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

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Abstract

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.

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Correspondence to M. C. Rocca.

Appendices

Appendix 1

1.1 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 \(\varvec{{\mathbb {R}}^n}\) (respectively \(\varvec{{\mathbb {C}}^n}\)) be the real (respectively complex) n-dimensional space whose points are denoted by \(x=(x_1,x_2,\ldots ,x_n)\) (resp. \(z=(z_1,z_2,\ldots ,z_n)\)). We shall use the following notations:

  1. (i)

    \(x+y=(x_1+y_1,x_2+y_2,\ldots ,x_n+y_n)\) ; \(\alpha x=(\alpha x_1,\alpha x_2,\ldots ,\alpha x_n)\)

  2. (ii)

    \(x\geqq 0\) means \(x_1\geqq 0, x_2\geqq 0,\ldots ,x_n\geqq 0\)

  3. (iii)

    \(x\cdot y=\sum \limits _{j=1}^n x_j y_j\)

  4. (iv))

    \(\mid x\mid =\sum \limits _{j=1}^n \mid x_j\mid \)

Consider the set of n-tuples of natural numbers \(\varvec{{\mathbb {N}}^n}\). If \(p\in \varvec{{\mathbb {N}}^n}\), then \(p=(p_1, p_2,\ldots ,p_n)\), where \(p_j\) is a natural number, \(1\leqq j\leqq n\). \(p+q\) denote \((p_1+q_1, p_2+q_2,\ldots , p_n+q_n)\) and \(p\geqq q\) means \(p_1\geqq q_1, p_2\geqq q_2,\ldots ,p_n\geqq q_n\). \(x^p\) means \(x_1^{p_1}x_2^{p_2}\ldots x_n^{p_n}\). We denote by \(\mid p\mid =\sum \limits _{j=1}^n p_j \) and by \(D^p\) we understand the differential operator \({\partial }^{p_1+p_2+\cdots +p_n}/\partial {x_1}^{p_1} \partial {x_2}^{p_2}\ldots \partial {x_n}^{p_n}\).

For any natural number k we define \(x^k=x_1^k x_2^k\ldots x_n^k\) and \({\partial }^k/\partial x^k= {\partial }^{nk}/\partial x_1^k\partial x_2^k\ldots \partial x_n^k\)

The space \(\varvec{\mathcal{H}}\) of test functions such that \(e^{p|x|}|D^q\phi (x)|\) is bounded for any natural numbers p and q is defined (Ref. [38]) by means of the countably set of norms:

$$\begin{aligned} {\Vert \hat{\phi }\Vert }_p=\sup _{0\le q\le p,\,x} e^{p|x|} \left| D^q \hat{\phi } (x)\right| \quad p=0,1,2, \ldots \end{aligned}$$
(7.1)

According to reference[54] \(\varvec{\mathcal{H}}\) is a space with:

$$\begin{aligned} M_p(x)=e^{(p-1)|x|}\quad p=1,2, \ldots \end{aligned}$$
(7.2)

complies condition \(\varvec{(}\mathcal{N})\) of Guelfand (Ref. [55]). It is a countable Hilbert and nuclear space:

(7.3)

where \(\varvec{\mathcal{H}_p}\) is obtained by completing \(\varvec{\mathcal{H}}\) with the norm induced by the scalar product:

$$\begin{aligned} {\langle \hat{\phi }, \hat{\psi }\rangle }_p = \int \limits _{-\infty }^{\infty } e^{2(p-1)|x|} \sum \limits _{q=0}^p D^q \overline{\hat{\phi }} (x) D^q \hat{\psi } (x)dx\quad p=1,2,\ldots \end{aligned}$$
(7.4)

where \(dx=dx_1\;dx_2\ldots dx_n\)

If we take the conventional scalar product:

$$\begin{aligned} <\hat{\phi }, \hat{\psi }>\,= \int \limits _{-\infty }^{\infty } \overline{\hat{\phi }}(x) \hat{\psi }(x)\;dx \end{aligned}$$
(7.5)

then \(\varvec{\mathcal{H}}\), completed with (7.5), is the Hilbert space \(\varvec{H}\) of square integrable functions.

By definition, the space of continuous linear functionals defined on \(\varvec{\mathcal{H}}\) is the space \(\varvec{{\Lambda }_{\infty }}\) of the distributions of the exponential type (Ref. [38]).

The Fourier transform of a distribution of exponential type \(\hat{F}\) is given by (see [37, 38]):

$$\begin{aligned} F(k)&=\int \limits _{-\infty }^{\infty } H[\mathfrak {I}(k)]H[\mathfrak {R}(x)-H[-\mathfrak {I}(k)]H[-\mathfrak {R}(x)]\hat{F}(x) e^{ikx}\;dx\nonumber \\&= H[\mathfrak {I}(k)]\int \limits _0^{\infty }\hat{F}(x)e^{ikx}- H[-\mathfrak {I}(k)]\int \limits _{-\infty }^0\hat{F}(x)e^{ikx} \end{aligned}$$
(7.6)

where F is the corresponding tempered ultradistribution (see the next subsection).

The triplet

(7.7)

is a rigged Hilbert space (or a Guelfand’s triplet [55]).

Moreover, we have: \(\varvec{\mathcal{H}}\subset \varvec{\mathcal{S}} \subset \varvec{H}\subset \varvec{\mathcal{S}^{'}}\subset \varvec{{\Lambda }_{\infty }}\), where \(\varvec{\mathcal{S}}\) is the Schwartz space of rapidly decreasing test functions (ref[40]).

Any Rigged Hilbert Space has the fundamental property that a linear and symmetric operator on \(\varvec{\Phi }\), which admits an extension to a self-adjoint operator in \(\varvec{H}\), has a complete set of generalized eigenfunctions in \(\varvec{{\Phi }^{'}}\) with real eigenvalues.

1.2 Tempered Ultradistributions

The Fourier transform of a function \(\hat{\phi }\in \varvec{\mathcal{H}}\) is

$$\begin{aligned} \phi (z)=\frac{1}{2\pi } \int \limits _{-\infty }^{\infty }\overline{\hat{\phi }}(x)\;e^{iz\cdot x}\;dx \end{aligned}$$
(7.8)

Here \(\phi (z)\) is entire analytic and rapidly decreasing on straight lines parallel to the real axis. We call \(\varvec{{\mathfrak H}}\) the set of all such functions.

$$\begin{aligned} \varvec{{\mathfrak H}}=\mathcal{F}\left\{ \varvec{\mathcal{H}}\right\} \end{aligned}$$
(7.9)

It is a countably normed and complete space (Ref. [54]), with:

$$\begin{aligned} M_p(z)= (1+|z|)^p \end{aligned}$$
(7.10)

\(\varvec{{\mathfrak H}}\) is a nuclear space defined with the norms:

$$\begin{aligned} {\Vert \phi \Vert }_{pn} = \sup _{z\in V_n} {\left( 1+|z|\right) }^p |\phi (z)| \end{aligned}$$
(7.11)

where \(V_k=\{z=(z_1,z_2,\ldots ,z_n)\in \varvec{{\mathbb {C}}^n}: \mid Im z_j\mid \leqq k, 1\leqq j \leqq n\}\)

We can define the habitual scalar product:

$$\begin{aligned} \langle \,\phi (z), \psi (z)\, \rangle =\int \limits _{-\infty }^{\infty } \phi (z) {\psi }_1(z)dz = \int \limits _{-\infty }^{\infty } \overline{\hat{\phi }}(x) \hat{\psi }(x)\;dx \end{aligned}$$
(7.12)

where:

$$\begin{aligned} {\psi }_1(z)=\int \limits _{-\infty }^{\infty } \hat{\psi }(x)\; e^{-iz\cdot x}\;dx \end{aligned}$$

and \(dz=dz_1\;dz_2\ldots dz_n\)

By completing \(\varvec{{\mathfrak H}}\) with the norm induced by (7.12) we obtain the Hilbert space of square integrable functions.

The dual of \(\varvec{{\mathfrak H}}\) is the space \(\varvec{\mathcal{U}}\) of tempered ultradistributions (Refs. [37, 38]). Namely, a tempered ultradistribution is a continuous linear functional defined on the space \(\varvec{{\mathfrak H}}\) of entire functions rapidly decreasing on straight lines parallel to the real axis.

The set is also a Rigged Hilbert Space.

Moreover, we have: \(\varvec{{\mathfrak H}}\subset \varvec{\mathcal{S}} \subset \varvec{H}\subset \varvec{\mathcal{S}^{'}}\subset \varvec{\mathcal{U}}\).

\(\varvec{\mathcal{U}}\) can also be characterized in the following way (Ref. [38]): let \(\varvec{\mathcal{A}_{\omega }}\) be the space of all functions F(z) such that:

\({\varvec{A)}}\) F(z) is analytic on the set \(\{z\in \varvec{{\mathbb {C}}^n}: |Im(z_1)|>p, |Im(z_2)|>p,\ldots ,|Im(z_n)|>p\}\).

\({\varvec{B)}}\) \(F(z)/z^p\) is bounded continuous in \(\{z\in \varvec{{\mathbb {C}}^n} :|Im(z_1)|\geqq p,|Im(z_2)|\geqq p,\ldots ,|Im(z_n)|\geqq p\}\), where \(p=0,1,2,\ldots \) depends on F(z).

Let \(\varvec{\Pi }\) be the set of all z-dependent pseudo-polynomials, \(z\in \varvec{{\mathbb {C}}^n}\). Then \(\varvec{\mathcal{U}}\) is the quotient space:

\({\varvec{C)}}\) \(\varvec{\mathcal{U}}=\varvec{\mathcal{A}_{\omega }/\Pi }\)

By a pseudo-polynomial we denote a function of z of the form

\(\sum _s z_j^s G(z_1,\ldots ,z_{j-1},z_{j+1},\ldots ,z_n)\) with \(G(z_1,\ldots ,z_{j-1},z_{j+1},\ldots ,z_n)\in \varvec{\mathcal{A}_{\omega }}\)

Due to these properties it is possible to represent any ultradistribution as (Ref. [38]):

$$\begin{aligned} F(\phi )=\langle F(z), \phi (z)\rangle =\oint \limits _{\Gamma } F(z) \phi (z)\;dz \end{aligned}$$
(7.13)

where \(\Gamma ={\Gamma }_1\cup {\Gamma }_2\cup \ldots {\Gamma }_n\) and where the path \({\Gamma }_j\) runs parallel to the real axis from \(-\infty \) to \(\infty \) for \(Im(z_j)>\zeta \), \(\zeta >p\) and back from \(\infty \) to \(-\infty \) for \(Im(z_j)<-\zeta \), \(-\zeta <-p\). (\(\Gamma \) 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]):

$$\begin{aligned} F(z)=\frac{1}{(2\pi i)^n}\int \limits _{-\infty }^{\infty } \frac{f(t)}{(t_1-z_1)(t_2-z_2)\ldots (t_n-z_n)}\;dt \end{aligned}$$
(7.14)

where the “density” f(t) is the cut of F(z) along the real axis and satisfy:

$$\begin{aligned} \oint \limits _{\Gamma } F(z) \phi (z)\;dz = \int \limits _{-\infty }^{\infty } f(t) \phi (t)\;dt \end{aligned}$$
(7.15)

While F(z) is analytic on \(\Gamma \), 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 \(\Gamma \), F(z) is bounded by a power of z (Ref. [38]):

$$\begin{aligned} |F(z)|\le C|z|^p \end{aligned}$$
(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:

$$\begin{aligned} \oint \limits _{\Gamma }\{F(z)+P(z)\}\phi (z)\;dz= \oint \limits _{\Gamma } F(z)\phi (z)\;dz+\oint \limits _{\Gamma } P(z)\phi (z)\;dz \end{aligned}$$

But:

$$\begin{aligned} \oint \limits _{\Gamma } P(z)\phi (z)\;dz=0 \end{aligned}$$

as \(P(z)\phi (z)\) is entire analytic in some of the variables \(z_j\) (and rapidly decreasing),

$$\begin{aligned} \therefore \;\;\;\;\oint \limits _{\Gamma } \{F(z)+P(z)\}\phi (z)\;dz= \oint \limits _{\Gamma } F(z)\phi (z)\;dz \end{aligned}$$
(7.17)

The inverse Fourier transform of (7.6) is given by:

$$\begin{aligned} \hat{F}(x)=\frac{1}{2\pi }\oint \limits _{\Gamma }F(k) e^{-ikx}\;dk =\int \limits _{-\infty }^{\infty }f(k)e^{-ikx}\;dx \end{aligned}$$
(7.18)

Appendix 2

1.1 Ultradistributions of Exponential Type

Consider the Schwartz space of rapidly decreasing test functions \(\mathcal{S}\). Let \({\Lambda }_j\) be the region of the complex plane defined as:

$$\begin{aligned} {\Lambda }_j=\left\{ z\in \varvec{\mathbb {C}} : |\mathfrak {I}(z)|< j : j\in \varvec{\mathbb {N}}\right\} \end{aligned}$$
(7.19)

According to Ref. [37, 39] be the space of test functions is constituted by the set of all entire analytic functions of \(\mathcal{S}\) for which

$$\begin{aligned} ||\hat{\phi } ||_j=\max _{k\le j}\left\{ \sup _{z\in {\Lambda }_j}\left[ e^{(j|\mathfrak {R}(z)|)} |{\hat{\phi }}^{(k)}(z)|\right] \right\} \end{aligned}$$
(7.20)

is finite.

The space is then defined as:

(7.21)

It is a complete countably normed space with the topology generated by the set of semi-norms \(\{||\cdot ||_j\}_{j\in \mathbb {N}}\). The topological dual of , denoted by , is by definition the space of ultradistributions of exponential type (Ref. [37, 39]). Let be the space of rapidly decreasing sequences. According to Ref. [55] is a nuclear space. We consider now the space of sequences generated by the Taylor development of

(7.22)

The norms that define the topology of are given by:

$$\begin{aligned} ||\hat{\phi } ||^{'}_p=\sup _n \frac{n^p}{n!} |{\hat{\phi }}^n(0)| \end{aligned}$$
(7.23)

is a subspace of and as consequence is a nuclear space. The norms \(||\cdot ||_j\) and \(||\cdot ||^{'}_p\) are equivalent, the correspondence

(7.24)

is an isomorphism and therefore is a countably normed nuclear space. We define now the set of scalar products

$$\begin{aligned} \langle \hat{\phi }(z),\hat{\psi }(z)\rangle _n&=\sum \limits _{q=0}^n\int \limits _{-\infty }^{\infty }e^{2n|z|} \overline{{\hat{\phi }}^{(q)}}(z){\hat{\psi }}^{(q)}(z)\;dz\nonumber \\&=\,\sum \limits _{q=0}^n\int \limits _{-\infty }^{\infty }e^{2n|x|} \overline{{\hat{\phi }}^{(q)}}(x){\hat{\psi }}^{(q)}(x)\;dx \end{aligned}$$
(7.25)

This scalar product induces the norm

$$\begin{aligned} \left| |\hat{\phi }\right| |_n^{''}=\left[ \langle \hat{\phi }(x),\hat{\phi }(x)\rangle _n\right] ^{\frac{1}{2}} \end{aligned}$$
(7.26)

The norms \(||\cdot ||_j\) and \(||\cdot ||^{''}_n\) are equivalent, and therefore is a countably hilbertian nuclear space. Thus, if we call now the completion of by the norm p given in (7.26), we have:

(7.27)

where

(7.28)

is the Hilbert space of square integrable functions.

As a consequence the triplet

(7.29)

is also a Guelfand’s triplet.

can also be characterized in the following way (refs. [37],tp8): let be the space of all functions \(\hat{F}(z)\) such that:

\({\varvec{A)}}\) \(\hat{F}(z)\) is an analytic function for \(\{z\in \varvec{\mathbb {C}} : |Im(z)|>p\}\).

\({\varvec{B)}}\)- \(\hat{F}(z)e^{-p|\mathfrak {R}(z)|}/z^p\) is a bounded continuous function in \(\{z\in \varvec{\mathbb {C}} :|Im(z)|\geqq p\}\), where \(p=0,1,2,\ldots \) depends on \(\hat{F}(z)\).

Let be: . Then is the quotient space:

\({\varvec{C)}}\)-

Due to these properties it is possible to represent any ultradistribution of exponential type as (Ref. [37, 39]):

$$\begin{aligned} \hat{F}(\hat{\phi })=<\hat{F}(z), \hat{\phi }(z)>=\oint \limits _{\Gamma } \hat{F}(z) \hat{\phi }(z)\;dz \end{aligned}$$
(7.30)

where the path \({\Gamma }\) runs parallel to the real axis from \(-\infty \) to \(\infty \) for \(Im(z)>\zeta \), \(\zeta >p\) and back from \(\infty \) to \(-\infty \) for \(Im(z)<-\zeta \), \(-\zeta <-p\). (\(\Gamma \) surrounds all the singularities of \(\hat{F}(z)\)).

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]):

$$\begin{aligned} \hat{F}(z)\equiv \frac{1}{2\pi i}\int \limits _{-\infty }^{\infty } \frac{\hat{f}(t)}{t-z}\;dt\equiv \frac{\cosh (\lambda z)}{2\pi i}\int \limits _{-\infty }^{\infty } \frac{\hat{f}(t)}{(t-z)\cosh (\lambda t)}\;dt \end{aligned}$$
(7.31)

where the “density” \(\hat{f}(t)\) is such that

$$\begin{aligned} \oint \limits _{\Gamma } \hat{F}(z) \hat{\phi }(z)\;dz = \int \limits _{-\infty }^{\infty } \hat{f}(t) \hat{\phi }(t)\;dt \end{aligned}$$
(7.32)

(7.31) should be used carefully. While \(\hat{F}(z)\) is analytic function on \(\Gamma \), the density \(\hat{f}(t)\) 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 \(\Gamma \), \(\hat{F}(z)\) is bounded by a exponential and a power of z (Ref. [37, 39]):

$$\begin{aligned} |\hat{F}(z)|\le C|z|^pe^{p|\mathfrak {R}(z)|} \end{aligned}$$
(7.33)

where C and p depend on \(\hat{F}\).

The representation (7.30) implies that the addition of any entire function to \(\hat{F}(z)\) does not alter the ultradistribution:

$$\begin{aligned} \oint \limits _{\Gamma }\left\{ \hat{F}(z)+\hat{G}(z)\right\} \hat{\phi }(z)\;dz= \oint \limits _{\Gamma } \hat{F}(z)\hat{\phi }(z)\;dz+\oint \limits _{\Gamma } \hat{G}(z)\hat{\phi }(z)\;dz \end{aligned}$$

But:

$$\begin{aligned} \oint \limits _{\Gamma } \hat{G}(z)\hat{\phi }(z)\;dz=0 \end{aligned}$$

as \(\hat{G}(z)\hat{\phi }(z)\) is an entire analytic function,

$$\begin{aligned} \therefore \;\;\;\;\oint \limits _{\Gamma } \{\hat{F}(z)+\hat{G}(z)\}\hat{\phi }(z)\;dz= \oint \limits _{\Gamma } \hat{F}(z)\hat{\phi }(z)\;dz \end{aligned}$$
(7.34)

Another very important property of is that is reflexive under the Fourier transform:

(7.35)

where the complex F ourier transform F(k) of is given by:

$$\begin{aligned} F(k)&=\,H[\mathfrak {I}(k)]\int \limits _{{\Gamma }_+}\hat{F}(z)e^{ikz}\;dz- H[-\mathfrak {I}(k)]\int \limits _{{\Gamma }_{-}}\hat{F}(z)e^{ikz}\;dz\nonumber \\&=\,\oint \limits _{\Gamma }\{H[\mathfrak {I}(k)H[\mathfrak {R}(z)]-H[-\mathfrak {I}(k)H[-\mathfrak {R}(z)]\} \hat{F}(z)e^{ikz}\;dz\nonumber \\&=\,H[\mathfrak {I}(k)]\int \limits _0^{\infty }\hat{f}(x)e^{ikx}\;dx- H[-\mathfrak {I}(k)]\int \limits _{-\infty }^0\hat{f}(x)e^{ikx}\;dx \end{aligned}$$
(7.36)

Here \({\Gamma }_+\) is the part of \(\Gamma \) with \(\mathfrak {R}(z)\ge 0\) and \({\Gamma }_{-}\) is the part of \(\Gamma \) with \(\mathfrak {R}(z)\le 0\) Using (7.36) we can interpret Dirac’s Formula as:

$$\begin{aligned} F(k)\equiv \frac{1}{2\pi i}\int \limits _{-\infty }^{\infty } \frac{f(s)}{s-k}\; ds\equiv \mathcal{F}_c\left\{ \mathcal{F}^{-1}\left\{ f(s)\right\} \right\} \end{aligned}$$
(7.37)

The inverse Fourier transform corresponding to (7.37) is given by:

$$\begin{aligned} \hat{F}(z)=\frac{1}{2\pi } \oint \limits _{\Gamma }\big \{H[\mathfrak {I}(z)]H[-\mathfrak {R}(k)]-H[-\mathfrak {I}(z)]H[\mathfrak {R}(k)]\big \} F(k)e^{-ikz}\;dk \end{aligned}$$
(7.38)

The treatment for ultradistributions of exponential type defined on \({\varvec{\mathbb {C}}}^n\) is similar to the case of one variable. Thus let \({\Lambda }_j\) be given as

$$\begin{aligned} {\Lambda }_j=\left\{ z=(z_1, z_2,\ldots ,z_n)\in {\varvec{\mathbb {C}}}^n : |\mathfrak {I}(z_k)|\le j\;\;\;1\le k\le n\right\} \end{aligned}$$
(7.39)

and

$$\begin{aligned} ||\hat{\phi } ||_j=\max _{k\le j}\left\{ \sup _{z\in {\Lambda }_j}\left[ e^{j\left[ \sum \limits _{p=1}^n|\mathfrak {R}(z_p)|\right] }\left| D^{(k)}\hat{\phi }(z)\right| \right] \right\} \end{aligned}$$
(7.40)

where \(D^{(k)}={\partial }^{(k_1)}{\partial }^{(k_2)}\cdot \cdot \cdot {\partial }^{(k_n)}\;\;\;\; k=k_1+k_2+\cdot \cdot \cdot +k_n\)

is characterized as follows. Let be the space of all functions \(\hat{F}(z)\) such that:

\({\varvec{A^{'})}}\) \(\hat{F}(z)\) is analytic for

\(\{z\in \varvec{{\mathbb {C}}^n} : |Im(z_1)|>p, |Im(z_2)|>p,\ldots ,|Im(z_n)|>p\}\).

\({\varvec{B^{'})}}\) \(\hat{F}(z)e^{-\left[ p\sum \limits _{j=1}^n|\mathfrak {R}(z_j)|\right] }/z^p\) is bounded continuous in \(\{z\in \varvec{{\mathbb {C}}^n} :|Im(z_1)|\geqq p,|Im(z_2)|\geqq p, \ldots ,|Im(z_n)|\geqq p\}\), where \(p=0,1,2,\ldots \) depends on \(\hat{F}(z)\).

Let be: is entire analytic function at minus in one of the variables \(\left. z_j\;\;\;1\le j\le n\right\} \) Then is the quotient space:

\({\varvec{C^{'})}}\) We have now

$$\begin{aligned} \hat{F}(\hat{\phi })=<\hat{F}(z), \hat{\phi }(z)>=\oint \limits _{\Gamma } \hat{F}(z) \hat{\phi }(z)\; dz \end{aligned}$$
(7.41)

where \(\Gamma ={\Gamma }_1\cup {\Gamma }_2\cup \ldots {\Gamma }_n\) and where the path \({\Gamma }_j\) runs parallel to the real axis from \(-\infty \) to \(\infty \) for \(Im(z_j)>\zeta \), \(\zeta >p\) and back from \(\infty \) to \(-\infty \) for \(Im(z_j)<-\zeta \), \(-\zeta <-p\). (Again the path \(\Gamma \) surrounds all the singularities of \(\hat{F}(z)\) ). The n-dimensional Dirac’s Formula is now

$$\begin{aligned} \hat{F}(z)=\frac{1}{(2\pi i)^n}\int \limits _{-\infty }^{\infty } \frac{\hat{f}(t)}{(t_1-z_1)(t_2-z_2)\ldots (t_n-z_n)}\;dt \end{aligned}$$
(7.42)

and the “density” \(\hat{f}(t)\) is such that

$$\begin{aligned} \oint \limits _{\Gamma } \hat{F}(z)\hat{\phi }(z)\;dz = \int \limits _{-\infty }^{\infty } \hat{f}(t) \hat{\phi }(t)\;dt \end{aligned}$$
(7.43)

The modulus of \(\hat{F}(z)\) is bounded by

$$\begin{aligned} |\hat{F}(z)|\le C|z|^p e^{\left[ p\sum \limits _{j=1}^n|\mathfrak {R}(z_j)|\right] } \end{aligned}$$
(7.44)

where C and p depend on \(\hat{F}\).

Appendix 3

1.1 Fractional Derivative

According to [32] the fractional derivative of a distribution of exponential type \(\hat{F}(x)\) is given by

$$\begin{aligned} \frac{d^{\lambda }\hat{F}(x)}{dx^{\lambda }}=\frac{1}{2\pi }\oint \limits _{\Gamma } (-ik)^{\lambda } F(k) e^{-ik x}\;dk+ \oint \limits _{\Gamma }(-ik)^{\lambda }a(k) e^{-ik x}\;dk \end{aligned}$$
(7.45)

Where a(k) is entire analytic and rapidly decreasing. If \(\lambda =-1\), \(d^{\lambda }/dx^{\lambda }\) is the inverse of the derivative (an integration). In this case the second term of the right side of (7.45) gives a primitive of \(\hat{f}(x)\). Using Cauchy’s theorem the additional term is

$$\begin{aligned} \oint \limits _{\Gamma } \frac{a(k)}{k}e^{-ik x}\;dk= 2\pi a(0) \end{aligned}$$
(7.46)

Of course, an integration should give a primitive plus an arbitrary constant. Analogously when \(\lambda =-2\) (a double iterated integration) we have

$$\begin{aligned} \oint \limits _{\Gamma } \frac{a(k)}{{k}^2}e^{-ik x}\;dk= \gamma +\delta x \end{aligned}$$
(7.47)

where \(\gamma \) and \(\delta \) are arbitrary constants.

For a ultradistribution of exponential type we have for the fractional derivative:

$$\begin{aligned} \frac{{\partial }^{\lambda }\hat{F}(z)}{\partial z^{\lambda }}&=\frac{1}{2\pi } \oint \limits _{\Gamma }\{H[\mathfrak {I}(z)H[-\mathfrak {R}(k)]-H[-\mathfrak {I}(z)H[\mathfrak {R}(k)]\} (-ik)^{\lambda }F(k)e^{-ikz}\;dk\nonumber \\&\quad +\,\oint \limits _{\Gamma }\{H[\mathfrak {I}(z)H[-\mathfrak {R}(k)]-H[-\mathfrak {I}(z)H[\mathfrak {R}(k)]\} (-ik)^{\lambda }a(k)e^{-ikz}\;dk\qquad \end{aligned}$$
(7.48)

where . This fractional derivative behaves similarly to the above-defined for distributions of exponential type.

Unlike all other definitions of fractional derivative, (7.45) and (7.48) are defined for all values of \(\lambda \), real o complex. Furthermore, are the only known definitions that unify derivation and integration in a single operation.

Appendix 4

1.1 Some Useful Formulas Related to the Hypergeometric Function

According to the result given in [57] we can obtain:

$$\begin{aligned} \int \limits _0^t\frac{t^{'n}}{(x+at^{'}\pm i0)^{\lambda n +1}}\;dt^{'}&= \frac{t^{n+1}}{(x\pm i0)^{\lambda n+1}}B(1,n+1)\nonumber \\&\quad \times \,F\left( \lambda n +1, n+1,n+2;-\frac{at}{x\pm i0}\right) . \end{aligned}$$
(7.49)

Using the transformation formula given in [58] for the hypergeometric function

$$\begin{aligned} F(\lambda +1,2;3;z)&=\frac{2\Gamma (1-\lambda )}{\Gamma (2-\lambda )} (-1)^{\lambda +1}z^{-\lambda -1} F\left( \lambda +1,\lambda -1;\lambda ;\frac{1}{z}\right) \nonumber \\&\quad +\, \frac{2\Gamma (\lambda -1)}{\Gamma (\lambda +1)} z^{-2} F\left( 2,0;2-\lambda ;\frac{1}{z}\right) , \end{aligned}$$
(7.50)

with the particular value

$$\begin{aligned} F(a,0;c;z)=1. \end{aligned}$$
(7.51)

we obtain the expression:

$$\begin{aligned} F(\lambda +1,2;3;z)&=\frac{2\Gamma (1-\lambda )}{\Gamma (2-\lambda )} (-1)^{\lambda +1}z^{-\lambda -1} F\left( \lambda +1,\lambda -1;\lambda ;\frac{1}{z}\right) \nonumber \\&\quad +\,\frac{2\Gamma (\lambda -1)}{\Gamma (\lambda +1)}z^{-2} \end{aligned}$$
(7.52)

Now by recourse to the transformation formula [59] we have:

$$\begin{aligned} F\left( \lambda +1,\lambda -1;\lambda ;\frac{1}{z}\right) = \left( 1-\frac{1}{z}\right) ^{-\lambda }F\left( -1,1;\lambda ;\frac{1}{z}\right) , \end{aligned}$$
(7.53)

or equivalently:

$$\begin{aligned} F\left( \lambda +1,\lambda -1;\lambda ;\frac{1}{z}\right) = \frac{z^{\lambda }}{(z-1)^{\lambda }}\left( \frac{\lambda z-1}{\lambda z}\right) . \end{aligned}$$
(7.54)

Thus, we get, finally,

$$\begin{aligned} F(\lambda +1,2;3;z)=\frac{2}{\lambda (\lambda -1)z^2} \left[ 1+\frac{\lambda z-1}{(1-z)^{\lambda }}\right] . \end{aligned}$$
(7.55)

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Rocca, M.C., Plastino, A.R., Plastino, A. et al. New Solution of Diffusion–Advection Equation for Cosmic-Ray Transport Using Ultradistributions. J Stat Phys 161, 986–1009 (2015). https://doi.org/10.1007/s10955-015-1359-x

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