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 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.

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.

scholarly journals Weak Estimates of Singular Integrals with Variable Kernel and Fractional Differentiation on Morrey-Herz Spaces

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Yanqi Yang ◽  
Shuangping Tao
Keyword(s):  
Integral Operator ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differentiation Operator ◽  
Fractional Differentiation ◽  
Herz Spaces ◽  
Variable Kernel ◽  
Harmonic Decomposition ◽  
Spherical Harmonic Decomposition ◽  
Fractional Differentiation Operator

Let T be the singular integral operator with variable kernel defined by Tf(x)=p.v.∫Rn(Ω(x,x-y)/x-yn)f(y)dy and let Dγ  (0≤γ≤1) be the fractional differentiation operator. Let T⁎and T♯ be the adjoint of T and the pseudoadjoint of T, respectively. In this paper, the authors prove that TDγ-DγT and (T⁎-T♯)Dγ are bounded, respectively, from Morrey-Herz spaces MK˙p,1α,λ(Rn) to the weak Morrey-Herz spaces WMK˙p,1α,λ(Rn) by using the spherical harmonic decomposition. Furthermore, several norm inequalities for the product T1T2 and the pseudoproduct T1∘T2 are also given.

scholarly journals Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents

2018 ◽  
Vol 16 (1) ◽  
pp. 326-345 ◽  
Author(s):  
Yanqi Yang ◽  
Shuangping Tao
Keyword(s):  
Integral Operator ◽  
Singular Integral ◽  
Hardy Spaces ◽  
Convolution Operator ◽  
Singular Integrals ◽  
Differentiation Operator ◽  
Variable Exponents ◽  
Fractional Differentiation ◽  
Variable Kernel ◽  
Fractional Differentiation Operator

AbstractLet T be the singular integral operator with variable kernel defined by $$\begin{array}{} \displaystyle Tf(x)= p.v. \int\limits_{\mathbb{R}^{n}}\frac{{\it\Omega}(x,x-y)}{|x-y|^{n}}f(y)\text{d}y \end{array} $$and Dγ(0 ≤ γ ≤ 1) be the fractional differentiation operator. Let T∗ and T♯ be the adjoint of T and the pseudo-adjoint of T, respectively. The aim of this paper is to establish some boundedness for TDγ − DγT and (T∗ − T♯)Dγ on the homogeneous Morrey-Herz-type Hardy spaces with variable exponents $\begin{array}{} HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda} \end{array} $ via the convolution operator Tm, j and Calderón-Zygmund operator, and then establish their boundedness on these spaces. The boundedness on $\begin{array}{} HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda} \end{array} $(ℝn) is shown to hold for TDγ − DγT and (T∗ − T♯)Dγ. Moreover, the authors also establish various norm characterizations for the product T1T2 and the pseudo-product T1 ∘ T2.

scholarly journals Cauchy Problem for a Linear System of Ordinary Differential Equations of the Fractional Order

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1475
Author(s):  
Murat Mamchuev
Keyword(s):  
Linear System ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Existence And Uniqueness ◽  
Differentiation Operator ◽  
Initial Problem ◽  
Fractional Differentiation ◽  
The Matrix ◽  
Constant Coefficients ◽  
Fractional Differentiation Operator

We investigate the initial problem for a linear system of ordinary differential equations with constant coefficients and with the Dzhrbashyan–Nersesyan fractional differentiation operator. The existence and uniqueness theorems of the solution of the boundary value problem under the study are proved. The solution is constructed explicitly in terms of the Mittag–Leffler function of the matrix argument. The Dzhrbashyan–Nersesyan operator is a generalization of the Riemann–Liouville, Caputo and Miller–Ross fractional differentiation operators. The obtained results as particular cases contain the results related to the study of initial problems for the systems of ordinary differential equations with Riemann–Liouville, Caputo and Miller–Ross derivatives and the investigated initial problem that generalizes them.

scholarly journals On One Evolution Equation of Parabolic Type with Fractional Differentiation Operator in S Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
V. V. Gorodetskiy ◽  
R. S. Kolisnyk ◽  
N. M. Shevchuk
Keyword(s):  
Fundamental Solution ◽  
Evolution Equation ◽  
Parabolic Type ◽  
Generalized Functions ◽  
Fractional Differentiation ◽  
Uniform Stabilization ◽  
Time Problem ◽  
Fixed Parameter ◽  
Space Of Generalized Functions ◽  
Fractional Differentiation Operator

In the paper, we investigate a nonlocal multipoint by a time problem for the evolution equation with the operator A=I−Δω/2, Δ=d2/dx2, and ω∈1;−2 is a fixed parameter. The operator A is treated as a pseudodifferential operator in a certain space of type S. The solvability of this problem is proved. The representation of the solution is given in the form of a convolution of the fundamental solution with the initial function which is an element of the space of generalized functions of ultradistribution type. The properties of the fundamental solution are investigated. The behavior of the solution at t⟶+∞ (solution stabilization) in the spaces of generalized functions of type S′ and the uniform stabilization of the solution to zero on ℝ are studied.

scholarly journals Energy Change of Particles or Photons at Crossing Gravity Fields of Galaxy Clusters or Single Stars: The Concept of Gravitons

2020 ◽  
Vol 3 (4) ◽  
Keyword(s):  
Moving Objects ◽  
Rational Solution ◽  
Terrestrial Planet ◽  
Theoretical Concept ◽  
Central Mass ◽  
Gravitational Fields ◽  
Orbital Decay ◽  
Gravity Fields ◽  
Centrally Symmetric ◽  
Cosmic Mass

We investigate the gravitational action of cosmic mass associations, like stars or stellar clusters, on moving massive objects. Hereby the relativistic effect of propagating field quanta communicating the position of gravity sources by means of so-called gravitons is taken into account. In case of moving objects this causes an aberration of the recognized actual location of the cosmic mass sources with respect to their positions in the cosmic rest frame. The astonishing effect of that position retardation is that a moving objekt, even if it moves right through the center of a centrally symmetric cluster mass association, experiences a net gravitational braking and energy loss. Applying this view to the problem of a planetary object orbiting around a central mass like the sun, then it turns out that the orbiting planet permanently reduces its orbital angular momentum, since permanently experiencing a gravitational force component antiparallel to its orbital motion. From that an orbital decay time can be derived which for a terrestrial planet would imply the spiralling-in period of only a few 103 years. Compared to the age of the planet earth of about 4. 5 Billion years this represents a big problem of understanding. In this article we cannot offer a rational solution of this problem and thus we simply end with the recommendation to perhaps reinvestigate the theoretical concept of gravitons thought to be the quantum messengers of gravitational fields.

Local interaction rules and collective motion in black neon tetra (Hyphessobrycon herbertaxelrodi) and zebrafish (Danio rerio).

2019 ◽  
Vol 133 (2) ◽  
pp. 143-155 ◽  
Author(s):  
Vicenç Quera ◽  
Elisabet Gimeno ◽  
Francesc S. Beltran ◽  
Ruth Dolado
Keyword(s):  
Danio Rerio ◽  
Collective Motion ◽  
Local Interaction
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