Explicit solutions to fractional diffusion equations via Generalized Gamma Convolution

In this paper we deal with Mellin convolution of generalized Gamma densities which leads to integrals of modified Bessel functions of the second kind. Such convolutions allow us to explicitly write the solutions of the time-fractional diffusion equations involving the adjoint operators of a square Bessel process and a Bessel process.


Introduction and main result
In the last years, the analysis of the compositions of processes and the corresponding governing equations has received the attention of many researchers. Many of them are interested in compositions involving subordinators, in other words, subordinated processes Y (T (t)), t > 0 (according to [9]) where T (t), t > 0 is a random time with non-negative, independent and homogeneous increments (see [4]). If the random time is a (symmetric or totally skewed) stable process we have results which are strictly related to the Bochner's subordination and the p.d.e.'s connections have been investigated, e.g., in [6; 7; 8; 22; 23]. If the random time is an inverse stable subordinator we shall refer to the governing equation of Y (T (t)) as a fractional equation considering that a fractional time-derivative must be taken into account. In the literature, several authors have studied the solutions to space-time fractional equations. In the papers by Wyss [30], Schneider and Wyss [29], the authors present solutions of the fractional diffusion equation ∂ λ t T = ∂ 2 x T in terms of Fox's functions (see Section 2). In the works by Mainardi et al., see e.g. [17; 18] the authors have shown that the solutions to space-time fractional equation x D α θ u = t D β * u can be represented by means of Mellin-Barnes integral representations (or Fox's functions) and M-Wright functions (see e.g. Kilbas et al. [13]). The fractional Cauchy problem D α t u = L u has been thoroughly studied by yet other authors and several representations of the solutions have been carried out, but an explicit form of the solutions has never been obtained. Nigmatullin [25] gave a physical derivation when L is the generator of some continuous Markov process. Zaslavsky [31] introduced the space-time fractional kinetic equation for Hamiltonian chaos. Kochubei [14,15] first introduced a mathematical approach while Baeumer and Meerschaert [1] established the connections between fractional problem and subordination by means of inverse stable subordinator when L is an infinitely divisible generator on a finite dimensional vector space. In particular, if ∂ t p = Lp is the governing equation of X(t), then under certain conditions, ∂ β t q = Lq +δ(x)t −β /Γ(1−β) is the equation governing the process X(V t ) where V t is the inverse or hitting time process to the β-stable subordinator, β ∈ (0, 1). Orsingher and Beghin [26,27] found explicit representations of the solutions to ∂ ν t u = λ 2 ∂ 2 x u only in some particlular cases: ν = (1/2) n , n ∈ N and ν = 1/3, 2/3, 4/3. Also, they represented the solutions to the fractional telegraph equations in terms of stable densities, see [3; 26]. In general, the solutions to fractional equations represent the probability densities of certain subordinated processes obtained by using a time clock (in the following we will refer to it as L ν t ) which is an inverse stable subordinator (see Section 4). For a short review on this field, see also Nane [24] and the references therein.
We will present the role of the Mellin convolution formula in finding solutions of fractional diffusion equations. In particular, our result allows us to write the distribution of both stable subordinator and its inverse process whose governing equations are respectively space-fractional or time-fractional equations. This result turns out to be useful for representing the solutions to the following fractional diffusion equation is the Riemann-Liouville fractional derivative, ν ∈ (0, 1] and G γ,µ is an operator to be defined below (see formula (3.4)). We present, for ν = 1/(2n + 1), n ∈ N ∪ {0}, the explicit solutions to (1.1) in terms of integrals of modified Bessel functions of the second kind (K ν ) whereas, for ν ∈ (0, 1], we obtain the solutions to (1.1) in terms of Fox's functions. After some preliminaries in Section 2, in Section 3 we recall the generalized Gamma density Q γ µ starting from which we define the distribution g γ µ of the (generalized Gamma) process G γ,µ t and the distribution e γ µ of the process E γ,µ t . The latter can be seen as the reciprocal Gamma process, indeed E γ,µ t = 1/G γ,µ t , or in a more striking interpretation, as the hitting time process for which (E γ,µ t < x) = (G γ,µ x > t). We shall refer to E γ,µ t as the reciprocal or equivalently the inverse process of G γ,µ t . It must be noticed that e γ µ = g −γ µ because G −γ,µ t = 1/G γ,µ t . Furthermore, we introduce the most important tool we deal with in this paper, the Mellin convolutions g γ,⋆n µ (see formula (3.14)) and e ⋆n µ (see formula (3.13)) where e ⋆n µ stands for e 1,⋆n µ . In Section 4 we draw some useful transforms of the distribution h ν of the stable subordinatorτ ν t and the distribution l ν of the inverse process L ν t . Similar calculations can be found in the paper by Schneider and Wyss [29]. The inverse (or hitting time) process is defined once again from the fact that (L ν t < x) = (τ ν x > t) (see also [1; 4]). In Section 5 we present our main contribution. We show that the following representations hold true: h ν (x, t) = e ⋆n µ (x, ϕ n+1 (t)), x > 0, t > 0, ν = 1/(n + 1), n ∈ N and l ν (x, t) = g (n+1),⋆n µ (x, ψ n+1 (t)), x > 0, t > 0, ν = 1/(n + 1), n ∈ N.

Preliminaries
The H functions were introduced by Fox [10] in 1996 as a very general class of functions. For our purpose, the Fox's H functions will be introduced as the class of functions uniquely identified by their Mellin transforms. A function f for which the following Mellin transform exists can be written in terms of H functions by observing that . (2. 2) The inverse Mellin transform is defined as at all points x where f is continuous and for some real θ. Thus, according to a standard notation, the Fox H function is defined as follows where P(D) is a suitable path in the complex plane C depending on the fundamental strip (D) such that the integral (2.1) converges. For an extensive discussion on this function see Fox [10]; Mathai and Saxena [20]. The Mellin convolution formula turns out to be very useful later on. Formula (2.3) is a convolution in the sense that Throughout the paper we will consider the integral (for some well-defined f 1 , f 2 ) which is not, in general, a Mellin convolution. We recall the following connections between Mellin transform and both integer and fractional order derivatives. In particular, we consider a rapidly decreasing function f : and, for 0 (see Kilbas et al. [13]; Samko et al. [28] for details). The fractional derivative appearing in (2.7) must be understood as follows that is the Dzerbayshan-Caputo sense. We also deal with the Riemann-Liouville fractional derivative and the fact that , n − 1 < α < n, (2.10) see Gorenflo and Mainardi [11] and Kilbas et al. [13]. We refer to Kilbas et al. [13]; Samko et al. [28] for a close examination of the fractional derivatives (2.8) and (2.9).

Mellin convolution of generalized Gamma densities
In this section we introduce and study the Mellin convolution of generalized gamma densities. In the literature, it is well-known that generalized Gamma r.v. possess density law given by Our discussion here concerns the function Let us introduce the convolution for which we have (see formula (2.4)) as a straightforward calculation shows. We now introduce the generalized Gamma process (GGP in short). Roughly speaking, the function (3.1) can be viewed as the distribution of a GGP {G γ,µ t , t > 0} in the sense that ∀t the distribution of the r.v. G γ,µ t is the generalized Gamma distribution (3.1). Thus, we make some abuse of language by considering a process without its covariance structure. In the literature there are several non-equivalent definitions of the distribution on R n + of Gamma distributions, see e.g. Kotz et al. [16] for a comprehensive discussion. In Section 5 (Corollary 1) we will show that the distribution (3.1) satisfies the p.d.e.
t/2 , t > 0} and, for γ = 2 we obtain the distribution of a 2µdimensional Bessel process {BES (2µ) t/2 , t > 0}, both starting from zero. Some interesting distributions can be realized through Mellin convolution of distribution g γ µ . Indeed, after some algebra we arrive at and where B(·, ·) is the Beta function (see e.g. Gradshteyn A further distribution arising from convolution can be presented. In particular, for γ = 0, we have which proves to be very useful further on. The function K ν appearing in (3.7) is the modified Bessel function of imaginary argument (see e.g [12, formula 8.432]). For the sake of completeness we have writen the following Mellin transforms: For the one-dimensional GGP we are able to define the inverse generalized Gamma process {E γ,µ t , t > 0} (IGGP in short) by means of the following relation P r{E γ,µ t < x} = P r{G γ,µ x > t}.
The density law e γ µ = e γ µ (x, t) of the IGGP can be carried out by observing that and, making use of the Mellin transform, we obtain The derivative under the integral sign in (3.9) is allowed from the fact that Ξ 1 (s) = ∂ ∂x g γ µ (s, x) ∈ L 1 (R + ) as a function of s. From (2.2) and the fact that for all c > 0 (see Mathai and Saxena [20]), we have that By observing that M e γ µ (·, t) (1) = 1, we immediately verify that (3.11) integrates to unity. The density law g γ µ can be expressed in terms of H functions as well, therefore we have In view of (3.11) and (3.12) we can argue that Remark 1. We notice that the inverse process {E 1,1/2 t , t > 0} can be written as where B is a standard Brownian motion. Thus, E 1,1/2 can be interpreted as the first-passage time of a standard Brownian motion through the level √ 2t.

Main results
In this section we consider compositions of processes whose governing equations are (generalized) fractional diffusion equations. When we consider compositions involving Markov processes and stable subordinators we still have Markov processes. Here we study Markov processes with random time which is the inverse of a stable subordinator. Such a process does not belong to the family of stable subordinators (see (4.12)) and the resultant composition is not, in general, a Markov process. This somehow explains the effect of the fractional derivative appearing in the governing equation, see Mainardi et al. [19]. Hereafter, we exploit the Mellin convolution of generalized Gamma densities in order to write explicitly the solutions to fractional diffusion equations. We first present a new representation of the density law h ν by means of the convolution e ⋆n µ introduced in Section 3. To do this we also introduce the time-stretching function ϕ m (s) = (s/m) m , m ≥ 1, s ∈ (0, ∞).