Asymptotic expansions for Riesz fractional derivatives of Airy functions and applications

Riesz fractional derivatives of a function, $D_{x}^{\alpha}f(x)$ (also called Riesz potentials), are defined as fractional powers of the Laplacian. Asymptotic expansions for large $x$ are computed for the Riesz fractional derivatives of the Airy function of the first kind, $Ai(x)$, and the Scorer function, $Gi(x)$. Reduction formulas are provided that allow one to express Riesz potentials of products of Airy functions, $D_{x}^{\alpha}\left\{ Ai(x)Bi(x)\right\}$ and $D_{x}^{\alpha}\left\{ Ai^{2}(x)\right\}$, via $D_{x}^{\alpha}Ai(x)$ and $D_{x}^{\alpha}Gi(x)$. Here $Bi(x)$ is the Airy function of the second type. Integral representations are presented for the function $A_{2}\left(a,b;x\right)=Ai\left(x-a\right)Ai\left(x-b\right)$ with $a, b\in\mathbb{R}$ and its Hilbert transform. Combined with the above asymptotic expansions they can be used for computing asymptotics of the Hankel transform of $D_{x}^{\alpha}\left\{ A_{2}\left(a,b;x\right)\right\}$. These results are used for obtaining the weak rotation approximation for the Ostrovsky equation (asymptotics of the fundamental solution of the linearized Cauchy problem as the rotation parameter tends to zero)

Asymptotic expansions for Riesz fractional derivatives of Airy functions and applications

Riesz fractional derivatives of a function, $D_{x}^{\alpha}f(x)$ (also called Riesz potentials), are defined as fractional powers of the Laplacian. Asymptotic expansions for large $x$ are computed for the Riesz fractional derivatives of the Airy function of the first kind, $Ai(x)$, and the Scorer function, $Gi(x)$. Reduction formulas are provided that allow one to express Riesz potentials of products of Airy functions, $D_{x}^{\alpha}\left\{ Ai(x)Bi(x)\right\}$ and $D_{x}^{\alpha}\left\{ Ai^{2}(x)\right\}$, via $D_{x}^{\alpha}Ai(x)$ and $D_{x}^{\alpha}Gi(x)$. Here $Bi(x)$ is the Airy function of the second type. Integral representations are presented for the function $A_{2}\left(a,b;x\right)=Ai\left(x-a\right)Ai\left(x-b\right)$ with $a, b\in\mathbb{R}$ and its Hilbert transform. Combined with the above asymptotic expansions they can be used for computing asymptotics of the Hankel transform of $D_{x}^{\alpha}\left\{ A_{2}\left(a,b;x\right)\right\}$. These results are used for obtaining the weak rotation approximation for the Ostrovsky equation (asymptotics of the fundamental solution of the linearized Cauchy problem as the rotation parameter tends to zero)

Asymptotic expansions for Riesz potentials of Airy functions and their products

Riesz potentials of a function are defined as fractional powers of the Laplacian. Asymptotic expansions for $x\to\pm\infty$ are derived for the Riesz potentials of the Airy function $Ai(x)$ and the Scorer function $Gi(x)$. Reduction formulas are provided that allow to compute Riesz potentials of the products of Airy functions $Ai^2(x)$ and $Ai(x)Bi(x)$, where $Bi(x)$ is the Airy function of the second type, via the Riesz potentials of $Ai(x)$ and $Gi(x)$. Integral representations are given for the function $A_{2}\left(a,b;x\right)=Ai\left(x-a\right)Ai\left(x-b\right)$ with $a, b\in \mathbf{R}$, and its Hilbert transform. Combined with the above asymptotic expansions they can be used for obtaining asymptotics of the Hankel transform of Riesz potentials of $A_{2}(a,b;x)$. The study of the above Riesz fractional derivatives can be used for establishing new properties of Korteweg-de Vries-type equations

Pseudoprocesses related to space-fractional higher-order heat-type equations

In this paper we construct pseudo random walks (symmetric and asymmetric) which converge in law to compositions of pseudoprocesses stopped at stable subordinators. We find the higher-order space-fractional heat-type equations whose fundamental solutions coincide with the law of the limiting pseudoprocesses. The fractional equations involve either Riesz operators or their Feller asymmetric counterparts. The main result of this paper is the derivation of pseudoprocesses whose law is governed by heat-type equations of real-valued order $\gamma>2$. The classical pseudoprocesses are very special cases of those investigated here

Time-changed processes governed by space-time fractional telegraph equations

In this work we construct compositions of processes of the form \bm{S}_n^{2\beta}(c^2 \mathpzc{L}^\nu (t) \r, t>0, \nu \in (0, 1/2], \beta \in (0,1], n \in \mathbb{N}, whose distribution is related to space-time fractional n-dimensional telegraph equations. We present within a unifying framework the pde connections of n-dimensional isotropic stable processes \bm{S}_n^{2\beta} whose random time is represented by the inverse \mathpzc{L}^\nu (t), t>0, of the superposition of independent positively-skewed stable processes, \mathpzc{H}^\nu (t) = H_1^{2\nu} (t) + (2\lambda \r^{\frac{1}{\nu}} H_2^\nu (t), t>0, (H_1^{2\nu}, H_2^\nu, independent stable subordinators). As special cases for n=1, \nu = 1/2 and \beta = 1 we examine the telegraph process T at Brownian time B (Orsingher and Beghin) and establish the equality in distribution B (c^2 \mathpzc{L}^{1/2} (t)) \stackrel{\textrm{law}}{=} T (|B(t)|), t>0. Furthermore the iterated Brownian motion (Allouba and Zheng) and the two-dimensional motion at finite velocity with a random time are investigated. For all these processes we present their counterparts as Brownian motion at delayed stable-distributed time.Comment: 34 page

Stable distributions and pseudo-processes related to fractional Airy functions

In this paper we study pseudo-processes related to odd-order heat-type equations composed with L\'evy stable subordinators. The aim of the article is twofold. We first show that the pseudo-density of the subordinated pseudo-process can be represented as an expectation of damped oscillations with generalized gamma distributed parameters. This stochastic representation also arises as the solution to a fractional diffusion equation, involving a higher-order Riesz-Feller operator, which generalizes the odd-order heat-type equation. We then prove that, if the stable subordinator has a suitable exponent, the time-changed pseudo-process becomes a genuine L\'evy stable process. This result permits us to obtain a power series representation for the probability density function of an arbitrary asymmetric stable process of exponent $\nu>1$ and skewness parameter $\beta$, with $0<|\beta|<1$. The methods we use in order to carry out our analysis are based on the study of a fractional Airy function which arises in the study of the higher-order Riesz-Feller operator

Spatially fractional-order viscoelasticity, non-locality and a new kind of anisotropy

Spatial non-locality of space-fractional viscoelastic equations of motion is studied. Relaxation effects are accounted for by replacing second-order time derivatives by lower-order fractional derivatives and their generalizations. It is shown that space-fractional equations of motion of an order strictly less than 2 allow for a new kind anisotropy, associated with angular dependence of non-local interactions between stress and strain at different material points. Constitutive equations of such viscoelastic media are determined. Explicit fundamental solutions of the Cauchy problem are constructed for some cases isotropic and anisotropic non-locality