New Solution of Diffusion–Advection Equation for Cosmic-Ray Transport Using Ultradistributions

In this paper we exactly solve the diffusion–advection equation (DAE) for cosmic-ray transport. For such a purpose we use the Theory of Ultradistributions of J. Sebastiao e Silva, to give a general solution for the DAE. From the ensuing solution, we obtain several approximations as limiting cases of various situations of physical and astrophysical interest. One of them involves Solar cosmic-rays’ diffusion.


The Problem at Hand
Fractional derivatives constitute a rather old subject, although not as familiar as the integerorder counterparts [1,2]. Fractional derivatives have recently been used with regards to many physical problems (for a small sample, see for instance [3][4][5][6][7][8][9][10]) and to hydrology [11][12][13]. Fractional derivatives have been recently applied to model super-diffusion of particles in astrophysical scenarios [14,15]. There is a considerable evidence emerging from data gathered by spacecrafts showing that the transport of energetic particles in the turbulent B M. C. Rocca rocca@fisica.unlp.edu.ar 1 Departamento de Física, Fac. de Ciencias Exactas, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina 2 IFLP-CCT-CONICET, C.C 727, 1900 La Plata, Argentina 3 CeBio y Secretaria de Investigacion, Univ. Nac. del Noroeste de la Prov. de Bs. As. UNNOBA and CONICET, R. Saenz Peña 456, Junín, Argentina heliospheric medium is super-diffusive [17,18]. An interesting work to be mentioned, in a different vein, is that of ref. [16] People employ fractional space derivatives so as to model anomalous diffusion or dispersion. Here, a particle plume spreads at a rate that is not the one of a classical Brownian model. If a fractional derivative takes the place of the second derivative in a diffusion or dispersion equation, this results in enhanced diffusion, or super-diffusion. In the case of a constant coefficients, one dimensional advection-dispersion equation, analytical solutions can be found by recourse to Fourier transform methods [11][12][13]. Many other problems require, instead, a treatment with variable coefficients [22].
In astrophysics, great activity revolves around the development of super diffusive models for the transport of electrons and protons in the heliosphere [19][20][21]. This sort of transport displays a power-law growth of the mean square displacement of the diffusing particles, x 2 ∝ t α , with α > 1 (see, for instance, [23]). The special case α = 2 is called ballistic. The limit case α → 1 is that of normal diffusion, described by a Gaussian propagator. Particles associated with violent solar events diffuse in the solar wind, a turbulent scenario that can be taken as statistically homogeneous at large distances from the sun [17]. This entails that the propagator P(x, x , t, t ), describing the probability of finding a particle that has been injected at (x , t ) at the space time location (x, t), depends solely on the differences x − x and t − t . In the super diffusive regime the propagator P(x, x , t, t ) is not Gaussian, and is characterized by power-law tails, emerging as the solution a non local diffusive process, governed by an integral equation. This equation can be cast as a diffusion one, in which the Laplacian is replaced by a term involving fractional derivatives [24]. See also [25][26][27][28][29][30][31], and references therein. An interesting step towards a more accurate analytical treatment of this problem was recently provided by Litvinenko and Effenberger (LE) in [14].

Ultradistributions
A series of papers [32][33][34][35][36] show that the Ultradistribution theory of Sebastiao e Silva [37][38][39] permits a significant advance in the treatment of quantum field theory. In particular, with the use of the convolution of Ultradistributions, one can show that it is possible to define a general product of distributions (a product in a ring with divisors of zero) that sheds new light on the question of the divergences in quantum field theory. Furthermore, ultradistributions of exponential type (UET) are adequate to describe Gamow States and exponentially increasing fields in quantum field theory [41][42][43].
Other papers [44][45][46] demonstrated that Ultradistributions of Exponential type provide an adequate framework for a consistent treatment of string and string field theories. In particular, a general state of the closed string is represented by UET of compact support, and as a consequence the string field is a linear combination of UET of compact support. Moreover, five recent papers [47][48][49][50][51] show that Ultradistributions can be used to develop in a consistent way the so-called non-extensive statistical mechanics, allowing for an adequate definition of q-Fourier and q-Laplace transforms, and for the removal of divergences of this theory.
Ultradistributions also have the advantage of being representable by means of analytic functions. In general, they are easy to work with and, as we shall see, have interesting properties. One of those properties is that Schwartz's tempered distributions are canonical and continuously injected into Ultradistributions. Another interesting property is that the space of UET is reflexive under the operation of Fourier transform (in a similar way of tempered distributions of Schwartz)

Our Goal
In this paper we wish to show that Ultradistributions provide an adequate tool for a consistent treatment of a fractional differential diffusion-advection equation.
A more conventional handling of this equation is given in [15]. The present treatment is of a much more general character. This paper is organized as follows: In Sect. 2, we summarize a set of mathematical concepts, while, in Sect. 3 we formulate the problem to be addressed herein. We obtain in Sect. 4 a general solution of the fractional diffusion-advection equation, our main result . In Sect. 5, we discuss the so-called weak diffusion approximation and in Sect. 6 we analyze an important change of variables. Some conclusions are drawn in Sect. 1. The appendices 1, 2, 3 and 4 give the fundamentals of the mathematical theory used in this work.

Formulation of the Problem
The authors of [14] have proposed the equation: where t > 0, for the distribution function f (x, t). In their specific case, f refers to solar cosmic-ray transport. They used the following definition of fractional derivative (see [23]): To solve (2.1) the authors use the Green function given by Using this Green function, the solution of (2.1), with the initial condition f (x, 0) = 0, can be written as The solution to the above problem is well posed, except for one major problem: the fractional derivative used is not defined for λ = 1 and does not coincide for this value of λ with the usual derivative defined by Newton and Leibniz. We will solve in this paper this serious problem by recourse to a definition of fractional derivative valid for all values of λ, both real or complex, and matching things for λ ∈ N (N = the set of natural numbers), with the usual derivative defined by Newton and Leibniz. To achieve this goal we use the definition given in [32] for distributions of exponential type and extended in our Appendix 3 to ultradistributions of exponential type.
An interesting property of this fractional derivative is that it unifies in a single operation the operations of derivation and indefinite integration, for any real or complex value of λ.

General Solutions
To solve (2.1) we divide the problem into two parts: Our solution will be valid for all values of λ such that |e κ(−ik) λ | ≤ |e κk |. For the remaining possible values of λ, the solution is obtained via analytic prolongation. In fact, for these values, the solutions of the equations above become exponentially growing ones, which forces one to appeal to ultradistributions of exponential type extending these equations to the complex plane. Thus, we have for 1) and for 2) Using now the complex Fourier transformation we can obtain the solution to our four equations. For 1) one has and for 2) We pass now to find explicit expressions for equations (3.9)-(3.12).

Case (1) x ≥ 0
Expanding e κ(−ik) λ t in power series we obtain, for (3.10), Each term of the sum in (3.13) is a tempered ultradistribution (see Appendix 1). We go then to the real axis and evaluate the cut along it. Thus, Eq. (3.14) can be cast in the following equivalent form The integrals given in (3.16) have been calculated in [53]. One has (3.17) By recourse to (2.4) we have for f (3.18) Using (7.49,) we obtain the general solution for From (3.12), and expanding e (ik) λ in power series, G adopts the form Each term of the sum in (3.20) is, again, a tempered ultradistribution. Thus, proceeding as in the case x ≥ 0, we obtain Eq. (3.22) can be rewritten as Using [53] we then have As it should, G vanishes for x ≥ 0. From (2.4), we can write f as and, according to (7.49), f is finally given by (3.26) Figure 1 depicts a typical graph for solutions of Eq. (3.19). We do this for different times t. The main result of this contribution is that these curves are very well approximated by power laws. Figure 2 (κ = 0.5), 3 (κ = 1, strong diffusion), and 4 (κ = 1.5, strong diffusion ) show how the above solutions can be adjusted by power laws of the form 1/x β : the corresponding lines rotate, for fixed λ = 3/2, in the plane f (x, t) versus 1/x β when at changes, as the plots clearly illustrate (Figs. 3, 4). In all our figures we have taken a = 1 and λ = 3/2. Figure 5 displays a possible quasi-linear relationship between the power-law exponent β and t.

A Useful Approximation
Following [14], we shall now consider a weak diffusion approximation. Within this approximation, we can treat κ as a small parameter and expand f up to order one [14]. Thus, we can write for where f 0 is given by By recourse to [56], we can express F(1, 1; 2; z) in terms of elementary functions, i.e., and obtain for f 0 the expression For f 1 we have Using again the result (7.55) one has f 1 and then Thus, we have for f in the weak diffusion approximation For x > 0 f this becomes simplified and one has (4.8) We can now distinguish two limiting cases. The first one is the asymptotic situation x >> at.
In this instance The second case is 0 < x << at. For it we have (4.10) For x < 0 we have for f 0 the same expression obtained for the case x ≥ 0, and thus, for f 1 , As a consequence, we have for f (4.12) For x + at < 0, (4.2) adopts the form (4.13) (4.14) Another special situation arises when x < 0, x + at > 0, and x << −at. In this case, from (4.12) we deduce the following expression for f : We can see below graphs corresponding to Eqs. (4.9), and (4.10). Note the similarity between these graphs and the graph displayed in Fig. 1, that represents the general solution. This fact confirms our assertion about the power-law behaviour of the general solution (Figs. 6, 7).

Change of Frame
Assume that, in the solar wind rest frame, the particles' transport is represented by the fractional-diffusion equation (FDE) without advection term (a = 0 in (2.1)). The shock front, started at x 0 = −V sh t 0 , moves with constant speed V sh . It is considered as highly localized in the x-coordinate) and constitutes the source of the particles. Then we face an FDE with a uniformly moving Dirac's delta source of the form δ(x − V sh t). So as to have a stationary delta source, we require performing a suitable coordinates-change, reformulating our task in a reference frame where the shock front is stationary. We also modify the timeorigin so that the source begins being active at t = 0. In such a modified reference frame, the transport equation acquires an advection term with velocity a = V sh , and a stationary source δ(0), that begins at t = 0. After solving the diffusion-advection equation in this frame, one expresses the solution in terms of the original coordinates associated with the solar wind rest frame. Such step is briefly described by the 3 correspondences a → v sh , t → t + t 0 , and x → x − v sh t. Consequently, Eqs. (3.19) and (3.26) acquire the form, for And for x < 0: Thus, in the weak diffusion approach of (4.7) and (4.12), we have for

Conclusions
By recourse to ultradistributions, we have provided here an explicit analytical solution for an advection-diffusion equation (ADE) involving fractional derivatives. First, we devised a generalized treatment for these derivatives that includes the normal case.
We also found the exact solution for the ADE both in the x-configuration space and in the associated k-space, that are related via a Fourier transform. Our solution allows us to obtain in a unified and systematic fashion all the different approximations that were introduced in [14], each one in a distinct manner. We achieve in this way a great degree of generality. We conclude from our analysis that the solutions of the fractional diffusion-advection equations are essentially power laws.

Distributions of Exponential Type
For the benefit of the reader, we present here a brief description of the main properties of Tempered Ultradistributions and of ultradistributions of exponential type.

Notations
The notations are almost textually taken from Ref. [38]. Let R n (respectively C n ) be the real (respectively complex) n-dimensional space whose points are denoted by x = (x 1 , x 2 , . . . , x n ) (resp. z = (z 1 , z 2 , . . . , z n )). We shall use the following notations: Consider the set of n-tuples of natural numbers N n . If p ∈ N n , then p = ( p 1 , p 2 , . . . , p n ), where p j is a natural number, 1 j n. p + q denote ( p 1 + q 1 , p 2 + q 2 , . . . , p n + q n ) and p q means p 1 q 1 , p 2 q 2 , . . . , p n q n . x p means x We denote by | p |= n j=1 p j and by D p we understand the differential operator The space H of test functions such that e p|x| |D q φ(x)| is bounded for any natural numbers p and q is defined (Ref. [38]) by means of the countably set of norms: According to reference [54] H is a K{M p } space with: K{e ( p−1)|x| } complies condition (N ) of Guelfand (Ref. [55]). It is a countable Hilbert and nuclear space: where H p is obtained by completing H with the norm induced by the scalar product: where dx = dx 1 dx 2 . . . dx n If we take the conventional scalar product: then H, completed with (7.5), is the Hilbert space H of square integrable functions. By definition, the space of continuous linear functionals defined on H is the space ∞ of the distributions of the exponential type (Ref. [38]).
The Fourier transform of a distribution of exponential typeF is given by (see [37,38]): (7.6) where F is the corresponding tempered ultradistribution (see the next subsection).
The triplet is a rigged Hilbert space (or a Guelfand's triplet [55]). Moreover, we have: H ⊂ S ⊂ H ⊂ S ⊂ ∞ , where S is the Schwartz space of rapidly decreasing test functions (ref [40]).
Any Rigged Hilbert Space G = ( , H, ) has the fundamental property that a linear and symmetric operator on , which admits an extension to a self-adjoint operator in H, has a complete set of generalized eigenfunctions in with real eigenvalues.

Tempered Ultradistributions
The Fourier transform of a functionφ ∈ H is Here φ(z) is entire analytic and rapidly decreasing on straight lines parallel to the real axis.
We call H the set of all such functions.
It is a Z {M p } countably normed and complete space (Ref. [54]), with: H is a nuclear space defined with the norms: where V k = {z = (z 1 , z 2 , . . . , z n ) ∈ C n :| I mz j | k, 1 j n} We can define the habitual scalar product: where: and dz = dz 1 dz 2 . . . dz n By completing H with the norm induced by (7.12) we obtain the Hilbert space of square integrable functions.
The dual of H is the space U of tempered ultradistributions (Refs. [37,38]). Namely, a tempered ultradistribution is a continuous linear functional defined on the space H of entire functions rapidly decreasing on straight lines parallel to the real axis.
The set U = (H, H, U ) is also a Rigged Hilbert Space.
Moreover, we have: H ⊂ S ⊂ H ⊂ S ⊂ U . U can also be characterized in the following way (Ref. [38]): let A ω be the space of all functions F(z) such that: Let be the set of all z-dependent pseudo-polynomials, z ∈ C n . Then U is the quotient space: C) U = A ω / By a pseudo-polynomial we denote a function of z of the form s z s j G(z 1 , . . . , z j−1 , z j+1 , . . . , z n ) with G(z 1 , . . . , z j−1 , z j+1 , . . . , z n ) ∈ A ω Due to these properties it is possible to represent any ultradistribution as (Ref. [38]): . . n and where the path j runs parallel to the real axis from −∞ to ∞ for I m(z j ) > ζ, ζ > p and back from ∞ to −∞ for I m(z j ) < −ζ , −ζ < − p. ( surrounds all the singularities of F(z)). Formula (7.13) will be our fundamental representation for a tempered ultradistribution. Sometimes use will be made of "Dirac Formula" for ultradistributions (Ref. [37]): where the "density" f (t) is the cut of F(z) along the real axis and satisfy: While F(z) is analytic on , the density f (t) is in general singular, so that the r.h.s. of (7.15) should be interpreted in the sense of distribution theory. Another important property of the analytic representation is the fact that on , F(z) is bounded by a power of z (Ref. [38]): |F(z)| ≤ C|z| p (7.16) where C and p depend on F. The representation (7.15) implies that the addition of a pseudo-polynomial P(z) to F(z) do not alter the ultradistribution: But: as P(z)φ(z) is entire analytic in some of the variables z j (and rapidly decreasing), The inverse Fourier transform of (7.6) is given by:

Appendix 2 Ultradistributions of Exponential Type
Consider the Schwartz space of rapidly decreasing test functions S. Let j be the region of the complex plane defined as: According to Ref. [37,39] be the space of test functionsφ ∈ V j is constituted by the set of all entire analytic functions of S for which is finite.
The space Z is then defined as: It is a complete countably normed space with the topology generated by the set of seminorms {|| · || j } j∈N . The topological dual of Z, denoted by B, is by definition the space of ultradistributions of exponential type (Ref. [37,39]). Let S be the space of rapidly decreasing sequences. According to Ref. [55] S is a nuclear space. We consider now the space of sequences P generated by the Taylor development ofφ ∈ Z The norms that define the topology of P are given by: P is a subspace of S and as consequence is a nuclear space. The norms || · || j and || · || p are equivalent, the correspondence Z ⇐⇒ P (7.24) is an isomorphism and therefore Z is a countably normed nuclear space. We define now the set of scalar products This scalar product induces the norm The norms || · || j and || · || n are equivalent, and therefore Z is a countably hilbertian nuclear space. Thus, if we call now Z p the completion of Z by the norm p given in (7.26), we have: is the Hilbert space of square integrable functions. As a consequence the triplet is also a Guelfand's triplet.
B can also be characterized in the following way (refs. [37],tp8): let E ω be the space of all functionsF(z) such that: A)F(z) is an analytic function for {z ∈ C : |I m(z)| > p}. B)-F(z)e − p| (z)| /z p is a bounded continuous function in {z ∈ C : |I m(z)| p}, where p = 0, 1, 2, . . . depends onF(z). Let N be: N = {F(z) ∈ E ω :F(z) is entire analytic}. Then B is the quotient space: Due to these properties it is possible to represent any ultradistribution of exponential type as (Ref. [37,39] where the path runs parallel to the real axis from −∞ to ∞ for I m(z) > ζ, ζ > p and back from ∞ to −∞ for I m(z) < −ζ , −ζ < − p. ( surrounds all the singularities ofF(z)). Formula (7.30) will be our fundamental representation for a ultradistribution of exponential type. The "Dirac Formula" for ultradistributions of exponential type is (Ref. [37,39]): where the "density"f (t) is such that (7.31) should be used carefully. WhileF(z) is analytic function on , the densityf (t) is in general singular, so that the right hand side of (7.32) should be interpreted again in the sense of distribution theory. Another important property of the analytic representation is the fact that on ,F(z) is bounded by a exponential and a power of z (Ref. [37,39]): |F(z)| ≤ C|z| p e p| (z)| (7.33) where C and p depend onF. The representation (7.30) implies that the addition of any entire functionĜ(z) ∈ N tô F(z) does not alter the ultradistribution: But: Ĝ (z)φ(z) dz = 0 asĜ(z)φ(z) is an entire analytic function, ∴ {F(z) +Ĝ(z)}φ(z) dz = F (z)φ(z) dz (7.34) Another very important property of B is that B is reflexive under the Fourier transform: where the complex F ourier transform F(k) ofF(z) ∈ B is given by: Here + is the part of with (z) ≥ 0 and − is the part of with (z) ≤ 0 Using (7.36) we can interpret Dirac's Formula as: The inverse Fourier transform corresponding to (7.37) is given by: The treatment for ultradistributions of exponential type defined on C n is similar to the case of one variable. Thus let j be given as j = z = (z 1 , z 2 , . . . , z n ) ∈ C n : | (z k )| ≤ j 1 ≤ k ≤ n (7.39) and where D (k) = ∂ (k 1 ) ∂ (k 2 ) · · · ∂ (k n ) k = k 1 + k 2 + · · · + k n B n is characterized as follows. Let E n ω be the space of all functionsF(z) such that: A )F(z) is analytic for {z ∈ C n : |I m(z 1 )| > p, |I m(z 2 )| > p, . . . , |I m(z n )| > p}. where = 1 ∪ 2 ∪ . . . n and where the path j runs parallel to the real axis from −∞ to ∞ for I m(z j ) > ζ, ζ > p and back from ∞ to −∞ for I m(z j ) < −ζ , −ζ < − p. (Again the path surrounds all the singularities ofF(z) ). The n-dimensional Dirac's Formula is nowF (7.42) and the "density"f (t) is such that

B )F(z)e
The modulus ofF(z) is bounded by |F(z)| ≤ C|z| p e p n j=1 | (z j )| (7.44) where C and p depend onF.