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 ON A NONLOCAL PROBLEM FOR PARTIAL DIFFERENTIAL EQUATIONS OF PARABOLIC TYPE

2020 ◽  
Vol 8 (2) ◽  
pp. 24-39
Author(s):  
V. Gorodetskiy ◽  
R. Kolisnyk ◽  
O. Martynyuk
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solution ◽  
Parabolic Type ◽  
Generalized Functions ◽  
Initial Condition ◽  
Partial Derivatives ◽  
Time Problem ◽  
The Cauchy Problem ◽  
Space Of Generalized Functions ◽  
Derivatives Of

Spaces of $S$ type, introduced by I.Gelfand and G.Shilov, as well as spaces of type $S'$, topologically conjugate with them, are natural sets of the initial data of the Cauchy problem for broad classes of equations with partial derivatives of finite and infinite orders, in which the solutions are integer functions over spatial variables. Functions from spaces of $S$ type on the real axis together with all their derivatives at $|x|\to \infty$ decrease faster than $\exp\{-a|x|^{1/\alpha}\}$, $\alpha > 0$, $a > 0$, $x\in \mathbb{R}$. The paper investigates a nonlocal multipoint by time problem for equations with partial derivatives of parabolic type in the case when the initial condition is given in a certain space of generalized functions of the ultradistribution type ($S'$ type). Moreover, results close to the Cauchy problem known in theory for such equations with an initial condition in the corresponding spaces of generalized functions of $S'$ type were obtained. The properties of the fundamental solution of a nonlocal multipoint by time problem are investigated, the correct solvability of the problem is proved, the image of the solution in the form of a convolution of the fundamental solution with the initial generalized function, which is an element of the space of generalized functions of $S'$ type.

scholarly journals A Nonlocal in Time Problem for Evolutionary Singular Equations in Generalized Spaces of Type S∘

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Hanna Verezhak ◽  
Vasyl Gorodetskyi
Keyword(s):  
Initial Function ◽  
Infinite Order ◽  
Parabolic Type ◽  
Generalized Functions ◽  
Bessel Operator ◽  
Evolutionary Equation ◽  
Correct Solvability ◽  
Singular Equations ◽  
Time Problem ◽  
Space Of Generalized Functions

In this paper, we establish the correct solvability of a nonlocal multipoint in time problem for the evolutionary equation of a parabolic type with the Bessel operator of infinite order in the case where the initial function is an element of the space of generalized functions of type S∘′.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

THE NON-LOCAL TIME PROBLEM FOR ONE CLASS OF PSEUDODIFFERENTIAL EQUATIONS WITH SMOOTH SYMBOLS

2021 ◽  
Vol 9 (1) ◽  
pp. 107-127
Author(s):  
R. Kolisnyk ◽  
V. Gorodetskyi ◽  
O. Martynyuk
Keyword(s):  
Fundamental Solution ◽  
Initial Function ◽  
Adjoint Operator ◽  
Differentiation Operator ◽  
The Self ◽  
Multipoint Problem ◽  
Time Problem ◽  
Sigma 1 ◽  
Non Local ◽  
Differential Operator Equation

In this paper we investigate the differential-operator equation $$ \partial u (t, x) / \partial t + \varphi (i \partial / \partial x) u (t, x) = 0, \quad (t, x) \in (0, + \infty) \times \mathbb {R} \equiv \Omega, $$ where the function $ \varphi \in C ^ {\infty} (\mathbb {R}) $ and satisfies certain conditions. Using the explicit form of the spectral function of the self-adjoint operator $ i \partial / \partial x $, in $ L_2 (\mathbb {R}) $ it is established that the operator $ \varphi (i \partial / \partial x) $ can be understood as a pseudodifferential operator in a certain space of type $ S $. The evolution equation $ \partial u / \partial t + \sqrt {I- \Delta} u = 0 $, $ \Delta = D_x ^ 2 $, with the fractionation differentiation operator $ \sqrt { I- \Delta} = \varphi (i \partial / \partial x) $, where $ \varphi (\sigma) = (1+ \sigma ^ 2) ^ {1/2} $, $ \sigma \in \mathbb {R} $ is attributed to the considered equation. Considered equation is a nonlocal multipoint problem with the initial function $ f $, which is an element of a space of type $ S $ or type $ S '$ which is a topologically conjugate with a space of type $ S $ space. The properties of the fundamental solution of such a problem are established, the correct solvability of the problem in the half-space $ t> 0 $ is proved, the representation of the solution in the form of a convolution of the fundamental solution with the initial function is found, the behavior of the solution $ u (t, \cdot) $ for $ t \to + \infty $ (solution stabilization) in spaces of type $ S '$.

scholarly journals Non-radial functions, nonlocal operators and Markov processes over p-adic numbers

2019 ◽  
Vol 24 (2) ◽  
pp. 381-406 ◽  
Author(s):  
Leonardo Fabio Chacón-Cortés ◽  
Oscar Francisco Casas-Sánchez
Keyword(s):  
Markov Processes ◽  
First Passage Time ◽  
Parabolic Type ◽  
Passage Time ◽  
Fundamental Solutions ◽  
Nonlocal Operators ◽  
Transition Functions ◽  
Time Problem ◽  
The Cauchy Problem ◽  
First Passage Time Problem

The main goal of this article is to study a new class of nonlocal operators and the Cauchy problem for certain parabolic-type pseudodifferential equations naturally associated with them. The fundamental solutions of these equations are transition functions of Markov processes on an n-dimensional vector space over the p-adic numbers. We also study some properties of these Markov processes, including the first passage time problem.

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.

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