scholarly journals THE MAXIMUM PRINCIPLE FOR THE EQUATION OF LOCAL FLUCTUATIONS OF RIESZ GRAVITATIONAL FIELDS OF PURELY FRACTIONAL ORDER

2021 ◽  
Vol 9 (2) ◽  
pp. 81-91
Author(s):  
V. Litovchenko
Keyword(s):  
Cauchy Problem ◽  
Maximum Principle ◽  
Fractional Order ◽  
Moving Objects ◽  
Riesz Potential ◽  
Pseudodifferential Equation ◽  
Time Interval ◽  
The Maximum Principle ◽  
Gravitational Fields ◽  
The Cauchy Problem

The parabolic pseudodifferential equation with the Riesz fractional differentiation operator of α ∈ (0; 1) order, which acts on a spatial variable, is considered in the paper. This equation naturally summarizes the known equation of fractal diffusion of purely fractional order. It arises in the mathematical modeling of local vortices of nonstationary Riesz gravitational fields caused by moving objects, the interaction between the masses of which is characterized by the corresponding Riesz potential. The fundamental solution of the Cauchy problem for this equati- on is the density distribution of the probabilities of the force of local interaction between these objects, it belongs to the class of Polya distributions of symmetric stable random processes. Under certain conditions, for the coefficient of local field fluctuations, an analogue of the maximum principle was established for this equation. This principle is important in particular for substantiating the unity of the solution of the Cauchy problem on a time interval where the fluctuation coefficient is a non-decreasing function.

ASYMPTOTIC DECAY TOWARD THE PLANAR RAREFACTION WAVES FOR A MODEL SYSTEM OF THE RADIATING GAS IN TWO DIMENSIONS

2008 ◽  
Vol 18 (04) ◽  
pp. 511-541 ◽  
Author(s):  
WENLIANG GAO ◽  
CHANGJIANG ZHU
Keyword(s):  
Cauchy Problem ◽  
Maximum Principle ◽  
Coupled System ◽  
Model System ◽  
Two Dimensions ◽  
Rarefaction Waves ◽  
Asymptotic Decay ◽  
The Maximum Principle ◽  
The Cauchy Problem ◽  
Asymptotic Decay Rate

In this paper, we consider the asymptotic decay rate towards the planar rarefaction waves to the Cauchy problem for a hyperbolic–elliptic coupled system called as a model system of the radiating gas in two dimensions. The analysis based on the standard L2-energy method, L1-estimate and the monotonicity of profile obtained by the maximum principle.

scholarly journals Pseudodifferential Equation of Fluctuations of Nonstationary Gravitational Fields

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Vladyslav Litovchenko
Keyword(s):  
Moving Objects ◽  
Riesz Potential ◽  
Differentiation Operator ◽  
Local Interaction ◽  
General Nature ◽  
Pseudodifferential Equation ◽  
Fractional Differentiation ◽  
Gravitational Fields ◽  
Gravitational Influence ◽  
Fractional Differentiation Operator

Developing Holtzmark’s idea, the distribution of nonstationary fluctuations of local interaction of moving objects of the system with gravitational influence, which is characterized by the Riesz potential, is constructed. A pseudodifferential equation with the Riesz fractional differentiation operator is found, which corresponds to this process. The general nature of symmetric stable random Lévy processes is determined.

scholarly journals ON THE NATURE OF A CLASSICAL PSEUDODIFFERENTIAL EQUATION

2020 ◽  
Vol 8 (2) ◽  
pp. 83-92
Author(s):  
V. Litovchenko
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solution ◽  
Moving Objects ◽  
Classical Case ◽  
Random Processes ◽  
Natural Generalization ◽  
General Nature ◽  
Pseudodifferential Equation ◽  
Fractional Differentiation ◽  
The Cauchy Problem

The work is devoted to the study of the general nature of one classical parabolic pseudodi- fferential equation with the operator M.Rice of fractional differentiation. At the corresponding values of the order of fractional differentiation, this equation is also known as the isotropic superdiffusion equation. It is a natural generalization of the classical diffusion equation. It is also known that the fundamental solution of the Cauchy problem for this equation is the density distribution of probabilities of stable symmetric random processes by P.Levy. The paper shows that the fundamental solution of this equation is the distribution of probabilities of the force of local influence of moving objects in a nonstationary gravitational field, in which the interaction between masses is subject to the corresponding potential of M.Rice. In this case, the classical case of Newton’s gravity corresponds to the known nonstationary J.Holtsmark distribution.

Mkhitar Djrbashian and his contribution to Fractional Calculus

2020 ◽  
Vol 23 (6) ◽  
pp. 1797-1809
Author(s):  
Sergei Rogosin ◽  
Maryna Dubatovskaya
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Approximation Theory ◽  
Fractional Derivatives ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Modern Theory ◽  
Survey Paper ◽  
The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

scholarly journals UNIQUENESS IN THE CAUCHY PROBLEM FOR PARABOLIC EQUATIONS

2003 ◽  
Vol 46 (2) ◽  
pp. 329-340 ◽  
Author(s):  
Elisa Ferretti
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equations ◽  
Growth Conditions ◽  
Parabolic Differential Equations ◽  
The Maximum Principle ◽  
The Cauchy Problem ◽  
Primary 35K15 ◽  
Uniqueness Of The Solution ◽  
Boot Strapping ◽  
Uniformly Bounded

AbstractWe discuss the problem of the uniqueness of the solution to the Cauchy problem for second-order, linear, uniformly parabolic differential equations. For most uniqueness theorems the solution must be uniformly bounded with respect to the time variable $t$, but some authors have shown an interest in relaxing the growth conditions in time.In 1997, Chung proved that, in the case of the heat equation, uniqueness holds under the restriction: $|u(x,t)|\leq C\exp[(a/t)^{\alpha}+a|x|^2]$, for some constants $C,a>0$, $0\lt\alpha\lt1$. The proof of Chung’s theorem is based on ultradistribution theory, in particular it relies heavily on the fact that the coefficients are constants and that the solution is smooth. Therefore, his method does not work for parabolic operators with arbitrary coefficients. In this paper we prove a uniqueness theorem for uniformly parabolic equations imposing the same growth condition as Chung on the solution $u(x,t)$. At the centre of the proof are the maximum principle, Gaussian-type estimates for short cylinders and a boot-strapping argument.AMS 2000 Mathematics subject classification: Primary 35K15

scholarly journals An Analytical Study of Smooth Solutions of the Bløtekjær Hydrodynamic Model of Electron Transport

VLSI Design ◽  
2002 ◽  
Vol 15 (4) ◽  
pp. 729-742 ◽  
Author(s):  
Joseph W. Jerome
Keyword(s):  
Heat Flux ◽  
Cauchy Problem ◽  
Hydrodynamic Model ◽  
Analytical Study ◽  
Time Interval ◽  
Well Posedness ◽  
Smooth Solutions ◽  
Self Consistent ◽  
The Cauchy Problem ◽  
Electron Carriers

We study the Bløtekjær hydrodynamic model from the standpoint of local well-posedness. We employ analytical methods, originally introduced by T. Kato for complex systems, to obtain the existence of unique local smooth solutions of the Cauchy problem, with smooth initial data. The time interval is invariant with respect to vanishing heat flux. The model is self-consistent, and is developed for onevalley electron carriers only. A symmetrizer is introduced for the system, and regularization is employed to avoid the formation of singularities due to vacuum regions. In the regime studied, it is not possible for shocks to form.

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