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Anatoly N. Kochubei

Publications and source records attributed to Anatoly N. Kochubei.

At least 37 records · Page 2Linked to original sources
arXiv2017

Linear and Nonlinear Heat Equations on a p-Adic Ball

We study the Vladimirov fractional differentiation operator $D^α_N$, $α>0, N\in \mathbb Z$, on a $p$-adic ball $B_N=\{ x\in \mathbb Q_p:\ |x|_p\le p^N\}$. To its known interpretations via restriction from a similar operator on $\mathbb Q_p$ and via a certain stochastic process on $B_N$, we add an interpretation as a pseudo-differential operator in terms of the Pontryagin duality on the additive group of $B_N$. We investigate the Green function of $D^α_N$ and a nonlinear equation on $B_N$, an analog the classical porous medium equation.

math.AP↗
arXiv2017

p-Adic Analogue of the Porous Medium Equation

We consider a nonlinear evolution equation for complex-valued functions of a real positive time variable and a p-adic spatial variable. This equation is a non-Archimedean counterpart of the fractional porous medium equation. Developing, as a tool, an $L^1$-theory of Vladimirov's p-adic fractional differentiation operator, we prove m-accretivity of the appropriate nonlinear operator, thus obtaining the existence and uniqueness of a mild solution. We give also an example of an explicit solution of the p-adic porous medium equation.

math.AP↗
arXiv2016

Fractional statistical dynamics and fractional kinetics

We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. As a by-product we obtain the evolution of the density of particles in the fractional kinetics in terms of a non-linear Vlasov-type kinetic equation. As an application we study the intermittency of the fractional mesoscopic dynamics.

math-ph↗
arXiv2016

Fractional kinetic hierarchies and intermittency

We consider general convolutional derivatives and related fractional statistical dynamics of continuous interacting particle systems. We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. Conditions for the intermittency property of fractional kinetic dynamics are obtained.

math-ph↗
arXiv2016

Non-Archimedean Duality: Algebras, Groups, and Multipliers

We develop a duality theory for multiplier Banach-Hopf algebras over a non-Archimedean field K. As examples, we consider algebras corresponding to discrete groups and zero-dimensional locally compact groups with K-valued Haar measure, as well as algebras of operators generated by regular representations of discrete groups.

math.RA↗
arXiv2014

Fractional contact model in the continuum

We consider the evolution of correlation functions in a non-Markov version of the contact model in the continuum. The memory effects are introduced by assuming the fractional evolution equation for the statistical dynamics. This leads to a behavior of time-dependent correlation functions, essentially different from the one known for the standard contact model.

math-ph↗
arXiv2014

Cauchy Problem for Fractional Diffusion-Wave Equations with Variable Coefficients

We consider an evolution equation with the Caputo-Dzhrbashyan fractional derivative of order $α\in (1,2)$ with respect to the time variable, and the second order uniformly elliptic operator with variable coefficients acting in spatial variables. This equation describes the propagation of stress pulses in a viscoelastic medium. Its properties are intermediate between those of parabolic and hyperbolic equations. In this paper, we construct and investigate a fundamental solution of the Cauchy problem, prove existence and uniqueness theorems for such equations.

math.AP↗
arXiv2014

Asymptotic Properties of Solutions of the Fractional Diffusion-Wave Equation

For the fractional diffusion-wave equation with the Caputo-Dzhrbashyan fractional derivative of order $α\in (1,2)$ with respect to the time variable, we prove an analog of the principle of limiting amplitude (well-known for the wave equation and some other hyperbolic equations) and a pointwise stabilization property of solutions (similar to a well-known property of the heat equation and some other parabolic equations).

math.AP↗
arXiv2014

Non-Archimedean Group Algebras with Baer Reductions

Within the concept of a non-Archimedean operator algebra with the Baer reduction (A. N. Kochubei, On some classes of non-Archimedean operator algebras, Contemporary Math. 596 (2013), 133--148), we consider algebras of operators on Banach spaces over non-Archimedean fields generated by regular representations of discrete groups.

math.FA↗
arXiv2013

Fractional-Hyperbolic Systems

We describe a class of evolution systems of linear partial differential equations with the Caputo-Dzhrbashyan fractional derivative of order $α\in (0,1)$ in the time variable $t$ and the first order derivatives in spatial variables $x=(x_1,...,x_n)$, which can be considered as a fractional analogue of the class of hyperbolic systems. For such systems, we construct a fundamental solution of the Cauchy problem having exponential decay outside the fractional light cone $\{(t,x):\ |t^{-α}x|\le 1\}$.

math.AP↗
arXiv2013

Radial Solutions of Non-Archimedean Pseudo-Differential Equations

We consider a class of equations with the fractional differentiation operator $D^α$, $α>0$, for complex-valued functions $x\mapsto f(|x|_K)$ on a non-Archimedean local field $K$ depending only on the absolute value $|\cdot |_K$. We introduce a right inverse $I^α$ to $D^α$, such that the change of an unknown function $u=I^αv$ reduces the Cauchy problem for an equation with $D^α$ (for radial functions) to an integral equation whose properties resemble those of classical Volterra equations. This contrasts much more complicated behavior of $D^α$ on other classes of functions.

math.CA↗
arXiv2012

Non-Archimedean Normal Operators

We describe some classes of linear operators on Banach spaces over non-Archimedean fields, which admit orthogonal spectral decompositions. Several examples are given.

math.FA↗
arXiv2012

Fractional-Parabolic Systems

We develop a theory of the Cauchy problem for linear evolution systems of partial differential equations with the Caputo-Dzrbashyan fractional derivative in the time variable $t$. The class of systems considered in the paper is a fractional extension of the class of systems of the first order in $t$ satisfying the uniform strong parabolicity condition. We construct and investigate the Green matrix of the Cauchy problem. While similar results for the fractional diffusion equations were based on the H-function representation of the Green matrix for equations with constant coefficients (not available in the general situation), here we use, as a basic tool, the subordination identity for a model homogeneous system. We also prove a uniqueness result based on the reduction to an operator-differential equation.

math.AP↗
arXiv2012

Cauchy Problem for Fractional Diffusion Equations

We consider an evolution equation with the regularized fractional derivative of an order $α\in (0,1)$ with respect to the time variable, and a uniformly elliptic operator with variable coefficients acting in the spatial variables. Such equations describe diffusion on inhomogeneous fractals. A fundamental solution of the Cauchy problem is constructed and investigated.

math.AP↗
arXiv2011

General Fractional Calculus, Evolution Equations, and Renewal Processes

We develop a kind of fractional calculus and theory of relaxation and diffusion equations associated with operators in the time variable, of the form $(Du)(t)=\frac{d}{dt}\int\limits_0^tk(t-τ)u(τ)\,dτ-k(t)u(0)$ where $k$ is a nonnegative locally integrable function. Our results are based on the theory of complete Bernstein functions. The solution of the Cauchy problem for the relaxation equation $Du=-λu$, $λ>0$, proved to be (under some conditions upon $k$) continuous on $[(0,\infty)$ and completely monotone, appears in the description by Meerschaert, Nane, and Vellaisamy of the process $N(E(t))$ as a renewal process. Here $N(t)$ is the Poisson process of intensity $λ$, $E(t)$ is an inverse subordinator.

math.CA↗