary diffusion which corre-sponds to a = 1. In the superdiffusion regime (1 < a < 2) particles on the average move faster than in theordinary diffusion. For dimensions higher than D = 1, solutions of Eq. (9) for 1 < a < 2 can be bimodal,and from a waiting time perspective correspond to the velocity model.

Eq. (9) is a mathematical model of important physical phenomena ranging from amorphous (Scher andMontroll, 1975), colloid (Weeks and Weitz, 2002), glassy (Bendler et al., 2002; Hilfer, 2002) and porous (Kochand Brady, 1988; Kimmich, 2002) materials through fractals (Ben-Avraham and Havlin, 2001; Even et al.,1984; Nigmatullin, 1986), percolation clusters (Kimmich, 2002), random (Giona and Roman, 1992) and dis-ordered (Ben-Avraham and Havlin, 2001) media to comb structures (Lubashevskii and Zemlyanov, 1998),dielectrics (Nigmatullin, 1984a) and semiconductors (Nigmatullin, 1984b), polymers (Cates, 1984; Paul,2002) and biological systems (Periasamy and Verkman, 1998).

Metzler and Nonnenmacher (2003) connect fractional relaxation to a generalized diffusion approach similarto the Zener model, and construct generalized rheological models based on fractional elements.

A quasi-static uncoupled theory of diffusive (or thermal) stresses based on Eq. (9) was proposed by theauthor (Povstenko, 2005a,b). The purpose of this paper is to study stresses in axially symmetric case (for planedeformation) in the framework of this theory. Because Eq. (9) in the case 1 6 a 6 2 interpolates the heat con-duction equation (a = 1) and the wave equation (a = 2) the proposed theory interpolates the classical thermo-elasticity and the thermoelasticity without energy dissipation introduced by Green and Naghdi (1993).

2. Essentials of the Riemann–Liouville fractional calculus

In this section we recall the main ideas of fractional calculus (Samko et al., 1993; Miller and Ross, 1993;Gorenflo and Mainardi, 1997, among others). It is common knowledge that integrating by parts n � 1 times


the calculation of the n-fold primitive of a function f(t) can be reduced to the calculation of a single integral ofthe convolution type

Inf ðtÞ ¼ 1

ðn� 1Þ!

Z t

0

ðt � sÞn�1f ðsÞds; ð14Þ

where n is a positive integer. The Riemann–Liouville fractional integral is introduced as a natural generaliza-tion of the convolution type form (14):

Iaf ðtÞ ¼ 1

CðaÞ

Z t

0

ðt � sÞa�1f ðsÞds; a > 0; ð15Þ

where C(a) is the gamma function. The Laplace transform rule for the fractional integral reads

LfIaf ðtÞg ¼ 1

saLff ðtÞg; ð16Þ

where s is the transform variable.The Riemann–Liouville derivative of the fractional order a is defined as left-inverse to Ia

DaRLf ðtÞ ¼ DnIn�af ðtÞ ¼

dn

dtn

1

Cðn� aÞ

Z t

0

ðt � sÞn�a�1f ðsÞds

� �; n� 1 < a < n;

dn

dtnf ðtÞ; a ¼ n

8>><>>: ð17Þ

and for its Laplace transform requires the knowledge of the initial values of the fractional integral In�af(t) andits derivatives of the order k = 1,2, . . . ,n � 1:

LfDaRLf ðtÞg ¼ saLff ðtÞg �

Xn�1

k¼0

DkIn�af ð0þÞsn�1�k; n� 1 < a < n: ð18Þ

An alternative definition of the fractional derivative was proposed by Caputo (1967, 1969):

DaCf ðtÞ ¼ In�aDnf ðtÞ ¼

1

Cðn� aÞ

Z t

0

ðt � sÞn�a�1 dnf ðsÞdsn

ds; n� 1 < a < n;

dn

dtnf ðtÞ; a ¼ n:

8>><>>: ð19Þ

For its Laplace transform rule the Caputo fractional derivative requires the knowledge of the initial values ofthe function f(t) and its integer derivatives of order k = 1,2, . . . ,n � 1:

LfDaCf ðtÞg ¼ saLff ðtÞg �

Xn�1

k¼0

Dkf ð0þÞsa�1�k; n� 1 < a < n: ð20Þ

The Caputo fractional derivative is a regularization in the time origin for the Riemann–Liouville fractionalderivative by incorporating the relevant initial conditions (Gorenflo and Mainardi, 1998). In this paper weshall use the Caputo fractional derivative omitting the index C. If care is taken, the results obtained usingthe Caputo formulation can be recast to the Riemann–Liouville version.

3. Diffusive stresses. Formulation of the problem

A quasi-static uncoupled theory of diffusive stress is governed by the equilibrium equation in terms ofdisplacements

lDuþ ðkþ lÞgraddivu ¼ bcKc gradc; ð21Þ

the stress–strain–concentration relation

r ¼ 2leþ ðk tre� bcKccÞI ð22Þ


and the time-fractional diffusion equation

oac

ota¼ aDcþ Q; 0 6 a 6 2; ð23Þ

where u is the displacement vector, r the stress tensor, e the linear strain tensor, c the concentration, Q themass source, a the diffusivity coefficient, k and l are Lame constants, Kc = k + 2l/3, bc is the diffusion coef-ficient of volumetric expansion, I denotes the unit tensor.

It should be pointed out that first theoretical investigation of the interaction between the processes of elas-ticity, heat and diffusion in elastic solid dates back to Podstrigach (1961, 1964, 1965). For additional refer-ences, further generalizations and discussion see Nowacki (1974), Podstrigach and Povstenko (1985),Nowacki and Olesiak (1991) and others.

Just as in classical theory we can use the representation of non-zero components of the stress tensor in termsof displacement potential U (Parkus, 1959; Nowacki, 1986)

r ¼ 2lð$$U� IDUÞ; ð24Þ
where $ is the gradient operator.
The displacement potential is determined from the following equation:

DU ¼ mc; m ¼ 1þ m1� m

bc

3; ð25Þ

where m is the Poisson ratio.If a bounded solid is considered the corresponding boundary conditions should be given; for unbounded

medium

limjxj!1

uðx; tÞ ¼ 0; ð26Þ

limjxj!1

cðx; tÞ ¼ 0: ð27Þ

Eq. (23) should also be subject to initial conditions

t ¼ 0 : c ¼ P ðxÞ; 0 < a 6 2; ð28Þ

t ¼ 0 :ocot¼ W ðxÞ; 1 < a 6 2: ð29Þ

In this paper we consider two-dimensional axisymmetric case. Hence, Eqs. (23), (28), (29) and (24) are rewrit-ten as

oacota¼ a

o2cor2þ 1

rocor

� �þ Qðr; tÞ; ð30Þ

t ¼ 0 : c ¼ P ðrÞ; 0 < a 6 2; ð31Þ

t ¼ 0 :ocot¼ W ðrÞ; 1 < a 6 2 ð32Þ

and

rzz ¼ rrr þ rhh ¼ �2lDU; ð33Þ

rrr � rhh ¼ 2lo2Uor2� 1

roUor

� �: ð34Þ

Solution of Eq. (30) under initial conditions (31) and (32) can be represented by the following formula:

cðr; tÞ ¼ 2pZ t

0

Z 1

0

RQðR; sÞEQðr;R; t � sÞdRdsþ 2pZ 1

0

RP ðRÞEP ðr;R; tÞdR

þ 2pZ 1

0

RW ðRÞEW ðr;R; tÞdR; ð35Þ


where we have three types of fundamental solutions EQ(r,R, t), EP(r,R, t) and EW(r,R, t) corresponding to

Qðr;R; tÞ ¼ q2pR

dðr � RÞdþðtÞ; ð36Þ

P ðr;RÞ ¼ p2pR

dðr � RÞ; ð37Þ

W ðr;RÞ ¼ w2pR

dðr � RÞ; ð38Þ

respectively. For the sake of convenience and to obtain the non-dimensional quantities we have introduced themultipliers q, p and w.

The purpose of this paper is to find the fundamental solutions EQ(r,R, t), EP(r,R, t) and EW(r,R, t) and tostudy stresses in axially symmetric case (for plane deformation) corresponding to these solutions.

4. The source problem

Consider the time-fractional diffusion equation with the source term and zero initial conditions

oacota¼ a

o2cor2þ 1

rocor

� �þ q

2pRdðr � RÞdþðtÞ; 0 6 r <1; 0 < t <1; ð39Þ

t ¼ 0 : c ¼ 0; 0 < a 6 2; ð40Þ

t ¼ 0 :ocot¼ 0; 1 < a 6 2: ð41Þ

Mathematical aspects concerning correctness of initial-value problem for anomalous diffusion equation withthe source term were discussed by Eidelman and Kochubei (2004) for 0 < a < 1 and Hanyga (2002a,b) for0 < a < 2.

Using the Laplace transform with respect to time t and the Hankel transform with respect to the radialcoordinate r we obtain

c� ¼ q2p

J 0ðRnÞ 1

sa þ an2; ð42Þ

where asterisk denotes the transforms, s is the Laplace transform variable, n is the Hankel transform variable.The most important special functions used in fractional calculus are the Mittag–Leffler functions (Erdelyi

et al., 1955)

EaðzÞ ¼X1n¼0

zn

Cðanþ 1Þ ; a > 0; z 2 C ð43Þ

providing a generalization of the exponential function and the generalized Mittag–Leffler functions in twoparameters a and b (Humbert, 1953; Agarwal, 1953; Humbert and Agarwal, 1953) which are described bythe following series representation:

Ea;bðzÞ ¼X1n¼0

zn

Cðanþ bÞ ; a > 0; b > 0; z 2 C: ð44Þ

The essential role of the Mittag–Leffler functions in fractional calculus results from the following formula forthe inverse Laplace transform (Podlubny, 1994)

L�1 sa�b

sa þ b

� �¼ tb�1Ea;bð�btaÞ: ð45Þ

To invert the Laplace transform of the concentration (42) the following formula

L�1 1

sa þ an2

� �¼ ta�1Ea;að�an2taÞ ð46Þ

is used.


Consequently, we obtain

c ¼ qta�1

2p

Z 1

0

Ea;að�an2taÞJ 0ðRnÞJ 0ðrnÞndn; ð47Þ

rrr þ rhh ¼ �2lmqta�1

2p

Z 1

0

Ea;að�an2taÞJ 0ðRnÞJ 0ðrnÞndn; ð48Þ

rrr � rhh ¼ �2lmqta�1

2p

Z 1

0

Ea;að�an2taÞJ 0ðRnÞJ 2ðrnÞndn ð49Þ

or

rrr ¼ �2lmqta�1

2pr

Z 1

0

Ea;að�an2taÞJ 0ðRnÞJ 1ðrnÞdn; ð50Þ

rhh ¼ �2lmc� rrr: ð51Þ

Let us consider several particular cases. The solutions for these particular values of the parameter a can be alsoused for testing the numerical algorithms in the general case of arbitrary values of 0 < a < 2.

It is convenient to introduce the following non-dimensional quantities:

q ¼ rR; j ¼

ffiffiffiap

ta=2

R; ð52Þ

�c ¼ 2pR2

qta�1c; �rij ¼

1

2lm2pR2

qta�1rij: ð53Þ

4.1. Normal diffusion (a = 1)

In this case

L�1 1

sþ an2

� �¼ e�an2t ð54Þ

and

c ¼ q2p

Z 1

0

e�an2tJ 0ðRnÞJ 0ðrnÞndn; ð55Þ

rrr ¼ �2lmq

2pr

Z 1

0

e�an2tJ 0ðRnÞJ 1ðrnÞdn: ð56Þ

Using integrals (A1) and (A6) from Appendix we obtain

�c ¼ 1

2j2exp � 1þ q2

4j2

� �I0

q2j2

; ð57Þ

�rrr ¼ �1

4j2

Z 1

0

exp � 1þ q2x4j2

� �I0

qffiffiffixp

2j2

� �dx; ð58Þ

�rhh ¼ ��c� �rrr: ð59Þ

The last equation follows from Eq. (51) and will be used for all considered particular cases.

4.2. Subdiffusion with a = 1/2

The inverse Laplace transform reads

L�1 1ffiffisp þ an2

� �¼ 1ffiffiffiffiffi

ptp � an2 expða2n4tÞerfcðan2

ffiffitpÞ ¼ 2ffiffiffiffiffi

ptp

Z 1

0

ve�v2�2affitp

n2v dv: ð60Þ


Inserting integral representation (60) into Eqs. (47) and (50), changing integration with respect to n and v andusing integrals (A1) and (A6) from Appendix we arrive at

�c ¼ 1

2ffiffiffipp

j2

Z 1

0

exp �v2 � 1þ q2

8j2v

� �I0

q4j2v

dv; ð61Þ

�rrr ¼ �1

4ffiffiffipp

j2

Z 1

0

e�v2

Z 1

0

exp � 1þ q2x8j2v

� �I0

qffiffiffixp

4j2v

� �dxdv: ð62Þ

4.3. Superdiffusion with a = 3/2

The inverse Laplace transform for this value of parameter a was obtained by Povstenko (2005b) and reads

L�1 1

s3=2 þ an2

� �¼ G3=2ðn; tÞ ¼

1

3a1=3n2=3�ec erfc

ffiffifficp þ 2e�c=2 cos

ffiffiffi3p

2c� p

3

!(

þ 4ffiffifficpffiffiffipp

Z 1

0

exp � 1

2cð1� v2Þ

� �cos

ffiffiffi3p

2cð1� v2Þ

" #dv

); ð63Þ

where c = a2/3n4/3t.The concentration and stress component are

c ¼ q2p

Z 1

0

G3=2ðn; tÞJ 0ðRnÞJ 0ðrnÞndn; ð64Þ

rrr ¼ �2lmq

2pr

Z 1

0

G3=2ðn; tÞJ 0ðRnÞJ 1ðrnÞdn: ð65Þ

4.4. Ballistic diffusion (a = 2)

In the case of the wave equation

L�1 1

s2 þ an2

� �¼ sinð ffiffiffiap tnÞffiffiffi

ap

nð66Þ

and

c ¼ q2p

ffiffiffiap

Z 1

0

sinðffiffiffiap

tnÞJ 0ðRnÞJ 0ðrnÞdn; ð67Þ

rrr ¼ �2lmq

2pffiffiffiap

r

Z 1

0

sinð ffiffiffiap tnÞn

J 0ðRnÞJ 1ðrnÞdn: ð68Þ

Using integrals (A2) and (A5) from Appendix we present the non-dimensional concentration �c and stress ten-sor component �rrr for different values of j.

4.4.1. 0 < j < 18
�c ¼
0; 0 6 q < 1� j;1

jpffiffiffiqp KðkÞ; 1� j < q < 1þ j;

0; 1þ j < q <1;

>><>>: ð69Þ

�rrr ¼

0; 0 6 q < 1� j;

� 1

jpq2

Z q

1�j

ffiffiffixp

K

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2 � ð1� xÞ2

q2ffiffiffixp

24

35dx; 1� j < q < 1þ j;

� 1

q2; 1þ j < q <1:

8>>>>>>><>>>>>>>:

ð70Þ


4.4.2. j = 18
�c ¼
1

pffiffiffiqp KðkÞ; 0 < q < 2;

0; 2 < q <1;

<: ð71Þ

�rrr ¼� 1

pq2

Z q

0

ffiffiffixp

K

ffiffiffiffiffiffiffiffiffiffiffi2� xp

2

!dx; 0 < q < 2;

� 1

q2; 2 < q <1:

8>>><>>>:

ð72Þ

4.4.3. j > 1 � �8
�c ¼
1

jpkffiffiffiqp K

1

k; 0 6 q < j� 1;

1

jpffiffiffiqp KðkÞ; j� 1 < q < 1þ j;

0; 1þ j < q <1;

>>>>><>>>>>:

ð73Þ

�rrr ¼

� 2

jpq2

Z q

0

xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2 � ðx� 1Þ2

q K2ffiffiffixpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

j2 � ðx� 1Þ2q264

375dx; 0 6 q < j� 1;

� 2

jpq2

Z j�1

0

xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2 � ðx� 1Þ2

q K2ffiffiffixpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

j2 � ðx� 1Þ2q264

375dx

� 1

jpq2

Z q

j�1

ffiffiffixp

K

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2 � ðx� 1Þ2

q2ffiffiffixp

24

35dx; j� 1 < q < 1þ j;

� 1

q2; 1þ j < q <1:

8>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>:

ð74Þ

Here and in what follows K(k) and E(k) are the complete elliptic integrals of the first and second kind,

k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2 � ðq� 1Þ2

q2ffiffiffiqp : ð75Þ

5. The first Cauchy problem

Consider the first Cauchy problem for the time-fractional diffusion equation

oacota¼ a

o2cor2þ 1

rocor

� �; 0 6 r <1; 0 < t <1; ð76Þ

t ¼ 0 : c ¼ p2pR

dðr � RÞ; 0 < a 6 2; ð77Þ

t ¼ 0 :ocot¼ 0; 1 < a 6 2: ð78Þ

Mathematical aspects relating to well-posedness of the Cauchy problems for anomalous diffusion equationwere considered by many authors. Here we refer to the papers of El-Sayed (1995, 1996) for 0 < a < 2, Umarovet al. (2000) (0 < a < 2), Hanyga (2002a,b) (0 < a < 2), Moustafa (2003) (1 < a < 2), Eidelman and Kochubei


(2004) (0 < a < 1) and references therein. Earlier works of Berens and Westphal (1968) (0 < a < 2), Kochubei(1990) (0 < a < 1) and Fujita (1990) (1 < a < 2) dealing with the one-dimensional case should also be mentioned.This remark (excluding papers in which the case (0 < a < 1) was studied) also concerns the second Cauchy prob-lem (see Eqs. (104)–(106) below).

Using the Laplace transform with respect to time t and the Hankel transform with respect to the radialcoordinate r we obtain

c� ¼ p2p

J 0ðRnÞ sa�1

sa þ an2: ð79Þ

The inverse Laplace transform is expressed in terms of the Mittag–Leffler functions

L�1 sa�1

sa þ an2

� �¼ Ea;1 �an2ta

� �� Eað�an2taÞ: ð80Þ

Inverting the Hankel transform leads to

c ¼ p2p

Z 1

0

Eað�an2taÞJ 0ðRnÞJ 0ðrnÞndn; ð81Þ

rrr ¼ �2lmp

2pr

Z 1

0

Eað�an2taÞJ 0ðRnÞJ 1ðrnÞdn: ð82Þ

Let us analyze some particular cases.

5.1. Localized diffusion

The limiting case a! 0 corresponds to the solution of the Helmholtz equation

lima!0

c� ¼ p2p

J 0ðRnÞ 1

1þ an2

1

sð83Þ

and

c ¼ p2p

Z 1

0

1

1þ an2J 0ðRnÞJ 0ðrnÞndn; ð84Þ

rrr ¼ �2lmp

2pr

Z 1

0

1

1þ an2J 0ðRnÞJ 1ðrnÞdn: ð85Þ

Using integrals (A4) and (A7) from Appendix we obtain

�c ¼

1

j2I0ðq=jÞK0ð1=jÞ; 0 6 q < 1;

1

j2I0ð1=jÞK0ðq=jÞ; 1 < q <1;

8><>: ð86Þ

�rrr ¼� 1

jqI1ðq=jÞK0ð1=jÞ; 0 < q < 1;

� 1

jq2I1ð1=jÞK0ð1=jÞ þ I0ð1=jÞK1ð1=jÞ � qI0ð1=jÞK1ðq=jÞ½ �; 1 < q <1;

8>><>>: ð87Þ

where the non-dimensional quantities (52) are used as well as

�c ¼ 2pR2

pc; �rij ¼

1

2lm2pR2

prij: ð88Þ


5.2. Normal diffusion (a = 1)

It is well-known that in the case of the diffusion equation solutions for the source problem and the Cauchyproblem coincide and in Eq. (35)

EQðr;R; tÞ ¼ EP ðr;R; tÞ: ð89Þ
Hence, we can use Eqs. (55) and (56) with q substituted by p.
5.3. Subdiffusion with a = 1/2

The inverse Laplace transform for this value of a is well-known. We obtain the integral representation

L�1 1ffiffisp ð ffiffisp þ an2Þ

( )¼ expða2n4tÞerfcðan2

ffiffitpÞ ¼ 2ffiffiffi

pp

Z 1

0

e�v2�2affitp

n2vdv ð90Þ

which is convenient to change the order of integration with respect to n and v after inserting into (81) and (82).Using integrals (A1) and (A6) from Appendix allows us to arrive at

�c ¼ 1

2ffiffiffipp

j2

Z 1

0

exp �v2 � 1þ q2

8j2v

� �I0

q4j2v

1

vdv; ð91Þ

�rrr ¼ �1

4ffiffiffipp

j2

Z 1

0

1

ve�v2

Z 1

0

exp � 1þ q2x8j2v

� �I0

qffiffiffixp

4j2v

� �dxdv: ð92Þ


The inverse Laplace transform for this value of parameter a was obtained by Povstenko (2005a) and reads

L�1

ffiffisp

s3=2 þ an2

� �¼ F 3=2ðn; tÞ ¼

1

3ec erfc

ffiffifficp þ 2e�c=2 cos

ffiffiffi3p

2c

!(


Z 1

0

exp � 1

2cð1� v2Þ

� �cos

ffiffiffi3p

2cð1� v2Þ þ p

3

" #dv

): ð93Þ

The concentration and stress component are

c ¼ p2p

Z 1

0

F 3=2ðn; tÞJ 0ðRnÞJ 0ðrnÞndn; ð94Þ

rrr ¼ �2lmp

2pr

Z 1

0

F 3=2ðn; tÞJ 0ðRnÞJ 1ðrnÞdn: ð95Þ


In the case of the wave equation

L�1 s

s2 þ an2

� �¼ cosð

ffiffiffiap

tnÞ; ð96Þ

c ¼ p2p

Z 1

0

cosðffiffiffiap

tnÞJ 0ðRnÞJ 0ðrnÞndn; ð97Þ

rrr ¼ �2lmp

2pr

Z 1

0

cosðffiffiffiap

tnÞJ 0ðRnÞJ 1ðrnÞdn: ð98Þ

The solution to the Cauchy problem can be received by differentiation of the solution (67) to the source prob-lem with respect to time. To obtain the stress component rrr, Eq. (A3) from Appendix is used.


5.5.1. 0 < j < 1 8
�c ¼ 1
2ffiffiffiffiffiffiffiffiffiffiffi1� jp dðq� 1þ jÞ þ 1

2ffiffiffiffiffiffiffiffiffiffiffi1þ jp dðq� 1� jÞ þ

0; 0 6 q < 1� j;

j4pq3=2

EðkÞ � k02KðkÞk2k02

; 1� j < q < 1þ j;

0; 1þ j < q <1;

>>><>>>:

ð99Þ

�rrr ¼

0; 0 6 q < 1� j;

� 1

q21� K0 arcsin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

1þ jþ q

s; k

!þ 1� j

pffiffiffiqp KðkÞ

" #; 1� j < q < 1þ j;

� 1

q2; 1þ j < q <1:

8>>>>>><>>>>>>:

ð100Þ

5.5.2. j = 1 ffiffiffiffiffiffiffiffiffiffiffis !" #8
�rrr ¼
� 1

q21� K0 arcsin

2

2þ q; k ; 0 < q < 2;

� 1

q2; 2 < q <1:

>>>><>>>>:

ð101Þ

5.5.3. j > 1 � �8

�c ¼ 1

2ffiffiffiffiffiffiffiffiffiffiffi1þ jp dðq� 1� jÞ þ

� 2j

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2 � ðq� 1Þ2

q½j2 � ðqþ 1Þ2�

E1

k; 0 6 q < j� 1;

j4pq3=2

EðkÞ � k02KðkÞk2k02

; j� 1 < q < 1þ j;

0; 1þ j < q <1;

>>>>>>><>>>>>>>:

ð102Þ

�rrr ¼

� 1

q21� K0 arcsin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ j� q1þ jþ q

s;1

k

!� 1

pkffiffiffiqp

K1

k

� �" #; 0 < q < j� 1;

� 1

q21� K0 arcsin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

1þ jþ q

s; k

!þ 1� j

pffiffiffiqp KðkÞ

" #; j� 1 < q < 1þ j;

� 1

q2; 1þ j < q <1;

8>>>>>>>>>><>>>>>>>>>>:

ð103Þ

where k is the same as in Eq. (75), k0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1� k2p

, Heuman’s Lambda function K0(u,k) is described in Appendix.

6. The second Cauchy problem

Consider the second Cauchy problem for the time-fractional diffusion equation

oac

ota¼ a

o2c

or2þ 1

rocor

� �; 0 6 r <1; 0 < t <1; ð104Þ

t ¼ 0 : c ¼ 0; 1 < a 6 2; ð105Þ

t ¼ 0 :ocot¼ w

2pRdðr � RÞ; 1 < a 6 2: ð106Þ


The Laplace transform with respect to time t and the Hankel transform with respect to the radial coordinate r

leads to

c� ¼ w2p

J 0ðRnÞ sa�2

sa þ an2; 1 < a 6 2: ð107Þ

It follows from Eq. (45) that

L�1 sa�2

sa þ an2

� �¼ tEa;2ð�an2taÞ ð108Þ

and

c ¼ wt2p

Z 1

0

Ea;2ð�an2taÞJ 0ðRnÞJ 0ðrnÞndn; ð109Þ

rrr ¼ �2lmwt2pr

Z 1

0

Ea;2ð�an2taÞJ 0ðRnÞJ 1ðrnÞdn: ð110Þ

In the case of the second Cauchy problem we use the following non-dimensional quantities

�c ¼ 2pR2

wtc; �rij ¼

1

2lm2pR2

wtrij: ð111Þ

For 1 < a 6 2 we consider two particular cases.


( ) ffiffiffip !(
L�1 1ffiffi
sp ðs3=2 þ an2Þ

¼ H 3=2ðn; tÞ ¼1

3a2=3n4=3ec erfc

ffiffifficp � 2e�c=2 cos

3

2cþ p

3


Z 1

0

exp � 1

2cð1� v2Þ

� �cos

ffiffiffi3p

2cð1� v2Þ � p

3

" #dv

)ð112Þ

and

c ¼ w2p

Z 1

0

H 3=2ðn; tÞJ 0ðRnÞJ 0ðrnÞndn; ð113Þ

rrr ¼ �2lmw

2pr

Z 1

0

H 3=2ðn; tÞJ 0ðRnÞJ 1ðrnÞdn: ð114Þ


In the case of the wave equation the fundamental solution to the second Cauchy problem coincides with thefundamental solution to the source problem, and in Eq. (35)

EQðr;R; tÞ ¼ EW ðr;R; tÞ: ð115Þ

Hence, we can use Eqs. (67) and (68) with q substituted by w.


7. Numerical results and discussion

In this section we present the numerical results for the non-dimensional concentration and stresses distri-butions for various values of order of fractional derivative a and the parameter j.

The series representation Eq. (44) of the Mittag–Leffler type functions is inconvenient for calculation ofintegrals (47), (50), (81), (82), (109) and (110). The integral representations of these functions suitable for suchcalculation were obtained by Gorenflo and Mainardi (1996, 1997). Using their results we get

L�1 1

sa þ an2

� �¼

a1=a�1n2=a�2Uð0Þa ðn; tÞ; 0 < a < 1;

a1=a�1n2=a�2½Uð0Þa ðn; tÞ �Wð0Þa ðn; tÞ�; 1 < a < 2;

8<: ð116Þ

L�1 sa�1

sa þ an2

� �¼

Uð1Þa ðn; tÞ; 0 < a < 1;

Uð1Þa ðn; tÞ þWð1Þa ðn; tÞ; 1 < a < 2;

(ð117Þ

L�1 sa�2

sa þ an2

� �¼ 1

a1=an2=a½�Uð2Þa ðn; tÞ þWð2Þa ðn; tÞ�; 1 < a < 2 ð118Þ

with

UðmÞa ðn; tÞ ¼sinðapÞ

p

Z 1

0

e�xa1=an2=at xa�m

x2a þ 2xa cosðapÞ þ 1dx; ð119Þ

WðmÞa ðn; tÞ ¼2

aea1=an2=at cosðp=aÞ cos a1=an2=at sin

pa

� ðm� 1Þ p

a

h i; ð120Þ

where m = 0 refers to the source problem, m = 1 and m = 2 correspond to the first and second Cauchy prob-lems, respectively.

The numerical results are shown in Figs. 1–16. The computations are carried out in the range 0 6 q 6 2 or0 6 q 6 4 according to the values of j reflecting the characteristic features of the curves for various order ofthe time-fractional derivative.

Fig. 1. Variation of concentration with distance (the source problem; j = 0.25).

Fig. 2. Variation of concentration with distance (the source problem; j = 1).

Fig. 3. Variation of concentration with distance (the source problem; j = 1.5).


In the case of subdiffusion (0 < a < 1) the anomalous diffusion equation interpolates the Helmholtz equa-tion and the ordinary diffusion equation. In the case of superdiffusion (1 < a < 2) the considered equationinterpolates the diffusion equation and the wave equation, and the proposed theory interpolates the classical

Fig. 5. Variation of stress rrr with distance (the source problem; j = 1.5).

Fig. 4. Variation of stress rrr with distance (the source problem; j = 0.25).


theory of diffusive stresses and that without energy dissipation introduced by Green and Naghdi (1993) interms of thermoelasticity.The solutions of fractional diffusion equation in the superdiffusion regime featurepropagating humps, underlininig the proximity to the standard wave equation in contrast to the shape ofcurves describing the subdiffusion regime. For this reason, in the case 1 < a < 2 the term ‘‘fractional waveequation’’ is also used in the literature (see e.g. Schneider and Wyss, 1989; Metzler and Klafter, 2000b).For some values of parameters the solution to this equation is not everywhere positive (in two- and three-dimensional cases). This might also be reflected in the fact that for 1 < a < 2 the Mittag–Leffler functionsare no more monotonically decaying with increasing argument, but contain oscillations (see Eq. (120)).

Fig. 7. Variation of stress rhh with distance (the source problem; j = 1.5).

Fig. 6. Variation of stress rhh with distance (the source problem; j = 0.25).


Fig. 9. Variation of stress rrr with distance (the first Cauchy problem; j = 0.25).

Fig. 8. Variation of concentration with distance (the first Cauchy problem; j = 0.25).


Three distinguishing values of the parameter j are considered: j = 0.25, j = 1 and j = 1.5. The particularcase R = 0 was studied in the previous papers (Povstenko, 2005a,b). The solution for this case can also beobtained supposing the large values of the parameter j (j!1).

Fig. 10. Variation of stress rhh with distance (the first Cauchy problem; j = 0.25).

Fig. 11. Variation of concentration with distance (the second Cauchy problem; j = 0.25).


In figures we have not displayed the curves for the same values of a because very different scales wereobtained. We try to show the transition of solutions from the Helmholtz equation through the diffusion equa-tion to the wave equation. In particular, it is evident from the figures how jumps and Dirac delta functions

Fig. 12. Variation of concentration with distance (the second Cauchy problem; j = 1.5).

Fig. 13. Variation of stress rrr with distance (the second Cauchy problem; j = 0.25).


arising in the case of the wave equation are approximated. Unfortunately, for reasons of space, it is impossibleto exhibit the whole spectrum of numerical results. Therefore, we have restricted ourselves to the most essen-tial and representative figures.

It should be noted that the results are presented in the non-dimensional form. The proposed theory is ageneralization of the classical theory of diffusive stresses accounting for the fractional diffusion equationinstead of the ordinary diffusion equations. To estimate stresses, as the first approximation we can use the

Fig. 14. Variation of stress rrr with distance (the second Cauchy problem; j = 1.5).

Fig. 15. Variation of stress rhh with distance (the second Cauchy problem; j = 0.25).


classical values of the shear modulus, Poisson’s ratio and diffusion expansion coefficient. There is a body ofdata concerning the value of a in Eq. (13) (see e.g. Luedtke and Landman, 1999; Upadhyaya et al., 2001; Kim-mich et al., 2001; Paul, 2002; Huc and Main, 2003). The solutions of specific problems for the time-fractionaldiffusion equation can be also of interest for experimental studies of the diffusivity coefficient.

Fig. 16. Variation of stress rhh with distance (the second Cauchy problem; j = 1.5).


Acknowledgement

The author is grateful to Prof. Oleksa Piddubniak for helpful discussion.

Appendix

Here we present integrals (Gradshtein and Ryzhik, 1980; Prudnikov et al., 1983) used in the paper:
Z 1
0

e�ax2

J 0ðbxÞJ 0ðcxÞxdx ¼ 1

2aexp � b2 þ c2

4a

� �I0

bc2a

� �; ðA1Þ

Z 1

0

sinðaxÞJ 0ðbxÞJ 0ðcxÞdx ¼

0; 0 < a < jb� cj;1

pffiffiffiffiffibcp KðkÞ; jb� cj < a < bþ c;

1

pkffiffiffiffiffibcp K

1

k

� �; bþ c < a <1;

8>>>>><>>>>>:

ðA2Þ

Z 1

0

cosðaxÞJ 1ðbxÞJ 0ðcxÞdn ¼¼

0; 0 < a < c� b;

1

b1� K0 u1; kð Þ þ c� a

pffiffiffiffiffibcp KðkÞ

� �; 0 < jc� bj < a < cþ b;

1

b1� K0 u2;

1

k

� �� 1

pk

ffiffiffibc

rK

1

k

� �" #; 0 < bþ c < a;

1

b; 0 < ja� cj < aþ c < b;

8>>>>>>>>>>><>>>>>>>>>>>:

ðA3Þ

Z 1

0

xx2 þ a2

J 0ðbxÞJ 0ðcxÞdx ¼I0ðabÞK0ðacÞ; 0 < b < c;

I0ðacÞK0ðabÞ; 0 < c < b;

(ðA4Þ


where In(x) and Kn(x) are the modified Bessel functions of order n, F(u,k) and E(u,k) are the incomplete ellip-tic integrals of the first and second kind, K(k) and E(k) are the complete elliptic integrals of the first and secondkind, respectively. Heuman’s Lambda function is expressed as (Abramowitz and Stegun, 1972)

K0ðu; kÞ ¼2

pEðkÞF ðu; k0Þ þ KðkÞEðu; k0Þ � KðkÞF ðu; k0Þ½ �:

Here

k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � ðb� cÞ2

q2ffiffiffiffiffibcp ; k0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� k2

p; a > 0; b > 0; c > 0;

u1 ¼ arcsin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2c

aþ bþ c

r; u2 ¼ arcsin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaþ c� baþ bþ c

r:

Using the well-known formula for the Bessel functions
Z c
0

yJ 0ðxyÞdy ¼ cx

J 1ðcxÞ ðA5Þ

allows us to obtain the additional integrals
Z 1
0

e�ax2

J 0ðbxÞJ 1ðcxÞdx ¼ 1

2ac

Z c

0

x exp � b2 þ x2

4a

� �I0

bx2a

� �dx; ðA6Þ

Z 1

0

1

x2 þ a2J 0ðbxÞJ 1ðcxÞdx ¼

1

aI1ðacÞK0ðabÞ; 0 < c < b;

1

ac½bI1ðabÞK0ðabÞ þ I0ðabÞðbK1ðabÞ � cK1ðacÞÞ�; 0 < b < c:

8><>: ðA7Þ

References

Abramowitz, M., Stegun, I.A., 1972. Handbook of Mathematical Functions. Dover, New York.Agarwal, R.P., 1953. A propos d’une note de M. Pierre Humbert. C. R. Acad. Sci. Paris 236, 2031–2032.Ben-Avraham, D., Havlin, S., 2001. Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press,

Cambridge.Bendler, J.T., Fontanella, J.J., Shlesinger, M.F., 2002. Anomalous defect diffusion near the glass transition. Chem. Phys. 284, 311–317.Berens, H., Westphal, U., 1968. A Cauchy problem for a generalized wave equation. Acta Sci. Math. (Szeged) 29, 93–106.Caputo, M., 1967. Linear models of dissipation whose Q is almost frequency independent, Part II. Geophys. J. Roy. Astron. Soc. 13, 529–

539.Caputo, M., 1969. Elasticita e Dissipazione. Zanichelli, Bologna (in Italian).Cates, M.E., 1984. Statics and dynamics of polymeric fractals. Phys. Rev. Lett. 53, 926–929.Cattaneo, C., 1958. Sur une forme de l’equation de la chaleur eliminant le paradoxe d’une propagation instantanee. C.R. Acad. Sci. 247,

431–433.Chandrasekharaiah, D.S., 1986. Thermoelasticity with second sound: a review. Appl. Mech. Rev. 39, 355–376.Chandrasekharaiah, D.S., 1998. Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51, 705–729.Chen, P.J., Gurtin, M.E., 1970. On second sound in materials with memory. ZAMP 21, 232–241.Eidelman, S.D., Kochubei, A.N., 2004. Cauchy problem for fractional diffusion equation. J. Diff. Equat. 199, 211–255.El-Sayed, A.M.A., 1995. Fractional order evolution equations. J. Fract. Calculus 7, 89–100.El-Sayed, A.M.A., 1996. Fractional-order diffusion-wave equation. Int. J. Theor. Phys. 35, 311–322.Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F., 1955. Higher Transcendental Functions, vol. 3. McGraw-Hill, New York.Even, U., Rademann, K., Jortner, J., Manor, N., Reisfeld, R., 1984. Electronic energy transfer on fractal. Phys. Rev. Lett. 52, 2164–2167.Fujita, Y., 1990. Integrodifferential equation which interpolates the heat equation and the wave equation. Osaka J. Math. 27, 309–321.Giona, M., Roman, H.E., 1992. Fractional diffusion equation for transport phenomena in random media. Physica A 211, 13–24.Gorenflo, R., Mainardi, F., 1996. Fractional oscillations and Mittag–Leffler functions. Preprint PR-A-96-14. Fachbereich Mathematik

und Informatik, Freie Universitat Berlin, pp. 1–22.Gorenflo, R., Mainardi, F., 1997. Fractional calculus: integral and differential equations of fractional order. In: Carpinteri, A., Mainardi, F.

(Eds.), Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York, pp. 223–276.Gorenflo, R., Mainardi, F., 1998. Fractional calculus and stable probability distributions. Arch. Mech. 50, 377–388.Gradshtein, I.S., Ryzhik, I.M., 1980. Tables of Integrals, Series and Products. Academic Press, New York.Green, A.E., Naghdi, P.M., 1993. Thermoelasticity without energy dissipation. J. Elast. 31, 189–208.


Gurtin, M.E., Pipkin, A.C., 1968. A general theory of heat conduction with finite wave speeds. Arch. Rational Mech. Anal. 31, 113–126.Hanyga, A., 2002a. Multidimensional solutions of space–time–fractional diffusion equations. Proc. R. Soc. Lond. A 458, 429–450.Hanyga, A., 2002b. Multidimensional solutions of time-fractional diffusion-wave equations. Proc. R. Soc. Lond. A 458, 933–957.Hetnarski, R.B., Ignaczak, J., 1999. Generalized thermoelasticity. J. Therm. Stress. 22, 451–476.Hilfer, R., 2002. Experimental evidence for fractional time evolution in glass forming materials. Chem. Phys. 284, 399–408.Huc, M., Main, I.G., 2003. Anomalous stress diffusion in earthquake triggering: correlation length, time dependence, and directionality.

J. Geophys. Res. 108 (B7), 2324.Humbert, P., 1953. Quelques resultats relatifs a la fonction de Mittag-Leffler. C.R. Acad. Sci. Paris 236, 1467–1468.Humbert, P., Agarwal, R.P., 1953. Sur la fonction de Mittag–Leffler et quelques-unes de ses generalisations. Bull. Sci. Math. 77, 180–185.Joseph, D.D., Preziosi, L., 1989a. Heat waves. Rev. Mod. Phys. 61, 41–73.Joseph, D.D., Preziosi, L., 1989b. Addendum to the paper ‘‘Heat waves’’. Rev. Mod. Phys. 62, 375–391.Kaliski, S., 1965. Wave equations of thermoelasticity. Bull. Acad. Polon. Sci. Ser. Sci. Tech. 13, 253–260.Kimmich, R., 2002. Strange kinetics, porous media, and NMR. Chem. Phys. 284, 253–285.Kimmich, R., Klemm, A., Weber, M., Seymour, J.D., 2001. Flow, diffusion, dispersion, and thermal convection in percolation clusters:

NMR experiments and numerical FEM/FVM simulations. Mater. Res. Soc. Symp. Proc. 651, T2.7.1–T2.7.12.Koch, D.L., Brady, J.F., 1988. Anomalous diffusion in heterogeneous porous media. Phys. Fluids 31, 965–973.Kochubei, A.N., 1990. Fractional order diffusion. Diff. Equat. 26, 485–492.Lord, H.W., Shulman, Y., 1967. A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309.Lubashevskii, I.A., Zemlyanov, A.A., 1998. Continuum description of anomalous diffusion on a comb structure. J. Exp. Theor. Phys. 87,

700–713.Luedtke, W.D., Landman, U., 1999. Slip diffusion and Levy flights of an adsorbed gold nanoclusters. Phys. Rev. Lett. 82, 3835–3838.Metzler, R., Compte, A., 1999. Stochastic foundation of normal and anomalous Cattaneo-type transport. Physica A 268, 454–468.Metzler, R., Klafter, J., 2000a. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77.Metzler, R., Klafter, J., 2000b. Accelerated Brownian motion: a fractional dynamics approach to fast diffusion. Europhys. Lett. 51,

492–498.Metzler, R., Nonnenmacher, T.F., 2003. Fractional relaxation process and fractional rheological models for the description of a class of

viscoelastic materials. Int. J. Plast. 19, 941–959.Metzler, R., Klafter, J., Sokolov, I.M., 1998. Anomalous transport in external fields: continuous time random walks and fractional

diffusion equation extended. Phys. Rev. E 58, 1621–1633.Miller, K.S., Ross, B., 1993. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York.Montroll, E.W., Shlesinger, M.F., 1984. On the wonderful world of random walks. In: Leibowitz, J., Montroll, E.W. (Eds.),

Nonequilibrium Phenomena II: from Stochastic to Hydrodynamics. Amsterdam, North-Holland, pp. 1–121.Moodi, T.B., Tait, R.J., 1983. On thermal transients with finite wave speeds. Acta Mech. 50, 97–104.Moustafa, O.L., 2003. On the Cauchy problem for some fractional order partial differential equations. Chaos, Solitons Fract. 18, 135–140.Nigmatullin, R.R., 1984a. On the theoretical explanation of the universal response. Phys. Status Solidi (b) 123, 739–745.Nigmatullin, R.R., 1984b. On the theory of relaxation with remnant temperature. Phys. Status Solidi (b) 124, 389–393.Nigmatullin, R.R., 1986. The realization of the generalized transfer in a medium with fractal geometry. Phys. Status Solidi (b) 133,

425–430.Norwood, F.R., 1972. Transient thermal waves in the general theory of heat conduction with finite wave speeds. J. Appl. Mech. 39,

673–676.Nowacki, W., 1974. Dynamical problems of thermodiffusion in solids. Bull. Acad. Polon. Sci., Ser. Sci. Tech. 23, 55–64, 129–135, 257–266.Nowacki, W., 1986. Thermoelasticity. Polish Scientific Publishers, Warszawa.Nowacki, W., Olesiak, Z.S., 1991. Thermodiffusion in Solids. Polish Scientific Publishers (PWN), Warszawa (in Polish).Parkus, H., 1959. Instationare Warmespannungen. Springer-Verlag, Wien.Paul, W., 2002. Anomalous diffusion in polymer melts. Chem. Phys. 284, 59–66.Periasamy, N., Verkman, A.S., 1998. Analysis of fluorophore diffusion by continuous distributions of diffusion coefficients: application to

photobleaching measurements of multicomponent and anomalous diffusion. Biophys. J. 75, 557–567.Podlubny, I., 1994. The Laplace transform method for linear differential equations of the fractional order. Preprint UEF-02-94. Inst. Exp.

Phys., Slovak Acad. Sci., Kosice.Podstrigach, Ya.S., 1961. Differential equations of thermodiffusion problem in isotropic deformable solid. Dop. Ukrain. Acad. Sci. (2),

169–172 (in Ukrainian).Podstrigach, Ya.S., 1964. Diffusional theory of deformation of isotropic continuum. Issues Mech. Real Solid 2, 71–99 (in Russian).Podstrigach, Ya.S., 1965. Diffusional theory of inelasticity of metals. J. Appl. Mech. Tech. Phys. (2), 67–72 (in Russian).Podstrigach, Ya.S., Kolyano, Yu.M., 1976. Generalized Thermomechanics. Naukova Dumka, Kiev (in Russian).Podstrigach, Ya.S., Povstenko, Y.Z., 1985. Introduction to Mechanics of Surface Phenomena in Deformable Elastic Solids. Naukova

Dumka, Kiev (in Russian).Povstenko, Y.Z., 2005a. Fractional heat conduction equation and associated thermal stress. J. Therm. Stress. 28, 83–102.Povstenko, Y.Z., 2005b. Stresses exerted by a source of diffusion in a case of a non-parabolic diffusion equation. Int. J. Eng. Sci. 43,

977–991.Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I., 1983. Integrals and Series: Special Functions. Nauka, Mocsow (in Russian).Samko, S.G., Kilbas, A.A., Marichev, O.I., 1993. Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach,

Amsterdam.


Scher, H., Montroll, E.W., 1975. Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12, 2455–2477.Schneider, W.R., 1990. Fractional diffusion. In: Lima, R., Streit, L., Viela Mendes, R. (Eds.), Dynamics and Stochastic Processes, Theory

and Applications, Lecture Notes in Physics, 355. Springer, Berlin, pp. 276–286.Schneider, W.R., Wyss, W., 1989. Fractional diffusion and wave equations. J. Math. Phys. 30, 134–144.Shlesinger, M.F., Zaslavsky, G.M., Klafter, J., 1993. Strange kinetics. Nature (London) 363, 31–37.Umarov, S.R., Luchko, Yu.F., Gorenflo, R., 2000. Partial pseudo-differential equations of fractional order: well-posedness of the Cauchy

and multi-point value problems. Preprint PR-A-00-05. Fachbereich Mathematik und Informatik, Freie Universitat Berlin, pp. 1–36.Upadhyaya, A., Rieu, L.-P., Glazier, J.A., Sawada, Y., 2001. Anomalous diffusion and non-Gaussian velocity distribution of Hydra cells

in cellular aggregates. Physica A 293, 549–558.Vernotte, P., 1958. Les paradoxes de la theorie continue de l’equation de la chaleur. C.R. Acad. Sci. 246, 3154–3155.Weeks, E.R., Weitz, D.A., 2002. Subdiffusion and the cage effect studied near the colloidal glass transition. Chem. Phys. 284, 361–377.Zaslavsky, G.M., 1992. Anomalous transport and fractal kinetics. In: Moffat, H.K., Zaslavsky, G.M., Comte, P., Tabor, M. (Eds.),

Topological Aspects of the Dynamics of Fluids and Plasmas. Kluwer Academic Publishers, Dordrecht, pp. 481–491.Zaslavsky, G.M., 2002. Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371, 461–580.Zumofen, G., Klafter, J., 1993. Scale-invariant motion in intermittent chaotic systems. Phys. Rev. E 47, 851–863.

Two-dimensional axisymmetric stresses exerted by ... · PDF fileThe numerical results for the...

Documents

Transcript of Two-dimensional axisymmetric stresses exerted by ... · PDF fileThe numerical results for the...