Construction of an Explicit Solution of a Time-Fractional Multidimensional Differential Equation

In this work, an explicit solution of the initial-boundary value problem for a multidimensional time-fractional differential equation is constructed. The possibility of obtaining this equation from an integro-differential wave equation with a Mittag–Leffler–type memory kernel is shown. An explicit solution to the problem under consideration is obtained using the Laplace and Fourier transforms, the properties of the Fox H-functions and the convolution theorem.


Introduction to the Problem and Its Setting
The investigation of many mathematical models that have a fractal structure have numerous applied applications. The presence of memory in such models indicates the dependence of its current state on a finite number of its previous states. This means the non-local properties of non-classical mathematical models, for example, in the mechanics of viscoelastic media when describing the action of aftereffect [1,2]. In materials science, material fatigue exists, which leads to the destruction of the material [3], in mathematical models of economics, the effect of dynamic memory is associated with the principle of causation of economic models [4].
As shown in [5], the memory property can be described using the mathematical apparatus of fractional calculus or using the operators fractional derivative. Fractional derivative operators have many definitions and have unique properties, but they all describe to one degree or another memory effect characterizing information about the previous states of the system.
If the memory functions are given and are power-law, then we can go to other types of equations that are based on derivatives of fractional orders, properties of which are considered in table books on fractional calculus [23]. Processes and systems that are described using fractional derivatives orders are called fractal; for example, oscillators are called fractal in the theory of hereditary fluctuations [24].
To solve initial-boundary value problems for fractional differential equations of diffusion type, various methods are used: the Green's function method, methods based on the integral transforms of Fourier, Laplace and Mellin, the method of separation of variables, methods of reduction to integral equations of the Volterra type, and other methods. At the same time, there are relatively few methods for obtaining explicit solutions of fractional differential equations.
In this paper, we consider the following n−dimensional differential equation with Caputo fractional derivative of order α > 0 where the initial and boundary conditions are: ∆ is the n-dimensional Laplace operator with respect to x and ∇ = ∂ ∂x 1 , ..., ∂ ∂x n . Equation (1) for ρ = 1, m = 1 describes the anomalously diffusive transport of solute in heterogeneous porous media [27]. This equation containing both the classical and fractional derivatives is more general and is of interest in the theory of the differential equations with fractional derivatives.
Our main goal is to obtain an explicit formula that gives a solution to the problem (1) and (2).

General Theories
In this section, we recall basic definitions and notations from integral transforms, special functions and fractional calculus.
Mittag-Leffler function. The Mittag-Leffler functions are defined as follows: respectively, where α, z, β ∈ C; R(α) > 0, R(α) denotes the real part of the complex number α. The Prabhakar function is [36]: where α, β, γ, z ∈ C, and (γ) n denotes the Pochammer symbol or the shifted factorial determined by Moreover, we can write (γ) n = Γ(γ + n)/Γ(γ), where Γ(γ) is the usual Gamma function. We have running special cases: The Mittag-Leffler function has numerous applications, and many authors have generalized it. In this category is the work of Haubold et al. [37].
Fox's H−function. Fox's H− function is a special function of fractional calculus and contains Mittag-Leffler functions. This function was introduced by Fox [38] as a generalization of the Meyer function. Here we give the definition and properties of this function [29] which we need, with minor changes in the notation.
The H−function is defined by means of a Mellin-Barnes-type integral in the following manner: where the integrand is In (2.3), an empty product is always interpreted as unity, m, n, p, q ∈ N 0 , with The contour Ω starting at the point σ − i∞ and going to σ + i∞, σ ∈ R such that all the poles of Γ(b j + B j s), j = 1, ..., m, are separated from those of The conditions for the convergence of the integral are given below: The H− function is studied in more detail in [29]. We only mentioned some properties and its Hankel transform that will be used in this paper.
• H m,n p,q z The inverse Hankel transform is given by where J ν (z) is a Bessel function of the first kind and defined by Note the relation between Prabhakar function and the Fox-H function [29]: We define the integral operator E γ α,β,ω;+ as follows [37]: It is noted that the integral operator (11) is nowadays known in the literature as the Prabhakar fractional integral. Lemma 1. The following Laplace transform of a Prabhakar function is given by [29,31]: where |ω/s α | < 1.

Lemma 3.
For ∀α > 0, β ∈ C, µ > 0 and a ∈ R, the following formula is valid [39]: Here J n 2 −1 (·) is a Bessel function and E For solvability of an integral equation of the Volterra type with a difference kernel the following assertion is true [40]: for a fixed T > 0 and k(t), r(t) are jointed by the integral equation then the solution of the integral equation is given by formula
here x ∈ R n , ξ ∈ R n , dξ = dξ 1 dξ 2 ...dξ n . (1) with respect to variable t taking into account the initial conditions from (2), the Laplace transform formula for the partial fractional derivative Caputo [23] and classical derivative

Proof. Applyin the Laplace transforms (15) to both sides of Equation
we obtain In sequence, applying to this equality the Fourier transform (16) and using well-known formula of Fourier transform of operator we come to the relation We calculate the inverse Laplace and Fourier transforms of the functionû(ξ, s) defined by (21). First, these operations we carry out for F(ξ, s). It may be performed by using the form 1 and expanding the second multiplier on the right side of this term into an infinitely decreasing geometric series: (−|ξ| 2 ) n s −αn (ρs m−α + 1) n for |ξ| 2 s −α ρs m−α +1 < 1. On bases of (22) from the last equality we have Then, according to Lemma 1, we note Taking these formulae into account, eventually F(ξ, s) can be expressed as Now, we apply the inverse transform the functions Φ j (ξ, s), j = 0, 1, ..., m − 1. For this, we note that the fractions at the functions ϕ j (ξ), j = 0, 1, ..., m − 1 in expressions for Φ j (ξ, s) as seen from (21) In view of the last relations, applying Lemma 1 to the frictions in the definition of Φ j (ξ, s), we obtain Further, in accordance with the Prabhakar function definition (3), from Equation (23) we obtain 1 By virtue of this fact we continue converting the function F(ξ, s) into According to the convolution property of the Laplace transform and the definition of integral operator E γ α,β,ω;+ ϕ by (11), the inverse Laplace transform of the function F(ξ, s) from last relations can be obtained as follows: Analogically, the inverse Laplace transform of functions Φ j (ξ, s) can be expressed as Considering the relationship between the generalized Mittag-Leffler function and the Fox H function (10), the last equalities can be rewritten in the form [29] In (27) we introduced the following notations: Now we compute the inverse Fourier transform of relations (27). Equation (28) can be further manipulated by employing an inverse Fourier transform; we have here j = 0, 1, ..., m − 1, k = 0, 1, 2, .... Using Lemma 3, we obtain the following results from formulae (29) and ( By same argument for Φ 2j we have Φ 2j (x, t) = 1 2π n/2 |x| n H 2,0 (1 + (m − α)(k + 1) + j, m/2) (n/2, 1/2), (1 + k, 1/2) .
Now, applying the inverse Fourier transform to both sides of (26) and (27), and substituting into the resulting equalities, formulae (31) and (32), we obtain where Continuing to convert the equalities (26) and (27) we can write formally In view of (27) and (34), employing an inverse Laplace transform to Equation (21) we finally obtaiñ To Equation (35) can be further applied inverse the Fourier transform and Fourier convolution property one by one. Therefore, we complete the proof of Theorem 1.

The Integro-Differential Diffusion Equation with the Mittag-Leffler Type Kernel
In this section we demonstrate the equivalence of one integro-differential wave equation with the Mittag-Leffler type kernel to the fractional wave equation.
Theorem 2. The integro-differential wave equation Proof. In general the Equation (36) is the Volterra integral equation of the second kind with respect to u for fixed x and employing Lemma 4, we obtain where r(t) is resolvent of k(t) and it satisfies the integral Equation (5). We use to both sides of (12) the Laplace and denoting by K(s) and R(s) the imagines of origins k(t) and r(t), respectively, we have R(s) = K(s) + K(s)R(s).

From this relation we obtain
Applying the inverse Laplace transformation to last equality (see [33]) Hence, if we suppose that the function k(t) is given by formula (39) then, (38) hlyields (37). From this comment it follows that the solution of Equation (36) with initial-boundary condition can be given by formula (16) for f (x, t) = 0.

Conclusions
Equation (36) (with different types of memory kernels) can be used to investigate a wide class of non-classical mathematical models in which the memory effect is present. In this paper, we show that the time-fractional multidimensional differential Equation (1) can be obtained from Equation (36) with the Mittag-Leffler kernel t m−1−α E m−α,m−α (−t m−α ). Applying the method of the Laplace transform in the time variable and the Fourier transform in the spatial variable, an explicit solution of problem (1) and (2) is obtained, which includes the fractional Prabhakar integral and the Fox H− functions. In many cases, due to the nonlocality of the integral term, it is not possible to obtain an exact solution to Equation (1). Therefore, in many cases, a more reasonable option is to find its numerical solution. In the future, we plan to numerically solve similar problems in one-dimensional and two-dimensional cases.

Conflicts of Interest:
The authors declare no conflict of interest.