Related papers: One-sided Fractional Integrals and Riesz Potential…
A solution to the more than 300-years old problem of geometric and physical interpretation of fractional integration and differentiation (i.e., integration and differentiation of an arbitrary real order) is suggested for the…
This paper detailedly discusses the locally one-dimensional numerical methods for efficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional…
In the articles [1] and [2] of D. Finch, M. Haltmeier, S. Patch and D. Rakesh inversion formulas were found in any dimension $n\geq2$ for recovering a smooth function with compact support in the unit ball from spherical means centered on…
We consider Sturm-Liouville operators on geometrical graphs without cycles (trees) with singular potentials from the class $W_2^{-1}$. We suppose that the potentials are known on a part of the graph, and study the so-called partial inverse…
The paper contains the inversion formula for the weighted spherical mean. The interest to reconstruction a function by its integral by sphere grews tremendously in the last six decades, stimulated by the spectrum of new problems and methods…
The fractional Laplacian $(-\triangle)^{\gamma/2}$ commutes with the primary coordination transformations in the Euclidean space $\RR^d$: dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes.…
We consider the inverse conductivity problem of identifying embedded objects in unbounded domains. The main tool is a set of special solutions to the Schroedinger equation, the complex spherical waves, which are constructed by a Carleman…
In this paper, a Sturm-Liouville boundary value problem equiped with conformable fractional derivates is considered. We give some uniqueness theorems for the solutions of inverse problems according to the Weyl function, two given spectra…
We prove an extension of the Stein-Weiss weighted estimates for fractional integrals, in the context of $L^{p}$ spaces with different integrability properties in the radial and the angular direction. In this way, the classical estimates can…
A method is described by which a function defined on a cubic grid (as from a finite difference solution of a partial differential equation) can be resolved into spherical harmonic components at some fixed radius. This has applications to…
We obtain $H^{p}_{w} - L^{q}_{w^{q/p}}$ estimates for certain fractional operators.
A general reformulation of classical sharp-edge diffraction theory is proposed within paraxial approximation. The, not so much known, Poincar\'e vector potential construction is employed directly inside Fresnel's 2D integral in order for it…
We formulate an inverse problem for an uncoupled space-time fractional Schr\"odinger equation on closed manifolds. Our main goal is to determine the fractional powers and the Riemannian metric (up to an isometry) simultaneously from the…
We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range power-wise interaction. The corresponding term in dynamical equations is proportional to $1/|n-m|^{\alpha+1}$. It is shown that the equation of…
In cluster physics a single particle potential to determine the microscopic part of the total energy of a collective configuration is necessary to calculate the shell- and pairing effects. In this paper we investigate the properties of the…
Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis…
In this paper we study the solutions of different forms of fractional equations on the unit sphere $\mathbb{S}_{1}^{2}$ $\subset \mathbb{R}^{3}$ possessing the structure of time-dependent random fields. We study the correlation functions of…
It is known that at least ten equivalent definitions of the fractional Laplacian exist in an unbounded domain. Here we derive a further equivalent definition that is based on the Mellin transform and it can be used when the fractional…
The fractional wave equation governs the propagation of mechanical diffusive waves in viscoelastic media which exhibits a power-law creep, and consequently provided a physical interpretation of this equation in the framework of dynamic…
Riesz potentials are well known objects of study in the theory of singular integrals that have been the subject of recent, increased interest from the numerical analysis community due to their connections with fractional Laplace problems…