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 order 0 < α ≤ 2 is considered in polar coordinates in a domain 0 ≤ r < ∞, 0 < ϕ < ϕ0 under Dirichlet and Neumann boundary conditions. The Laplace integral transform with respect to time, the finite sin- and cosFourier transforms with respect to the angular coordinate, and the Hankel transform with respect to the radial coordinate are used. The numerical results are illustrated graphically. MSC 2010 : Primary 26A33; Secondary 35K05, 35L05, 45K05, 44Axx Key Words and Phrases: fractional calculus, Caputo derivative, diffusion-wave equation, Mittag-Leffler functions, Dirichlet boundary condition, Neumann boundary condition
1. Introduction The time-fractional diffusion-wave equation ∂αT = aΔ T, ∂tα
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., Sofia pp. 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 t m 1 dα f (t) m−α−1 d f (τ ) = (t − τ ) dτ, m − 1 < α < m, (1.2) dtα Γ(m − α) 0 dτ m with Γ(α) being the gamma function. Starting from the pioneering papers [7], [10], [11], [30], [33], considerable interest has been shown in solutions to time-fractional diffusion-wave equation. Several problems in polar or cylindrical coordinates were studied in [8], [15]–[18], [20]–[25], [28]. In this paper, the time-fractional diffusionwave equation with the Caputo derivative of the order 0 < α ≤ 2 is considered in a wedge domain 0 ≤ r < ∞, 0 < ϕ < ϕ0 under Dirichlet and Neumann boundary conditions. 2. Mathematical preliminaries The integral transform techniques (see, e.g., [3], [5], [31]) allow us to remove the partial derivatives from the considered differential equations and to obtain algebraic equations in a transform domain. The Laplace transform is defined as ∞ ∗ f (t) e−st dt, (2.1) L {f (t)} = f (s) = 0
where s is the transform variable. The inverse Laplace transform is carried out according to the Fourier–Mellin formula c+i∞ 1 f ∗ (s) est ds, t > 0, (2.2) L−1 {f ∗ (s)} = f (t) = 2πi c−i∞ 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: L f (x) sin(xηn ) dx, (2.3) F{f (x)} = f(ηn ) = 0
F −1 {f(ηn )} = f (x) = where
∞
2 f (ηn ) sin(xηn ), L n=1
(2.4)
nπ . (2.5) L The finite sin-Fourier transform is used in the case of Dirichlet boundary condition, as for the second derivative of a function we have 2 d f (x) (2.6) = −ηn2 f(ηn ) + ηn f (0) − (−1)n f (L) . F 2 dx ηn =
124
Y. Povstenko
The finite cos-Fourier transform is the convenient reformulation of the cos-Fourier series in the domain 0 ≤ x ≤ L: L f (x) cos(xηn ) dx, (2.7) F{f (x)} = f (ηn ) = 0
F
−1
∞
2 {f(ηn )} = f (x) = f (ηn ) cos(xηn ), L
(2.8)
n=0
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 boundary conditions, as 2 df df d f (x) + (−1)n . (2.9) = −ηn2 f(ηn ) − F 2 dx dx dx x=0
x=L
The Hankel transform is defined as ∞ f (r) Jν (rξ) r dr, H{f (r)} = f (ξ) =
(2.10)
0
H
−1
{f (ξ)} = f (r) =
∞ 0
f (ξ) Jν (rξ) ξ dξ,
where Jν (r) is the Bessel function of the order ν. The basic equation for this integral transform reads: 2 d f (r) 1 df (r) ν 2 − 2 f (r) = −ξ 2 f (ξ). + H dr 2 r dr r
(2.11)
(2.12)
3. The Dirichlet boundary condition. Statement of the problem Consider the time-fractional diffusion-wave equation in polar coordinates in a domain 0 ≤ r < ∞, 0 < ϕ < ϕ0 2
1 ∂2T ∂ T 1 ∂T ∂αT + 2 =a + + Φ(r, ϕ, t) (3.1) ∂tα ∂r 2 r ∂r r ∂ϕ2 under initial conditions t=0:
T = f (r, ϕ),
∂T = F (r, ϕ), ∂t and Dirichlet boundary conditions t=0:
0 < α ≤ 2,
(3.2)
1 < α ≤ 2,
(3.3)
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 lim T (r, ϕ, t) = 0.
r→∞
The solution reads: ϕ0 T (r, t, ϕ) = 0
0
∞
ϕ0
∞
+ 0
0
t ϕ0
f (ρ, φ) Gf (r, ϕ, ρ, φ, t) ρ dρ dφ F (ρ, φ) GF (r, ϕ, ρ, φ, t) ρ dρ dφ
+ 0
0 t ∞
+ 0
0 t ∞
+ 0
0
(3.6)
0
∞
Φ(ρ, φ, τ ) GΦ (r, ϕ, ρ, φ, t − τ ) ρ dρ dφ dτ
(3.7)
g1 (ρ, τ ) Gg1 (r, ϕ, ρ, t − τ ) ρ dρ dτ g2 (ρ, τ ) Gg2 (r, ϕ, ρ, t − τ ) ρ dρ dτ,
where Gf (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 problem under zero Dirichlet boundary condition In this case we have δ(r − ρ) δ(ϕ − φ), F (r, ϕ) = 0, Φ(r, ϕ, t) = 0, f (r, ϕ) = r g1 (r, t) = 0, g2 (r, t) = 0, where δ(x) is the Dirac delta function. It should be noted that the twodimensional Dirac delta function in Cartesian coordinates after passing to 1 δ(r), but for the sake of simplicity we polar coordinates takes the form 2πr 1 have omitted the factor 2π in the delta term as well as the factor 2π in the solution (3.7). The Laplace transform with respect to time t applied to (3.1) gives
2G∗ ∗ 2G ∗ ∂ ∂G ∂ 1 1 δ(r − ρ) f f f δ(ϕ − φ) = a + 2 + , sα Gf∗ − sα−1 r ∂r 2 r ∂r r ∂ϕ2
126
Y. Povstenko
ϕ = 0 : Gf∗ = 0, ϕ = ϕ0 : Gf∗ = 0. Next we use the finite sin-Fourier transform (2.3) with respect to the angular coordinate ϕ, thus obtaining
∂ 2 Gf∗ 1 ∂ Gf∗ (nπ/ϕ0 )2 ∗ nπφ α ∗ α−1 δ(r − ρ) sin − + Gf . =a s Gf − s r ϕ0 ∂r 2 r ∂r r2 The Hankel transform (2.10) with respect to the radial variable r with ν = nπ/ϕ0 leads to the solution in the transform domain
nπφ sα−1 ∗ . (3.8) G f = Jnπ/ϕ0 (ρξ) sin ϕ0 sα + aξ 2 The inverse integral transforms result in
∞ ∞ 2 nπϕ nπφ sin Eα −aξ 2 tα sin G f (r, ϕ, ρ, φ, t) = ϕ0 ϕ0 ϕ0 0 (3.9) n=1
× Jnπ/ϕ0 (rξ) Jnπ/ϕ0 (ρξ) ξ dξ, where Eα (z) is the Mittag-Leffler function in one parameter α (e.g. [6], [9], [19]): ∞ zn , α > 0, z ∈ C. Eα (z) = Γ(αn + 1) n=0 3.2. The fundamental solution to the second Cauchy problem under zero Dirichlet boundary condition This solution is obtained for f (r, ϕ) = 0,
δ(r − ρ) δ(ϕ − φ), r g1 (r, t) = 0, g2 (r, t) = 0,
F (r, ϕ) =
and has the form ∞
GF (r, ϕ, ρ, φ, t) =
2t sin ϕ0 n=1
×
0
∞
nπϕ ϕ0
Eα,2 −aξ 2 t
sin
α
nπφ ϕ0
Φ(r, ϕ, t) = 0,
(3.10)
Jnπ/ϕ0 (rξ) Jnπ/ϕ0 (ρξ) ξ dξ,
where Eα,β (z) is the generalized Mittag-Leffler function in two parameters α and β described by the following series representation ([6], [9], [19]): ∞ zn , α > 0, β > 0, z ∈ C. Eα,β (z) = Γ(αn + β) n=0
SOLUTIONS TO THE FRACTIONAL DIFFUSION-WAVE . . . 127 3.3. The fundamental solution to the source problem under zero Dirichlet boundary condition In this case f (r, ϕ) = 0,
F (r, ϕ) = 0,
Φ(r, ϕ, t) =
g1 (r, t) = 0,
δ(r − ρ) δ(ϕ − φ) δ(t), r
g2 (r, t) = 0,
and
∞ 2tα−1 nπϕ nπφ sin sin GΦ (r, ϕ, ρ, φ, t) = ϕ0 ϕ0 ϕ0 n=1 ∞ Eα,α −aξ 2 tα Jnπ/ϕ0 (rξ) Jnπ/ϕ0 (ρξ) ξ dξ. ×
(3.11)
0
3.4. The fundamental solution to the first Dirichlet problem under zero initial conditions This solution corresponds to the choice f (r, ϕ) = 0, g1 (r, t) = g0
F (r, ϕ) = 0,
Φ(r, ϕ, t) = 0,
δ(r − ρ) δ(t), r
g2 (r, t) = 0,
and is expressed as
∞ 2atα−1 nπ nπϕ sin Gg1 (r, ϕ, ρ, φ, t) = ϕ0 ρ2 ϕ0 ϕ0 n=1 (3.12) ∞ Eα,α −aξ 2 tα Jnπ/ϕ0 (rξ) Jnπ/ϕ0 (ρξ) ξ dξ. × 0
The fundamental solution to the second Dirichlet problem under zero initial conditions is obtained from (3.12) by multiplying each term in the series by (−1)n+1 (see Eq. (2.6)). 4. The Neumann boundary condition The time-fractional diffusion-wave equation (3.1) is solved under initial conditions (3.2) and (3.3) and the Neumann boundary conditions ϕ=0: ϕ = ϕ0 :
−
1 ∂T = g1 (r, t), r ∂ϕ
1 ∂T = g2 (r, t). r ∂ϕ
(4.1) (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 problem under zero Neumann boundary condition To solve the problems under Neumann boundary condition the Laplace transform (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) of the order ν = nπ/ϕ0 with respect to the radial coordinate r are used. The solution has the following form
∞ 2 nπϕ nπφ cos cos Gf (r, ϕ, ρ, φ, t) = ϕ0 n=0 ϕ0 ϕ0 (4.3) ∞ Eα −aξ 2 tα Jnπ/ϕ0 (rξ) Jnπ/ϕ0 (ρξ) ξ dξ. × 0
Recall that the prime near the summation symbol means that the term with n = 0 is multiplied by 1/2. 4.2. The fundamental solution to the second Cauchy problem under zero Neumann boundary condition ∞
2t cos GF (r, ϕ, ρ, φ, t) = ϕ0 n=0 ×
∞ 0
nπϕ ϕ0
cos
nπφ ϕ0
(4.4)
Eα,2 −aξ 2 tα Jnπ/ϕ0 (rξ) Jnπ/ϕ0 (ρξ) ξ dξ.
4.3. The fundamental solution to the source problem under zero Neumann boundary condition ∞
2tα−1 cos GΦ (r, ϕ, ρ, φ, t) = ϕ0 n=0
×
0
∞
nπϕ ϕ0
cos
nπφ ϕ0
Eα,α −aξ 2 tα 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 mathematical Neumann 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 δ(r − ρ) 1 ∂Gg1 = g0 δ(t). (4.6) ϕ=0: − r ∂ϕ r In Eq. (4.6), we have introduced the constant multiplier g0 to obtain the nondimensional quantities used in numerical calculations. The solution has the form:
∞ ∞ 2ag0 tα−1 nπϕ cos Eα,α −aξ 2 tα Gg1 (r, ϕ, ρ, t) = ϕ0 ρ n=0 ϕ0 0 (4.7) × Jnπ/ϕ0 (rξ) Jnπ/ϕ0 (ρξ) ξ dξ. For classical diffusion equation (α = 1), using Eq. (A.1) from Appendix, we obtain [1] 2
∞
rρ r + ρ2 nπϕ g0 exp − cos Inπ/ϕ0 . (4.8) Gg1 (r, ϕ, ρ, t) = ρϕ0 t 4at 2at ϕ0 n=0
In the particular case of the wave equation (α = 2), taking into account Eq. (A.2) from Appendix, we get
√ ∞ 2 ag0 nπϕ cos Ψ(r, ρ, t), (4.9) Gg1 (r, ϕ, ρ, t) = ϕ0 ρ n=0 ϕ0 where Ψ(r, ρ, t) is expressed in terms of the Legendre functions: √ at < ρ a) ⎧ 0, ⎪ ⎪ ⎪ ⎪ ⎪
2 ⎨ r + ρ2 − at2 1 Ψ(r, ρ) = , √ P ⎪ 2 rρ nπ/ϕ0 −1/2 2rρ ⎪ ⎪ ⎪ ⎪ ⎩ 0, √ b) at = ρ
⎧ r 1 ⎪ ⎨ √ Pnπ/ϕ0 −1/2 , 2 rρ 2ρ Ψ(r, ρ) = ⎪ ⎩ 0,
0≤r ρ
2
2 ⎧ nπ at − r 2 − ρ2 1 ⎪ ⎪ , Qnπ/ϕ0 −1/2 − √ cos ⎪ ⎪ π rρ ϕ0 2rρ ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎪ 0 ≤ r < at − ρ, ⎪ ⎪ ⎪ ⎪ ⎨
2 1 r + ρ2 − at2 Ψ(r, ρ) = ⎪ , √ P ⎪ ⎪ 2 rρ nπ/ϕ0 −1/2 2rρ ⎪ ⎪ ⎪ √ √ ⎪ ⎪ at − ρ < r < ρ + at, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ √ 0, ρ + at < r < ∞. The dependence of nondimensional fundamental solution (4.7) G¯g1 = G on nondimensional distance r/ρ is presented in Fig. 4.1 for ϕ = 0 atα−1 g0 g1 and different values of the order of √ fractional derivative α. In calculations we have taken ϕ0 = π/4 and κ = atα/2 /ρ = 0.35. ρ3
4.5. The fundamental solution to the first physical Neumann problem under zero initial conditions For the physical Neumann problem, the boundary condition is formulated in terms of the normal component of the heat flux (see [23], [26]): ϕ=0:
δ(r − ρ) 1 1−α ∂Gg1 = g0 δ(t), − DRL r ∂ϕ r
0 < α ≤ 1,
(4.10)
ϕ=0:
∂Gg1 δ(r − ρ) 1 = g0 δ(t), − I α−1 r ∂ϕ r
1 < α ≤ 2,
(4.11)
1−α f (t) and I α−1 f (t) are the Riemann-Liouville fractional derivawhere DRL tive and fractional integral, respectively (see e.g. [6], [9], [19], [29]).
The solution is expressed as ∞
2ag0 cos Gg1 (r, ϕ, ρ) = ϕ0 ρ n=0
nπϕ ϕ0
∞
0
× Jnπ/ϕ0 (rξ) Jnπ/ϕ0 (ρξ) ξ dξ.
Eα −aξ 2 tα (4.12)
SOLUTIONS TO THE FRACTIONAL DIFFUSION-WAVE . . . 131 2.5
α = 1.5
2.0
α=1
1.5
G¯g1 1.0
0.5
α = 0.5
0.0 0.0
0.5
1.0
1.5
2.0
2.5
r¯
Fig. 4.1: The fundamental solution to the mathematical Neumann problem under zero initial conditions.
2.5
α = 1.15
2.0
α=1
1.5
G¯g1
α = 0.85
1.0
0.5
0.0 0.0
0.5
1.0
1.5
2.0
2.5
r¯
Fig. 4.2: The fundamental solution to the physical Neumann problem under zero initial conditions.
132
Y. Povstenko
In the limiting case α → 0 corresponding to the so-called localized diffusion the solution is obtained using Eq. (A.3) from Appendix and reads:
∞ 2g0 nπϕ cos Gg1 (r, ϕ, ρ, t) = ϕ0 ρ n=0 ϕ0
⎧ r ρ ⎪ ⎪ ⎪ ⎨ Inπ/ϕ0 √a Knπ/ϕ0 √a , 0 ≤ r < ρ, ×
⎪ ρ r ⎪ ⎪ ⎩ Inπ/ϕ0 √ Knπ/ϕ0 √ , ρ ≤ r < ∞. a a
Figure 4.2 presents the dependence of the nondimensional fundamental ρ3 Gg1 on distance for ϕ = 0 and different values solution (4.12) G¯g1 = ag 0 of the order of fractional derivative α. In the calculations we have taken ϕ0 = π/4 and κ = 0.35. The fundamental solutions to the second mathematical and physical Neumann problems, when the nonzero bounadry conditions are given at the boundary ϕ = ϕ0 , are obtained from Eqs. (4.7) and (4.12) by multiplying each term in the series by (−1)n (see Eq. (2.9) with taking into account that 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 spectrum from the localized diffusion (the Helmholtz equation when the order of the time-derivative α → 0) through the standard diffusion equation (represented by the particular case α = 1) to the ballistic diffusion (the wave equation when α = 2). We have derived the analytical solutions to the timefractional diffusion-wave equation in a wedge under Dirichlet, mathematical and physical Neumann boundary conditions. The integral transform technique has been used. It should be emphasized that the obtained fundamental solutions are expressed in terms of the Mittag-Leffler functions, in particular in the fundamental solutions to the mathematical and physical Neumann problems there appear the Mittag-Leffler functions Eα,α (−x) and Eα (−x), respectively. The difference between the mathematical and physical Neumann boundary conditions (as well as the difference between the solutions) disappear in the case of standard diffusion equation corresponding to α = 1. In this case the solutions (4.7) and (4.12) coincide and are 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: 2
∞ b + c2 1 bc −ax2 exp − Iν ; (A1) e Jν (bx) Jν (cx) x dx = 2a 4a 2a 0 ∞ sin ax Jν (bx) Jν (cx) dx 0
⎧
2 b + c2 − a2 ⎪ 1 ⎪ ⎪ √ Pν−1/2 , ⎪ ⎪ 2bc 2 bc ⎪ ⎪ ⎨
2 2 − c2 = − b a cos νπ ⎪ − √ Qν−1/2 , ⎪ ⎪ 2bc ⎪ π bc ⎪ ⎪ ⎪ ⎩ 0,
|b − c| < a < b + c,
(A2)
b + c < a, a < |b − c|,
where Pν (x) and Qν (x) are the Legendre functions of the first and second kinds, respectively; ∞ Iν (ac) Kν (ab), 0 < c < b, x (A3) J (bx) J (cx) dx = ν ν x2 + a2 0 0 < b < c, Iν (ab) Kν (ac), where Iν (x) and Kν (x) are the modified Bessel functions. In the above equations, we assume a > 0, b > 0, c > 0, and ν ≥ 0. References [1] B.M. Budak, A.A. Samarskii, A.N. Tikhonov, A Collection of Problems in Mathematical Physics. Pergamon Press, Oxford, 1964. [2] B. Datsko, V. Gafiychuk, Complex nonlinear dynamics in subdiffusive activator-inhibitor systems. Commun. Nonlinear Sci. Numer. Simulat. 17 (2012) 1673–1680. [3] G. Doetsch, Anleitung zum praktischen Gebrauch der LaplaceTransformation und der Z-Transformation. Springer, M¨ unchen, 1967. [4] V. Gafiychuk, B. Datsko, Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems. Comput. Math. Appl. 59 (2010), 1101–1107. [5] A.S. Galitsyn, A.N. Zhukovsky, Integral Transforms and Special Functions in Heat Conduction Problems. Naukova Dumka, Kiev, 1976 (In Russian). [6] R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order. In: Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York (1997), 223–276.
134
Y. Povstenko
[7] Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation. Osaka J. Math. 27 (1990), 309–321. [8] X.Y. Jiang, M.Y. Xu, The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems. Physica A 389 (2010), 3368–3374. [9] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006. [10] F. Mainardi, The fundamental solutions for the fractional diffusionwave equation. Appl. Math. Lett. 9, No 6 (1996), 23–28. [11] F. Mainardi, Fractional relaxation-oscillation and fractional diffusionwave phenomena. Chaos, Solitons Fractals 7 (1996), 1461–1477. [12] F. Mainardi, Applications of fractional calculus in mechanics. In: Transform Methods and Special Functions (Proc. Internat. Workshop, Varna, 1996), Bulgarian Academy of Sciences, Sofia (1998), 309–334. [13] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1–77. [14] R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37 (2004), R161–R208. [15] B.N. Narahari Achar, J.W. Hanneken, Fractional radial diffusion in a cylinder. J. Mol. Liq. 114 (2004), 147–151. ¨ [16] N. Ozdemir, D. Karadeniz, Fractional diffusion-wave problem in cylindrical coordinates. Phys. Lett. A 372 (2008), 5968–5972. ¨ [17] N. Ozdemir, D. Karadeniz, B.B. Iskender, Fractional optimal control problem of a distributed system in cylindrical coordinates. Phys. Lett. A 373 (2009), 221–226. ¨ [18] N. Ozdemir, O.P. Agrawal, D. Karadeniz, B.B. Iskender, Fractional optimal control problem of an axis-symmetric diffusion-wave propagation. Phys. Scr. T 136 (2009), # 014024. [19] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, 1999. [20] Y.Z. Povstenko, Fractional heat conduction equation and associated thermal stress. J. Thermal Stresses 28 (2005), 83–102. [21] Y.Z. Povstenko, Two-dimensional axisymmetric stresses exerted by instantaneous pulses and sources of diffusion in an infinite space in a case of time-fractional diffusion equation. Int. J. Solids Structures 44 (2007), 2324–2348. [22] Y.Z. Povstenko, Fractional radial diffusion in a cylinder. J. Mol. Liq. 137 (2008), 46–50. [23] Y. Povstenko, Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder. Fract. Calc.
SOLUTIONS TO THE FRACTIONAL DIFFUSION-WAVE . . . 135 Appl. Anal. 14 (2011), 418–435; DOI: 10.2478/s13540-011-0026-4; http://link.springer.com/article/10.2478/s13540-011-0026-4. [24] Y. Povstenko, Solutions to time-fractional diffusion-wave equation in cylindrical coordinates. Adv. Difference Eqs 2011 (2011), Article ID 930297, 14 pp. [25] Y. Povstenko, Time-fractional radial heat conduction in a cylinder and associated thermal stresses. Arch. Appl. Mech. 82 (2012), 345–362. [26] Y.Z. Povstenko, Fractional heat conduction in infinite one-dimensional composite medium. J. Thermal Stresses 36 (2013), 351–363. [27] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series. Special Functions. Nauka, Moscow, 1983 (In Russian). [28] H. Qi, J. Liu, Time-fractional radial diffusion in hollow geometries. Meccanica 45 (2010), 577–583. [29] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Amsterdam, 1993. [30] W.R. Schneider, W. Wyss, Fractional diffusion and wave equations. J. Math. Phys. 30 (1989), 134–144. [31] I.N. Sneddon, The Use of Integral Transforms. McGraw-Hill, New York, 1972. [32] V.V. Uchaikin, Fractional Derivatives for Physicists and Engineers. Springer, Berlin, 2013. [33] W. Wyss, The fractional diffusion equation. J. Math. Phys. 27 (1986), 2782–2785. 1
Institute of Mathematics and Computer Science Jan Dlugosz University in Cz¸estochowa al. Armii Krajowej 13/15 42-200 Cz¸estochowa, POLAND 2
Department of Computer Science European University of Informatics and Economics (EWSIE) ul. Bialostocka 22 03-741 Warsaw, POLAND e-mail:
[email protected]
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
