First and Second Fundamental Solutions of the Time-Fractional Telegraph Equation of Order 2 α ∗

In this work we obtain the ﬁrst and second fundamental solutions of the multidimensional time-fractional equation of order 2 α , α ∈ ]0 , 1], where the two time-fractional derivatives are in the Caputo sense. We obtain representations of the fundamental solutions in terms of Hankel transform, double Mellin-Barnes integral, and H-functions of two variables. As an application, the fundamental solutions are used to solve a Cauchy problem, and to study telegraph process with Brownian time.


Introduction
The telegraph equation is used as an alternative to the diffusion equation, since it has the potential to describe both diffusive and wave-like phenomena, due to the simultaneous presence of first and second order time derivatives. For example, in the case of the transport of energetic charged particle in turbulent magnetic fields such as low-energy cosmic rays in the solar wind, the diffusion equation can not be used to describe the transport for early times because it leads to a non-zero probability density everywhere, which would correspond to an infinite propagation speed. Using the telegraph equation in this case we get a more realistic model for the early phase transport because it combines diffusion with a finite propagation speed (see [18]). Telegraph equations have also an extraordinary importance in electrodynamics (the scalar Maxwell equations are of this type), in the theory of damped vibrations, and in probability because they are connected with finite velocity random motions (see [12,16]).
One of the first works studying the time-fractional telegraph equation is the paper of Cascaval et al. (see [4]). Here, the authors discussed some properties of the time-fractional telegraph equation in R × R + such as the well-posedness and the asymptotic behavior of the solutions, by using the Riemann-Liouville approach. In [15], Orsingher and Beghin obtained the fundamental solution of the time-fractional telegraph equation of order 2α in R × R + and gave a representation of their inverses in terms of stable densities. For the special case α = 1 2 , the authors showed that the fundamental solution is the probability density of a telegraph process with Brownian time. In [3] it was discussed the solution of a general space-time fractional telegraph equation by means of the Laplace and Fourier transforms in the variables x ∈ R and t ∈ R + , respectively. In [19] it was obtained the solutions of the space-time fractional telegraph equation in R × R + in terms of Mittag-Leffler functions, using an operational approach. In [14], Mamchuev considered the inhomogeneous time-fractional telegraph equation with Caputo derivatives, and obtained a general representation of regular solution in rectangular domain in terms of fundamental solution and appropriate Green functions. Regarding the multidimensional case, in [5] the authors discussed and derived the solution of the time-fractional telegraph equation in R n × R + with three kinds of nonhomogeneous boundary conditions, by the method of separation of variables. Very recently, in [9,10] the authors found several representations of the fundamental solution of the time-fractional telegraph and telegraph Dirac equations in R n × R + , in terms of integrals, special functions, and series.
The aim of this paper is to representations for the first and second fundamental solutions of the timefractional telegraph equation of order 2α in terms of Hankel transforms, double Mellin-Barnes integral and H-functions of two variables. Moreover, the first fundamental solution is used in the law of telegraph process with Brownian time. This connection is motivated by the fact that the iterated Brownian motion and telegraph process with Brownian time are governed by time-fractional telegraph equations (see [15]).
The structure of the paper reads as follows: in the Preliminaries section we recall some basic facts about some special functions and fractional calculus, which are necessary for the development of the work. In the following section we obtain the first and second fundamental solutions of the time-fractional telegraph equation , ∆ x is the Laplace operator in space, and the two time-fractional derivatives of orders α and 2α are in the Caputo sense, with α ∈]0, 1]. This section ends with an application to the resolution of a Cauchy problem. In the last section we present an application of our results to the law of a telegraph process with Brownian time.

Preliminaries
Here we recall the main tools concerning fractional derivatives and special functions that will be used in our work. We start by recalling the definition of the multivariate Mittag-Leffler function (see [13]).
Lemma 2.2 Let z 1 , z 2 ∈ C, and a 1 , a 2 , b ∈ C (with positive real parts). Then it holds For general properties of the Mittag-Leffler function see [11,13]. Now we recall the definition of the H-function of two complex variables.

Definition 2.3 (see [1]) The H-function of two complex variables is defined via a double Mellin-Barnes integral of the form
and where an empty product is interpreted as 1, , (e j ) and (f j ) are restricted that none of the poles of the integrand coincide. The contour L 1 in the complex s-plane, and the contour L 2 in the complex w-plane, are of Mellin-Barnes type with indentations, if necessary, to ensure that they separate one set of poles from the other.
In [1] the author proved that if and with the points x = 0 and y = 0 being tacitly excluded, the double Mellin-Barnes integral converges absolutely inside the sector given by For E (a1,a2),b we have the following two results (see [9]).

Lemma 2.4
The bivariate Mittag-Leffler function E (a1,a2),b has the following representation in the form of double Mellin-Barnes integral (1 − b; a 2 , a 1 ) ; (0, 1) ; (0, 1) In (5) and throughout the paper a horizontal line in the H-function means the absence of parameters. Now we recall the definition of the spaces C α , α ∈ R, and C m α , m ∈ N, given in [13].
Definition 2.6 (see [13]) A real or complex-valued function f (t), t > 0, is said to be in the space C α , α ∈ R, if there exists a real number p > α such that f (t) = t p f 1 (t) for some function f 1 ∈ C[0, ∞).
It is easy to see that C α is a vector space and the set of spaces C α is ordered by inclusion according to Definition 2.7 (see [13]) A function f (t), t > 0, is said to be in the space C m α , m ∈ N, if and only if f (m) ∈ C α .
Let D γ t denotes the Caputo fractional derivative of order γ > 0 defined by: where u (m) := d m u dt m , m ∈ N (see [2]). The next theorem will be used in our analysis and allow us to solve general linear fractional differential equations with constant coefficients and Caputo derivatives.

First and second fundamental solution of the time-fractional telegraph equation of order 2α
In this section we obtain the first and second fundamental solution of a particular case of the previous equation, where β = 2α and 0 < α ≤ 1, i.e., we look for a function G α (x, t) that satisfies the following Cauchy problem where x ∈ R n , t > 0, a ≥ 0, c > 0, C 0 , C 1 ∈ R, δ(x) = n j=1 δ(x j ) is the distributional Dirac delta function in R n . Applying the Fourier transform in R n to the Cauchy problem we get the following initial-value problem To solve the problem (9), we apply Theorem 2.8 with λ 1 = −a, λ 2 = −c 2 |κ| 2 , µ = 2α, µ 1 = α, µ 2 = 0, n = 2, m = 2, g(t) = 0, and k = 1, obtaining the following solution where Taking into account Lemma 2.2 we have the following alternative representation of u 0 and u 1 From (2), we have the following representation of u 0 and u 1 in the form of a double series From Lemma 2.4 we have the following representations of u 0 and u 1 in the form of a double Mellin-Barnes integral (−α; 2α, α) ; (0, 1) ; (0, 1) (−1; 2α, α) ; (0, 1) ; (0, 1) Inverting the Fourier transform in (10) we get We will denote by G α 1 (x, t) = u 0 (x, t) the first fundamental solution of the first equation of (8) that satisfies the initial conditions G α 1 (x, 0) = δ(x) and ∂G α 1 ∂t (x, 0) = 0. Additionally, we denote by G α 2 (x, t) = u 1 (x, t) the second fundamental solution of the first equation of (8) that satisfies the initial conditions G α 2 (x, 0) = 0 and ∂G α 2 ∂t (x, 0) = δ(x). For obtaining the explicit expressions for G α 1 and G α 2 we recall the following formula (see [17]) where J ν represents the Bessel function of first kind with index ν, and the right hand side can be seen as a Hankel transform. We need also the following integral formula (see Formula (7) in [6, p. 22 which is valid under the condition −Re(ν) − 3 2 < Re(µ) < − 1 2 . Since the expressions for G α 1 are longer then the ones for G α 2 , we are going to present only the calculations for G α 2 . Applying the inverse Fourier transform to (12) and using (20), we get Making use of Lemma 2.4, and interchanging the integrals due to the convergence, we obtain Finally, using (21) and Corollary 2.5, we obtain, under the condition − n 2 < (s) < − n 4 , the representation of the second fundamental solution in terms of double Mellin-Barnes integral and H-function of two variables: where the double Mellin-Barnes integral is convergent because the conditions (4) are fulfilled. For G α 1 we obtain, analogously, under the condition − n 2 < (s) < − n 4 , the following representations: We end this section solving a fractional Cauchy problem.
Theorem 3.1 Let x ∈ R n , t > 0, 0 < α ≤ 1, a ≥ 0, and c > 0, then the fractional Cauchy problem is solvable, and its solution has the form where G α 1 and G α 2 are the first and second fundamental solutions given by (28) and (25), respectively, and provided that the integrals in the right-hand side of (29) are convergent.  = 2α). Moreover, considering a = 0 the first fundamental solution G α 1 coincides with the expression presented in [7] for the fundamental solution of the time-fractional diffusion-wave operator (with β = 2α).

Telegraph process with Brownian time
Now we consider in (8) n = 1, a = 2λ ≥ 0, C 0 = 1 and C 1 = 0, i.e., we consider the following time-fractional telegraph equation of order 2α where x ∈ R, t > 0, c > 0, and subject to the initial condition u(x, 0) = δ(x) for 0 < α ≤ 1 2 , while, for 1 2 < α ≤ 1, besides the previous condition, also u t (x, 0) = 0 is assumed. Equation (30) was already studied in [15], where the authors presented only an integral representation for the Fourier transform of the fundamental solution. Moreover, for α = 1 2 it was obtained an integral representation of the fundamental solution based on the Fourier inversion transform (see Theorem 4.2 in [15]).
Physically, if we consider α = 1 2 in (30), we obtain a heat equation with damping term which depends on all values of u in [0, t] and assigning an overwhelming weight to those close to t (see [15]). The damping effect of the fractional derivative reverberates on the distribution u, where the solution of the heat equation (governing term) is perturbed by the telegraph distribution (which represents the impact of the fractional derivative).
Moreover, since the fundamental solution u of (30) reduces, in this particular case, to G 1 2 1 (see (19)), it can be understood as the distribution of a particle moving back and forth the real line with velocities ±c for a random