232 Yuriy Povstenko

fractional generalizations of the one-dimensional equation (27) were considered. It should be noted that interesting results were also obtained for fractional reaction-diffusion systems (see Datsko and Gafiychuk (2012), Gafiychuk and Datsko (2010)).

produces the following fractional telegraph equation with the advection term:

3.Theory of Diffusive Stresses

The theory of diffusive strain and stress is governed by a system of partial differential equations consisting of the equations of elasticity with gradient of concentration taken into account. The problem can be treated as an uncoupled in the case when the influence of the time change of strain on the matter flux is neglected. The first theoretical investigation of diffusive stresses dates back to Podstrigach (1961, 1963, 1965). For additional references, further generalizations and discussion see Nowacki (1974), Nowacki and Olesiak (1991), Podstrigach and Povstenko (1985), among others.

Povstenko (2005) proposed the theory of thermoelasticity based on the time-fractional heat conduction equation. A comprehesive review of further generalizations of fractional thermoelasticity (fractional theory of diffusive stresses) can be found in Povstenko (2013).

The stressed-strained state of a solid is governed by the equilibrium equation in terms of displacements

and the corresponding equation describing diffusion. In this paper we will use the timefractional advection-diffusion equation

∂αc

∂tα = a c − v · grad c. (41) Here u is the displacement vector, σ the stress tensor, e the linear strain tensor, λ and µ are Lam´e constants, K = λ + 2µ/3, βc is the diffusion coefficient of volumetric expansion, I denotes the unit tensor.

As in the case of classical thermoelasticity or the standard theory of diffusive stresses we can use the representation of stresses in terms of the displacement potential Φ (Parkus (1959))

where r stands for the gradient operator. The displacement potential is determined from the equation

with ν being the Poisson ratio.

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3.1.The Time-fractional Advection-diffusion Equation in a Plane

Here δ(x) is the Dirac delta function. In (45) we have introduced the constant multiplier p0 to obtain the nondimensional quantities displayed in Figures.

The zero conditions at infinity are also imposed:

In this paper, we will use the integral transform technique. Recall that the Caputo fractional derivative for its Laplace transform rule requires the knowledge of the initial values of the function f (t) and its integer derivatives of order k = 1, 2, . . . , m − 1 (see Kilbas et al. (2006), Podlubny (1999)):

Applying to Eq. (44) the Laplace transform with respect to time t and the double Fourier transform with rspect to the space coordinates x and y gives

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k=0

Changing variables w = sin θ and taking into account the following integral (Prudnikov et al. (1981))

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x¯

Figure 1. Dependence of the fundamental solution to the Cauchy problem on distance (the classical advection-diffusion equation, α = 1).

Let us analyze several particular cases of the solution (59). In the case of standard advection-diffusion equation (α = 1 and E1(−x) = e−x), evaluating the integral (Prudnikov et al. (1983))

The results of numerical computations of concentration for y = 0 are presented in Fig. 1

have been introduced.

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