RESEARCH PAPER

SOLUTIONS TO THE FRACTIONAL DIFFUSION-WAVE

EQUATION IN A WEDGE

Yuriy Povstenko 1,2

Abstract

The diffusion-wave equation with the Caputo derivative of the order0 < α ≤ 2 is considered in polar coordinates in a domain 0 ≤ r < ∞,0 < ϕ < ϕ0 under Dirichlet and Neumann boundary conditions. TheLaplace integral transform with respect to time, the finite sin- and cos-Fourier transforms with respect to the angular coordinate, and the Hankeltransform with respect to the radial coordinate are used. The numericalresults are illustrated graphically.

MSC 2010 : Primary 26A33; Secondary 35K05, 35L05, 45K05, 44AxxKey Words and Phrases: fractional calculus, Caputo derivative, diffu-

sion-wave equation, Mittag-Leffler functions, Dirichlet boundary condition,Neumann boundary condition

1. Introduction

The time-fractional diffusion-wave equation

∂αT

∂tα= aΔ T, 0 < α ≤ 2, (1.1)

describes many important physical phenomena in different media (see [2],[4], [12], [13], [14], [32] and the references therein).

c© 2014 Diogenes Co., Sofiapp. 122–135 , DOI: 10.2478/s13540-014-0158-4

SOLUTIONS TO THE FRACTIONAL DIFFUSION-WAVE . . . 123

We consider Eq. (1.1) with the Caputo fractional derivative

dαf(t)dtα

=1

Γ(m − α)

∫ t

0(t − τ)m−α−1 dmf(τ)

dτmdτ, m − 1 < α < m, (1.2)

with Γ(α) being the gamma function.Starting from the pioneering papers [7], [10], [11], [30], [33], consider-

able interest has been shown in solutions to time-fractional diffusion-waveequation. Several problems in polar or cylindrical coordinates were studiedin [8], [15]–[18], [20]–[25], [28]. In this paper, the time-fractional diffusion-wave equation with the Caputo derivative of the order 0 < α ≤ 2 is con-sidered in a wedge domain 0 ≤ r < ∞, 0 < ϕ < ϕ0 under Dirichlet andNeumann boundary conditions.

2. Mathematical preliminaries

The integral transform techniques (see, e.g., [3], [5], [31]) allow us toremove the partial derivatives from the considered differential equationsand to obtain algebraic equations in a transform domain.

The Laplace transform is defined as

L{f(t)} = f∗(s) =∫ ∞

0f(t) e−st dt, (2.1)

where s is the transform variable. The inverse Laplace transform is carriedout according to the Fourier–Mellin formula

L−1 {f∗(s)} = f(t) =1

2πi

∫ c+i∞

c−i∞f∗(s) est ds, t > 0, (2.2)

where c is a positive fixed number.The finite sin-Fourier transform is a convenient reformulation of the

sin-Fourier series in the domain 0 ≤ x ≤ L:

F{f(x)} = f(ηn) =∫ L

0f(x) sin(xηn) dx, (2.3)

F−1{f(ηn)} = f(x) =2L

∞∑n=1

f(ηn) sin(xηn), (2.4)

whereηn =

nπ

L. (2.5)

The finite sin-Fourier transform is used in the case of Dirichlet boundarycondition, as for the second derivative of a function we have

F{

d2f(x)dx2

}= −η2

nf(ηn) + ηn

[f(0) − (−1)nf(L)

]. (2.6)

124 Y. Povstenko

The finite cos-Fourier transform is the convenient reformulation of thecos-Fourier series in the domain 0 ≤ x ≤ L:

F{f(x)} = f(ηn) =∫ L

0f(x) cos(xηn) dx, (2.7)

F−1{f(ηn)} = f(x) =2L

∞∑n=0

′f(ηn) cos(xηn), (2.8)

where ηn is defined by (2.5). The prime near the summation symbol in(2.8) denotes that the term with n = 0 should be multiplied by 1/2.

The finite cos-Fourier transform is used in the case of Neumann bound-ary conditions, as

F{

d2f(x)dx2

}= −η2

nf(ηn) −(

df

dx

)∣∣∣∣∣x=0

+ (−1)n(

df

dx

)∣∣∣∣∣x=L

. (2.9)

The Hankel transform is defined as

H{f(r)} = f(ξ) =∫ ∞

0f(r)Jν(rξ) r dr, (2.10)

H−1{f(ξ)} = f(r) =∫ ∞

0f(ξ)Jν(rξ) ξ dξ, (2.11)

where Jν(r) is the Bessel function of the order ν.The basic equation for this integral transform reads:

H{

d2f(r)dr2

+1r

df(r)dr

− ν2

r2f(r)

}= −ξ2f(ξ). (2.12)

3. The Dirichlet boundary condition. Statement of the problem

Consider the time-fractional diffusion-wave equation in polar coordi-nates in a domain 0 ≤ r < ∞, 0 < ϕ < ϕ0

∂αT

∂tα= a

(∂2T

∂r2+

1r

∂T

∂r+

1r2

∂2T

∂ϕ2

)+ Φ(r, ϕ, t) (3.1)

under initial conditions

t = 0 : T = f(r, ϕ), 0 < α ≤ 2, (3.2)

t = 0 :∂T

∂t= F (r, ϕ), 1 < α ≤ 2, (3.3)

and Dirichlet boundary conditions

ϕ = 0 : T = g1(r, t), (3.4)

ϕ = ϕ0 : T = g2(r, t). (3.5)

SOLUTIONS TO THE FRACTIONAL DIFFUSION-WAVE . . . 125

The condition at infinity is also assumed

limr→∞T (r, ϕ, t) = 0. (3.6)

The solution reads:

T (r, t, ϕ) =∫ ϕ0

0

∫ ∞

0f(ρ, φ)Gf (r, ϕ, ρ, φ, t) ρdρdφ

+∫ ϕ0

0

∫ ∞

0F (ρ, φ)GF (r, ϕ, ρ, φ, t) ρdρdφ

+∫ t

0

∫ ϕ0

0

∫ ∞

0Φ(ρ, φ, τ)GΦ(r, ϕ, ρ, φ, t − τ) ρdρdφdτ

+∫ t

0

∫ ∞

0g1(ρ, τ)Gg1(r, ϕ, ρ, t − τ) ρdρdτ

+∫ t

0

∫ ∞

0g2(ρ, τ)Gg2(r, ϕ, ρ, t − τ) ρdρdτ,

(3.7)

whereGf (r, ϕ, ρ, φ, t) is the fundamental solution to the first Cauchy problem,GF (r, ϕ, ρ, φ, t) is the fundamental solution to the second Cauchy problem,GΦ(r, ϕ, ρ, φ, t) is the fundamental solution to the source problem,Gg1(r, ϕ, ρ, t) is the fundamental solution to the first Dirichlet problem,Gg2(r, ϕ, ρ, t) is the fundamental solution to the second Dirichlet problem.

3.1. The fundamental solution to the first Cauchy problemunder zero Dirichlet boundary condition

In this case we have

f(r, ϕ) =δ(r − ρ)

rδ(ϕ − φ), F (r, ϕ) = 0, Φ(r, ϕ, t) = 0,

g1(r, t) = 0, g2(r, t) = 0,where δ(x) is the Dirac delta function. It should be noted that the two-dimensional Dirac delta function in Cartesian coordinates after passing topolar coordinates takes the form 1

2πrδ(r), but for the sake of simplicity wehave omitted the factor 1

2π in the delta term as well as the factor 2π in thesolution (3.7).

The Laplace transform with respect to time t applied to (3.1) gives

sαG∗f − sα−1 δ(r − ρ)

rδ(ϕ − φ) = a

(∂2G∗

f

∂r2+

1r

∂G∗f

∂r+

1r2

∂2G∗f

∂ϕ2

),

126 Y. Povstenko

ϕ = 0 : G∗f = 0,

ϕ = ϕ0 : G∗f = 0.

Next we use the finite sin-Fourier transform (2.3) with respect to theangular coordinate ϕ, thus obtaining

sαG∗f − sα−1 δ(r − ρ)

rsin(

nπφ

ϕ0

)= a

[∂2G∗

f

∂r2+

1r

∂G∗f

∂r− (nπ/ϕ0)2

r2G∗

f

].

The Hankel transform (2.10) with respect to the radial variable r withν = nπ/ϕ0 leads to the solution in the transform domainGf

∗ = Jnπ/ϕ0(ρξ) sin

(nπφ

ϕ0

)sα−1

sα + aξ2. (3.8)

The inverse integral transforms result in

Gf (r, ϕ, ρ, φ, t) =2ϕ0

∞∑n=1

sin(

nπϕ

ϕ0

)sin(

nπφ

ϕ0

)∫ ∞

0Eα

(−aξ2tα

)× Jnπ/ϕ0

(rξ)Jnπ/ϕ0(ρξ) ξ dξ,

(3.9)

where Eα(z) is the Mittag-Leffler function in one parameter α (e.g. [6], [9],[19]):

Eα(z) =∞∑

n=0

zn

Γ(αn + 1), α > 0, z ∈ C.

3.2. The fundamental solution to the second Cauchy problemunder zero Dirichlet boundary condition

This solution is obtained for

f(r, ϕ) = 0, F (r, ϕ) =δ(r − ρ)

rδ(ϕ − φ), Φ(r, ϕ, t) = 0,

g1(r, t) = 0, g2(r, t) = 0,and has the form

GF (r, ϕ, ρ, φ, t) =2tϕ0

∞∑n=1

sin(

nπϕ

ϕ0

)sin(

nπφ

ϕ0

)

×∫ ∞

0Eα,2

(−aξ2tα

)Jnπ/ϕ0

(rξ)Jnπ/ϕ0(ρξ) ξ dξ,

(3.10)

where Eα,β(z) is the generalized Mittag-Leffler function in two parametersα and β described by the following series representation ([6], [9], [19]):

Eα,β(z) =∞∑

n=0

zn

Γ(αn + β), α > 0, β > 0, z ∈ C.

SOLUTIONS TO THE FRACTIONAL DIFFUSION-WAVE . . . 127

3.3. The fundamental solution to the source problem under zeroDirichlet boundary condition

In this case

f(r, ϕ) = 0, F (r, ϕ) = 0, Φ(r, ϕ, t) =δ(r − ρ)

rδ(ϕ − φ) δ(t),

g1(r, t) = 0, g2(r, t) = 0,

and

GΦ(r, ϕ, ρ, φ, t) =2tα−1

ϕ0

∞∑n=1

sin(

nπϕ

ϕ0

)sin(

nπφ

ϕ0

)

×∫ ∞

0Eα,α

(−aξ2tα

)Jnπ/ϕ0

(rξ)Jnπ/ϕ0(ρξ) ξ dξ.

(3.11)

3.4. The fundamental solution to the first Dirichlet problemunder zero initial conditions

This solution corresponds to the choice

f(r, ϕ) = 0, F (r, ϕ) = 0, Φ(r, ϕ, t) = 0,

g1(r, t) = g0δ(r − ρ)

rδ(t), g2(r, t) = 0,

and is expressed as

Gg1(r, ϕ, ρ, φ, t) =2atα−1

ϕ0ρ2

∞∑n=1

(nπ

ϕ0

)sin(

nπϕ

ϕ0

)

×∫ ∞

0Eα,α

(−aξ2tα

)Jnπ/ϕ0

(rξ)Jnπ/ϕ0(ρξ) ξ dξ.

(3.12)

The fundamental solution to the second Dirichlet problem under zero initialconditions is obtained from (3.12) by multiplying each term in the seriesby (−1)n+1 (see Eq. (2.6)).

4. The Neumann boundary condition

The time-fractional diffusion-wave equation (3.1) is solved under initialconditions (3.2) and (3.3) and the Neumann boundary conditions

ϕ = 0 : −1r

∂T

∂ϕ= g1(r, t), (4.1)

ϕ = ϕ0 :1r

∂T

∂ϕ= g2(r, t). (4.2)

128 Y. Povstenko

The solution is expressed by a formula similar to (3.7) with Gg1 and Gg2

being the fundamental solutions to the first and second Neumann problems,respectively.

4.1. The fundamental solution to the first Cauchy problemunder zero Neumann boundary condition

To solve the problems under Neumann boundary condition the Laplacetransform (2.1) with respect to time t, the finite cos-Fourier transform (2.7)we respect to the angular coordinate ϕ, and the Hankel transform (2.10) ofthe order ν = nπ/ϕ0 with respect to the radial coordinate r are used. Thesolution has the following form

Gf (r, ϕ, ρ, φ, t) =2ϕ0

∞∑n=0

′cos(

nπϕ

ϕ0

)cos(

nπφ

ϕ0

)

×∫ ∞

0Eα

(−aξ2tα

)Jnπ/ϕ0

(rξ)Jnπ/ϕ0(ρξ) ξ dξ.

(4.3)

Recall that the prime near the summation symbol means that the termwith n = 0 is multiplied by 1/2.

4.2. The fundamental solution to the second Cauchy problemunder zero Neumann boundary condition

GF (r, ϕ, ρ, φ, t) =2tϕ0

∞∑n=0

′cos(

nπϕ

ϕ0

)cos(

nπφ

ϕ0

)

×∫ ∞

0Eα,2

(−aξ2tα

)Jnπ/ϕ0

(rξ)Jnπ/ϕ0(ρξ) ξ dξ.

(4.4)

4.3. The fundamental solution to the source problem under zeroNeumann boundary condition

GΦ(r, ϕ, ρ, φ, t) =2tα−1

ϕ0

∞∑n=0

′cos(

nπϕ

ϕ0

)cos(

nπφ

ϕ0

)

×∫ ∞

0Eα,α

(−aξ2tα

)Jnπ/ϕ0

(rξ)Jnπ/ϕ0(ρξ) ξ dξ.

(4.5)

SOLUTIONS TO THE FRACTIONAL DIFFUSION-WAVE . . . 129

4.4. The fundamental solution to the first mathematicalNeumann problem under zero initial conditions

In this case

f(r, ϕ) = 0, F (r, ϕ) = 0, Φ(r, ϕ, t) = 0, g2(r, t) = 0,

and the boundary condition at ϕ = 0 is formulated as

ϕ = 0 : −1r

∂Gg1

∂ϕ= g0

δ(r − ρ)r

δ(t). (4.6)

In Eq. (4.6), we have introduced the constant multiplier g0 to obtain thenondimensional quantities used in numerical calculations.

The solution has the form:

Gg1(r, ϕ, ρ, t) =2ag0t

α−1

ϕ0ρ

∞∑n=0

′cos(

nπϕ

ϕ0

)∫ ∞

0Eα,α

(−aξ2tα

)× Jnπ/ϕ0

(rξ)Jnπ/ϕ0(ρξ) ξ dξ.

(4.7)

For classical diffusion equation (α = 1), using Eq. (A.1) from Appendix,we obtain [1]

Gg1(r, ϕ, ρ, t) =g0

ρϕ0texp

(−r2 + ρ2

4at

) ∞∑n=0

′Inπ/ϕ0

( rρ

2at

)cos(

nπϕ

ϕ0

). (4.8)

In the particular case of the wave equation (α = 2), taking into accountEq. (A.2) from Appendix, we get

Gg1(r, ϕ, ρ, t) =2√

ag0

ϕ0ρ

∞∑n=0

′cos(

nπϕ

ϕ0

)Ψ(r, ρ, t), (4.9)

where Ψ(r, ρ, t) is expressed in terms of the Legendre functions:

a)√

at < ρ

Ψ(r, ρ) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

0, 0 ≤ r < ρ −√at,

12√

rρPnπ/ϕ0−1/2

(r2 + ρ2 − at2

2rρ

), ρ −

√at < r < ρ +

√at,

0, ρ +√

at < r < ∞;

b)√

at = ρ

Ψ(r, ρ) =

⎧⎪⎨⎪⎩1

2√

rρPnπ/ϕ0−1/2

(r

2ρ

), 0 < r < 2ρ,

0, 2ρ < r < ∞;

130 Y. Povstenko

c)√

at > ρ

Ψ(r, ρ) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

− 1π√

rρcos(

nπ2

ϕ0

)Qnπ/ϕ0−1/2

(at2 − r2 − ρ2

2rρ

),

0 ≤ r <√

at − ρ,

12√

rρPnπ/ϕ0−1/2

(r2 + ρ2 − at2

2rρ

),

√at − ρ < r < ρ +

√at,

0, ρ +√

at < r < ∞.

The dependence of nondimensional fundamental solution (4.7) Gg1 =ρ3

atα−1g0Gg1 on nondimensional distance r/ρ is presented in Fig. 4.1 for ϕ = 0

and different values of the order of fractional derivative α. In calculationswe have taken ϕ0 = π/4 and κ =

√atα/2/ρ = 0.35.

4.5. The fundamental solution to the first physical Neumannproblem under zero initial conditions

For the physical Neumann problem, the boundary condition is formu-lated in terms of the normal component of the heat flux (see [23], [26]):

ϕ = 0 : −1r

D1−αRL

∂Gg1

∂ϕ= g0

δ(r − ρ)r

δ(t), 0 < α ≤ 1, (4.10)

ϕ = 0 : −1r

Iα−1 ∂Gg1

∂ϕ= g0

δ(r − ρ)r

δ(t), 1 < α ≤ 2, (4.11)

where D1−αRL f(t) and Iα−1f(t) are the Riemann-Liouville fractional deriva-

tive and fractional integral, respectively (see e.g. [6], [9], [19], [29]).

The solution is expressed as

Gg1(r, ϕ, ρ) =2ag0

ϕ0ρ

∞∑n=0

′cos(

nπϕ

ϕ0

)∫ ∞

0Eα

(−aξ2tα

)× Jnπ/ϕ0

(rξ)Jnπ/ϕ0(ρξ) ξ dξ.

(4.12)

SOLUTIONS TO THE FRACTIONAL DIFFUSION-WAVE . . . 131

α = 1.5�

����

α = 1�

��

����

α = 0.5�

��

����

Gg1

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5r

Fig. 4.1: The fundamental solution to the mathematical Neumannproblem under zero initial conditions.

α = 1.15�

��

���α = 1

��

��� α = 0.85

��

���

Gg1

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5r

Fig. 4.2: The fundamental solution to the physical Neumann problemunder zero initial conditions.

132 Y. Povstenko

In the limiting case α → 0 corresponding to the so-called localizeddiffusion the solution is obtained using Eq. (A.3) from Appendix and reads:

Gg1(r, ϕ, ρ, t) =2g0

ϕ0ρ

∞∑n=0

′cos(

nπϕ

ϕ0

)

×

⎧⎪⎪⎪⎨⎪⎪⎪⎩Inπ/ϕ0

(r√a

)Knπ/ϕ0

(ρ√a

), 0 ≤ r < ρ,

Inπ/ϕ0

(ρ√a

)Knπ/ϕ0

(r√a

), ρ ≤ r < ∞.

Figure 4.2 presents the dependence of the nondimensional fundamentalsolution (4.12) Gg1 = ρ3

ag0Gg1 on distance for ϕ = 0 and different values

of the order of fractional derivative α. In the calculations we have takenϕ0 = π/4 and κ = 0.35.

The fundamental solutions to the second mathematical and physicalNeumann problems, when the nonzero bounadry conditions are given at theboundary ϕ = ϕ0, are obtained from Eqs. (4.7) and (4.12) by multiplyingeach term in the series by (−1)n (see Eq. (2.9) with taking into accountthat in the left-hand sides of equations corresponding to (4.6), (4.10) and(4.11) there appears the sign “plus”).

5. Concluding remarks

The time fractional diffusion-wave equation covers the whole spectrumfrom the localized diffusion (the Helmholtz equation when the order ofthe time-derivative α → 0) through the standard diffusion equation (rep-resented by the particular case α = 1) to the ballistic diffusion (the waveequation when α = 2). We have derived the analytical solutions to the time-fractional diffusion-wave equation in a wedge under Dirichlet, mathemat-ical and physical Neumann boundary conditions. The integral transformtechnique has been used. It should be emphasized that the obtained fun-damental solutions are expressed in terms of the Mittag-Leffler functions,in particular in the fundamental solutions to the mathematical and physi-cal Neumann problems there appear the Mittag-Leffler functions Eα,α (−x)and Eα (−x), respectively. The difference between the mathematical andphysical Neumann boundary conditions (as well as the difference betweenthe solutions) disappear in the case of standard diffusion equation corre-sponding to α = 1. In this case the solutions (4.7) and (4.12) coincide andare equal to the solution (4.8).

SOLUTIONS TO THE FRACTIONAL DIFFUSION-WAVE . . . 133

Appendix

To obtain the particular cases of our results, the following integrals [27]are used:∫ ∞

0e−ax2

Jν(bx)Jν(cx)xdx =12a

exp(−b2 + c2

4a

)Iν

(bc

2a

); (A1)∫ ∞

0sin axJν(bx)Jν(cx) dx

=

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

12√

bcPν−1/2

(b2 + c2 − a2

2bc

), |b − c| < a < b + c,

−cos νπ

π√

bcQν−1/2

(a2 − b2 − c2

2bc

), b + c < a,

0, a < |b − c|,

(A2)

where Pν(x) and Qν(x) are the Legendre functions of the first and secondkinds, respectively;∫ ∞

0

x

x2 + a2Jν(bx)Jν(cx) dx =

{Iν(ac)Kν(ab), 0 < c < b,

Iν(ab)Kν(ac), 0 < b < c,(A3)

where Iν(x) and Kν(x) are the modified Bessel functions. In the aboveequations, we assume a > 0, b > 0, c > 0, and ν ≥ 0.

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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: July 17, 2013

Please cite to this paper as published in:Fract. Calc. Appl. Anal., Vol. 17, No 1 (2014), pp. 122–135;DOI: 10.2478/s13540-014-0158-4