RESEARCH PAPER

TIME-FRACTIONAL HEAT CONDUCTION

IN AN INFINITE MEDIUM WITH A SPHERICAL HOLE

UNDER ROBIN BOUNDARY CONDITION

Yuriy Povstenko 1,2

Dedicated to the 70th anniversaryof Professor Francesco Mainardi

Abstract

The time-fractional heat conduction equation with the Caputo deriv-ative of the order 0 < α ≤ 2 is considered in an infinite medium witha spherical hole in the central symmetric case under two types of Robinboundary condition: the mathematical one with the prescribed linear com-bination of the values of temperature and the values of its normal derivativeat the boundary and the physical condition with the prescribed linear com-bination of the values of temperature and the values of the heat flux at theboundary. The integral transforms techniques are used. Several particularcases of the obtained solutions are analyzed. The numerical results areillustrated graphically.

MSC 2010 : Primary 26A33; Secondary 35K05, 35L05, 45K05Key Words and Phrases: fractional calculus, diffusion-wave equation,

Mittag-Leffler functions, Robin boundary condition

c© 2013 Diogenes Co., Sofiapp. 354–369 , DOI: 10.2478/s13540-013-0022-y

TIME-FRACTIONAL HEAT CONDUCTION . . . 355

1. Introduction

The classical theory of heat conduction is based on the Fourier lawwhich relates the heat flux vector q to the temperature gradient grad T . Incombination with the law of conservation of energy, the Fourier law leadsto the standard parabolic heat conduction equation. The time-fractionalheat conduction equation (the time-fractional diffusion-wave equation) withCaputo derivative

∂αT

∂tα= aΔT, 0 < α ≤ 2, (1.1)

is obtained as a consequence of the law of conservation of energy and thetime-nonlocal generalization of the Fourier law with the “long-tail memory”power kernel [15], [18] (see also [6]):

q(t) = −kD1−αRL grad T (t), 0 < α ≤ 1,

q(t) = −kIα−1grad T (t), 1 < α ≤ 2,(1.2)

where Iα−1 and D1−αRL are the Riemann-Liouville fractional integral and

derivative, respectively.Recall that the Riemann–Liouville fractional integral Iαf(t) and deriv-

ative DαRLf(t) are defined as (see [9], [14], [22]):

Iαf(t) =1

Γ(α)

∫ t

0(t − τ)α−1f(τ) dτ, α > 0, (1.3)

DαRLf(t) =

dm

dtm

[1

Γ(m − α)

∫ t

0(t − τ)m−α−1f(τ) dτ

], m − 1 < α < m,

(1.4)whereas the Caputo fractional derivative has the following form (see [5],[9], [14]):

dαf(t)dtα

=1

Γ(m − α)

∫ t

0(t − τ)m−α−1 dmf(τ)

dτmdτ, m − 1 < α < m. (1.5)

Here Γ(α) if the gamma function.A detailed explanation of the derivation of time-fractional heat con-

duction equation (1.1) from the constitutive equations (1.2) and the law ofconservation of energy can be found in [19].

If heat conduction is studied in a domain with a boundary, the cor-responding boundary conditions should be imposed. The main types ofboundary conditions are the following.

The Dirichlet boundary condition specifies the value of temperatureover the surface of the body under consideration

T |S = F (xS , t). (1.6)

356 Y. Povstenko

For time-fractional heat conduction equations, two types of Neumann bound-ary condition can be considered: the mathematical condition with the pre-scribed boundary value of the normal derivative of temperature

∂T

∂n

∣∣∣∣∣S

= F (xS , t), (1.7)

and the physical condition with the prescribed boundary value of the heatflux

D1−αRL

∂T

∂n

∣∣∣∣∣S

= F (xS , t), 0 < α ≤ 1,

Iα−1 ∂T

∂n

∣∣∣∣∣S

= F (xS , t), 1 < α ≤ 2.

(1.8)

In the case of the classical heat conduction equation corresponding to α = 1,these two types of boundary conditions are identical, but for fractional heatconduction equation they are essentially different.

Similarly, the mathematical Robin boundary condition is a specificationof a linear combination of the values of temperatute and the values of itsnormal derivative at the boundary of the domain(

c1T + c2∂T

∂n

) ∣∣∣∣∣S

= F (xS , t) (1.9)

with some nonzero constants c1 and c2, while the physical Robin boundarycondition specifies a linear combination of the values of temperature andthe values of the heat flux at the boundary of the domain. For example, thecondition of convective heat exchange between a body and the environmentwith the temperature Te

q · n∣∣∣S

= h(T∣∣∣S− Te

), (1.10)

where h is the convective heat transfer coefficient, leads to(hT + kD1−α

RL

∂T

∂n

) ∣∣∣∣∣S

= hTe(xS , t), 0 < α ≤ 1,

(hT + kIα−1 ∂T

∂n

) ∣∣∣∣∣S

= hTe(xS , t), 1 < α ≤ 2.

(1.11)

Starting from the pioneering papers [3], [10], [11], [12], [13], [23], [24],considerable interest has been shown in solutions to time-fractional diffusion-wave equation. Additional references concerning this subject can be foundin [17]. There are several papers in which the fractional heat conduction

TIME-FRACTIONAL HEAT CONDUCTION . . . 357

equation [1], [8] and the fractional telegraph equation [2], [7] are investi-gated under the mathematical Robin boundary condition. In the previouspublications, problems for a cylinder [20] and a sphere [21] under mathe-matical and physical Neumann boundary conditions were considered. In[16] fractional heat conduction in an infinite solid with a spherical cav-ity was considered under Dirichlet and mathematical Neumann boundaryconditions.

In the present paper, the central symmetric time-fractional heat conduc-tion equation in an infinite medium with a spherical hole is studied underboth the mathematical and physical Robin boundary conditions with par-ticular cases corresponding to mathematical and physical Neumann bound-ary conditions.

2. Solution to the problem under mathematicalRobin boundary condition

Consider the central symmetric time-fractional heat conduction equa-tion in an infinite medium with a spherical hole of radius R:

∂ αT

∂tα= a

(∂2T

∂r2+

2r

∂T

∂r

),

R < r < ∞, 0 < t < ∞, 0 < α ≤ 2,(2.1)

under zero initial conditions

t = 0 : T = 0, 0 < α ≤ 2, (2.2)

t = 0 :∂T

∂t= 0, 1 < α ≤ 2, (2.3)

and the mathematical Robin boundary condition

r = R : −∂T

∂r+ HT = F (t). (2.4)

The zero condition at infinity is also assumed:

limr→∞T (r, t) = 0. (2.5)

The solution to the initial-boundary-value problem (2.1)–(2.5) can bewritten in a convolution form

T (r, t) =∫ t

0F (t − τ)Gm(r, τ) dτ, (2.6)

where Gm(r, t) is the fundamental solution being the solution of the follow-ing problem:

∂ αGm

∂tα= a

(∂2Gm

∂r2+

2r

∂Gm

∂r

),

R < r < ∞, 0 < t < ∞, 0 < α ≤ 2,(2.7)

t = 0 : Gm = 0, 0 < α ≤ 2, (2.8)

358 Y. Povstenko

t = 0 :∂Gm

∂t= 0, 1 < α ≤ 2, (2.9)

r = R : −∂Gm

∂r+ HGm = g0 δ(t), (2.10)

limr→∞Gm(r, t) = 0, (2.11)

where δ(t) is the Dirac delta function. We have introduced the constantmultiplier g0 in the delta term to obtain the nondimensional quantity Gm

displayed in the figures.Usually, for the considered geometry the auxiliary function v = rGm

and the auxiliary spatial variable x = r − R are used, and the problem(2.7)–(2.11) is reformulated as:

∂ αv

∂tα= a

∂2v

∂x2, 0 < x < ∞, 0 < t < ∞, 0 < α ≤ 2, (2.12)

t = 0 : v = 0, 0 < α ≤ 2, (2.13)

t = 0 :∂v

∂t= 0, 1 < α ≤ 2, (2.14)

x = 0 : −∂v

∂x+(

1R

+ H

)v = g0R δ(t), (2.15)

limx→∞ v(x, t) = 0. (2.16)

The Laplace transform with respect to time t and the sin-cos-Fouriertransform with respect to the auxuliary spatial coordinate x will be usedto solve the problem (2.12)–(2.16). In the case under consideration, thesin-cos-Fourier transform has the following form:

F {f(x)} = f(ξ) =∫ ∞

0

ξ cos(xξ) + (1/R + H) sin(xξ)√ξ2 + (1/R + H)2

f(x) dx, (2.17)

F−1{

f(ξ)}

= f(x)

=2π

∫ ∞

0

ξ cos(xξ) + (1/R + H) sin(xξ)√ξ2 + (1/R + H)2

f(ξ) dξ,(2.18)

F{

d2f(x)dx2

}= −ξ2f(ξ)

+ξ√

ξ2 + (1/R + H)2

[−df(x)

dx+(

1R

+ H

)f(x)

]x=0

.(2.19)

TIME-FRACTIONAL HEAT CONDUCTION . . . 359

In the transform domain we obtain

v∗(ξ, s) =aRg0ξ√

ξ2 + (1/R + H)21

sα + aξ2, (2.20)

where the Laplace transform is denoted by the asterisk, the sin-cos-Fouriertransform is denoted by the tilde, s is the Laplace transform variable, ξ isthe sin-cos-Fourier transform variable.

Inverting the integral transforms, we get

Gm(r, t) =2aRg0t

α−1

πr

∫ ∞

0Eα,α

(−aξ2tα

)× R2ξ2 cos[(r − R)ξ] + (1 + RH)Rξ sin[(r − R)ξ]

R2ξ2 + (1 + RH)2dξ.

(2.21)

Here Eα,β(z) is the Mittag-Leffler function in two parameters α and βdefined by the series representation

Eα,β(z) =∞∑

m=0

zm

Γ(αm + β), α > 0, β > 0, z ∈ C, (2.22)

and the following formula [5], [9], [14] has been used:

L−1

{sα−β

sα + b

}= tβ−1Eα,β (−btα) , b ∈ C. (2.23)

Consider several particular cases of the solution (2.21). The case H = 0corresponds to the mathematical Neumann boundary condition with theprescribed boundary value of the normal derivative of temperature, andthe solution reads ([16]):

Gm(r, t) =2aRg0t

α−1

πr

∫ ∞

0Eα,α

(−aξ2tα

)

× R2ξ2 cos[(r − R)ξ] + Rξ sin[(r − R)ξ]R2ξ2 + 1

dξ.

(2.24)

In the case of classical heat conduction (α = 1), we get from Eq. (2.21)

Gm(r, t) =aRg0

r

{1√πat

exp[−(r − R)2

4at

]

− 1 + RH

Rexp

[1 + RH

R(r − R) +

(1 + RH)2

R2at

]

× erfc(

r − R

2√

at+

1 + RH

R

√at

)},

(2.25)

where erfc z is the complementary error function.

360 Y. Povstenko

For the wave equation (α = 2), Eq. (2.21) leads to

Gm(r, t) =

⎧⎪⎨⎪⎩

√aRg0

rexp

[−1 + RH

R(√

at − r + R)]

, R < r < R +√

at,

0 R +√

at < r < ∞.(2.26)

Of particular interest is also the case α = 1/2 for which

E1/2, 1/2(−z) =1√π− zez2

erfc z =2√π

∫ ∞

0e−u2−2zuudu, (2.27)

and

Gm(r, t) =2aRg0√

πtr

∫ ∞

0ue−u2

{1√

2πuat1/4exp

[−(r − R)2

8ua√

t

]

− 1 + RH

Rexp

[1 + RH

R(r − R) + 2

(1 + RH)2

R2ua

√t

]

× erfc(

r − R

2√

2uat1/4+

1 + RH

R

√2uat1/4

)}du.

(2.28)

The dependence of the nondimensional fundamental solution (2.21)Gm = t

Rg0Gm on nondimensional distance r/R is presented in Figures

2.1 and 2.2 for α = 0.5 and α = 1.95, respectively, for various valuesof H = RH. In both cases κ =

√atα/2

R = 1.

3. Solution to the problem under physicalRobin boundary condition

Now we consider the central symmetric time-fractional heat conductionequation in an infinite medium with a spherical hole of radius R:

∂ αT

∂tα= a

(∂2T

∂r2+

2r

∂T

∂r

),

R < r < ∞, 0 < t < ∞, 0 < α ≤ 2,(3.1)

with zero initial conditions

t = 0 : T = 0, 0 < α ≤ 2, (3.2)

t = 0 :∂T

∂t= 0, 1 < α ≤ 2, (3.3)

TIME-FRACTIONAL HEAT CONDUCTION . . . 361

H = 1��

����

H = 0.5�

��

���

H = 0�

��

���

Gm

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1.0 1.5 2.0 2.5 3.0

r/R

Fig. 2.1: Dependence of the fundamental solution under themathematical Robin boundary condition on distance (α = 0.5)

H = 1�

��

���

H = 0.5������

H = 0�����

Gm

0.0

0.1

0.2

0.3

0.4

0.5

1.0 1.5 2.0 2.5 3.0

r/R

Fig. 2.2: Dependence of the fundamental solution under themathematical Robin boundary condition on distance (α = 1.95)

362 Y. Povstenko

under the physical Robin boundary condition

r = R : −D1−αRL

∂T

∂r+ HT = F (t), 0 < α < 1, (3.4)

r = R : −Iα−1 ∂T

∂r+ HT = F (t), 1 ≤ α ≤ 2, (3.5)

and zero condition at infinity

limr→∞T (r, t) = 0. (3.6)

The solution to the initial-boundary-value problem (3.1)–(3.6) can bealso written in a convolution form

T (r, t) =∫ t

0F (t − τ)Gp(r, τ) dτ, (3.7)

where Gp(r, t) is the fundamental solution being the solution of the followingproblem:

∂ αGp

∂tα= a

(∂2Gp

∂r2+

2r

∂Gp

∂r

),

R < r < ∞, 0 < t < ∞, 0 < α ≤ 2,(3.8)

t = 0 : Gp = 0, 0 < α ≤ 2, (3.9)

t = 0 :∂Gp

∂t= 0, 1 < α ≤ 2, (3.10)

r = R : −D1−αRL

∂Gp

∂r+ HGp = g0 δ(t), 0 < α ≤ 1, (3.11)

r = R : −Iα−1 ∂Gp

∂r+ HGp = g0 δ(t), 1 < α ≤ 2, (3.12)

limr→∞Gm(r, t) = 0. (3.13)

In terms of the auxiliary function v = rGp and the auxiliary spatialvariable x = r − R, the problem (3.8)–(3.13) takes the form

∂ αv

∂tα= a

∂2v

∂x2, 0 < x < ∞, 0 < t < ∞, 0 < α ≤ 2, (3.14)

t = 0 : v = 0, 0 < α ≤ 2, (3.15)

t = 0 :∂v

∂t= 0, 1 < α ≤ 2, (3.16)

x = 0 : −D1−αRL

∂v

∂x+(

1R

D1−αRL + H

)v = g0R δ(t), 0 < α ≤ 1, (3.17)

x = 0 : −Iα−1 ∂v

∂x+(

1R

Iα−1 + H

)v = g0R δ(t), 1 < α ≤ 2, (3.18)

limx→∞ v(x, t) = 0. (3.19)

TIME-FRACTIONAL HEAT CONDUCTION . . . 363

Applying the Laplace transform with respect to time t results in thefollowing boundary-value problem

sαv∗ = a∂2v∗

∂x2, (3.20)

x = 0 : −∂v∗

∂x+(

1R

+ Hsα−1

)v∗ = g0Rsα−1, 0 < α ≤ 2. (3.21)

In this case the kernel of the sin-cos-Fourier transform with respect tothe auxuliary coordinate x depends on the Laplace transform variable s:

F {f(x)} = f(ξ)

=∫ ∞

0

ξ cos(xξ) +(1/R + Hsα−1

)sin(xξ)√

ξ2 + (1/R + Hsα−1)2f(x) dx,

(3.22)

F−1{f(ξ)

}= f(x)

=2π

∫ ∞

0

ξ cos(xξ) +(1/R + Hsα−1

)sin(xξ)√

ξ2 + (1/R + Hsα−1)2f(ξ) dξ,

(3.23)

F{

d2f(x)dx2

}= −ξ2f(ξ)

+ξ√

ξ2 + (1/R + Hsα−1)2

[−df(x)

dx+(

1R

+ Hsα−1

)f(x)

]x=0

.

(3.24)In the transform domain we get

v∗(ξ, s) =aRg0ξ√

ξ2 + (1/R + Hsα−1)2sα−1

sα + aξ2. (3.25)

After inversion of the sin-cos-Fourier transform we arrive at

G∗p(r, s) =

2aRg0

πr

∫ ∞

0

sα−1

sα + aξ2

× R2ξ2 cos[(r − R)ξ] + (1 + RHsα−1)Rξ sin[(r − R)ξ]R2ξ2 + (1 + RHsα−1)2

dξ.

(3.26)

The solution to the corresponding problem under physical Neumannboundary condition is obtained when H = 0:

Gp(r, t) =2aRg0

πr

∫ ∞

0Eα

(−aξ2tα

)

× R2ξ2 cos[(r − R)ξ] + Rξ sin[(r − R)ξ]R2ξ2 + 1

dξ.

(3.27)

Inversion of the Laplace transform in Eq. (3.26) depends on the valueof α. For 0 < α < 1 this equation is rewritten as

364 Y. Povstenko

G∗p(r, s) =

2aRg0

πr

∫ ∞

0

1sα + aξ2

× s1−αR2ξ2 cos[(r − R)ξ] + (s1−α + RH)Rξ sin[(r − R)ξ](1 + R2ξ2)(s1−α)2 + 2RHs1−α + R2H2

dξ.

(3.28)

Next, we use the following decompositions into the sum of partial frac-tions:

1(1 + R2ξ2)(s1−α)2 + 2RHs1−α + R2H2

=i

2HR2ξ

⎡⎣ 1

s1−α + RH(1+iRξ)R2ξ2+1

− 1

s1−α + RH(1−iRξ)R2ξ2+1

⎤⎦ ,

(3.29)

s1−α

(1 + R2ξ2)(s1−α)2 + 2RHs1−α + R2H2

=1

2Rξ(R2ξ2 + 1)

⎡⎣ Rξ − i

s1−α + RH(1+iRξ)R2ξ2+1

+Rξ + i

s1−α + RH(1−iRξ)R2ξ2+1

⎤⎦ ,

(3.30)

where i =√−1.

Utilizing the convolution theorem, the solution is written as

Gp(r, t) =aRg0

πr

∫ ∞

0

∫ t

0

Rξ(t − τ)α−1τ−α

R2ξ2 + 1Eα,α

[−aξ2(t − τ)α

]

×{

(Rξ − i) ei(r−R)ξ E1−α,1−α

[−RH(1 + iRξ)τ1−α

R2ξ2 + 1

]

+ (Rξ + i) e−i(r−R)ξ E1−α,1−α

[−RH(1 − iRξ)τ1−α

R2ξ2 + 1

]}dτ dξ.

(3.31)

It should be emphasized that the fundamental solution (3.31) is a real-valued function and can be rewritten as

Gp(r, t) =2aRg0

πr

∫ ∞

0

∫ t

0

Rξ(t − τ)α−1τ−α√R2ξ2 + 1

Eα,α

[−aξ2(t − τ)α

]

×∞∑

m=0

(−1)m

Γ[(m + 1)(1 − α)]

(RHτ1−α√R2ξ2 + 1

)m

× sin [(r − R)ξ + (m + 1) arctan(Rξ)] dτ dξ.

(3.32)

The solution (3.31) simplifies significantly for α = 1/2 with taking intoaccount the representation (2.27):

TIME-FRACTIONAL HEAT CONDUCTION . . . 365

Gp(r, t) =4aRg0

π3/2r

∫ ∞

0

Rξ

R2ξ2 + 1

∫ t

0

1√τ(t − τ)

×∫ ∞

0uE1/2,1/2

(−aξ2

√t − τ

)exp

(−u2 − 2RHu

√τ

R2ξ2 + 1

)

×{

Rξ cos[(

r − R − 2R2Hu√

τ

R2ξ2 + 1

)ξ

]

+ sin[(

r − R − 2R2Hu√

τ

R2ξ2 + 1

)ξ

]}dudτ dξ.

(3.33)

Similarly, returning to (3.26) for 1 < α ≤ 2, we get the required decom-positions into the sum of partial fractions

1R2ξ2 + (1 + RHsα−1)2

=i

2Rξ

(1

RHsα−1 + 1 + iRξ− 1

RHsα−1 + 1 − iRξ

),

(3.34)

1 + RHsα−1

R2ξ2 + (1 + RHsα−1)2

=12

(1

RHsα−1 + 1 + iRξ+

1RHsα−1 + 1 − iRξ

).

(3.35)

The convolution theorem allows us to invert the Laplace transform and toobtain the solution:

Gp(r, t) =aRg0i

πHr

∫ ∞

0

∫ t

0ξτα−2 Eα

[−aξ2(t − τ)α

]

×[e−i(r−R)ξ Eα−1,α−1

(−1 + iRξ

RHτα−1

)

− ei(r−R)ξ Eα−1,α−1

(−1 − iRξ

RHτα−1

)]dτ dξ

(3.36)

366 Y. Povstenko

which can be rewritten in the real-valued form as

Gp(r, t) =2aRg0

πHr

∫ ∞

0

∫ t

0ξτα−2 Eα

[−aξ2(t − τ)α

]

×∞∑

m=0

(−1)m

Γ[(m + 1)(α − 1)]

(τ1−α

√1 + R2ξ2

RH

)m

× sin [(r − R)ξ − m arctan(Rξ)] dτ dξ.

(3.37)

The particular case corresponding to the value α = 3/2 is also obtainedusing the representation (2.27):

Gp(r, t) =4aRg0

π3/2Hr

∫ ∞

0

∫ t

0

∫ ∞

0

uξ√τ

E3/2

[−aξ2(t − τ)3/2

]

× exp(−u2 − 2u

√τ

RH

)sin[(

r − R +2u

√τ

H

)ξ

]dudτ dξ.

(3.38)

H = 0�

��

����

H = 0.5�

��

���

H = 1�

����

Gp

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1.0 1.5 2.0 2.5 3.0

r/R

Fig. 3.1: Dependence of the fundamental solution under the physicalRobin boundary condition on distance (α = 0.5)

TIME-FRACTIONAL HEAT CONDUCTION . . . 367

Figure 3.1 presents the dependence of the nondimensional fundamentalsolution Gp = R

ag0Gp under the physical Robin boundary condition on nondi-

mensional distance r/R for α = 0.5 and different values of H = RHt1−α.In calculations we have taken κ = 1.

4. Concluding remarks

We have derived the analytical solutions to central symmetric time-fractional heat conduction equation in an infinite medium with a sphericalcavity under the mathematical and physical Robin boundary conditions.The Laplace integral transform with respect to time and the sin-cos-Fouriertransform with respect to the auxiliary spatial coordinate have been used.It should be emphasized that in the case of physical Robin boundary con-dition the order of integral transforms is important as the kernel of thesin-cos-Fourier transform depends on the Laplace transform variable. Thelimiting case H = 0 corresponds to the solutions of problems under math-ematical and physical Neumann boundary conditions with the prescribedboundary value of the normal derivative and with the prescribed boundaryvalue of the heat flux, respectively. The difference between mathematicaland physical boundary conditions (as well as the difference between thesolutions) disappears in the case of the standard heat conduction equationwhen α = 1. The solutions were obtained in terms of convolution of Mittag-Leffler functions Eα,β(z); for their evaluation the algorithm suggested in [4]was applied. The solutions simplify for two typical values of the order ofthe time derivative: α = 1/2 and α = 3/2.

References

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1 Institute of Mathematics and Computer ScienceJan D�lugosz University in Czestochowaal. Armii Krajowej 13/1542-200 Czestochowa, POLAND

2 Department of Computer ScienceEuropean University of Informatics and Economics (EWSIE)ul. Bia�lostocka 2203-741 Warsaw, POLAND

e-mail: j.povstenko@ajd.czest.pl Received: October 12, 2012

Please cite to this paper as published in:Fract. Calc. Appl. Anal., Vol. 16, No 2 (2013), pp. 354–369;DOI:10.2478/s13540-013-0022-y