Higher order or fractional order Hardy-Sobolev type equations

In this paper we consider the following higher order or fractional order Hardy-Sobolev type equation

Восстановление динамически искаженных сигналов на основе теории оптимального управления решениями уравнений соболевского типа в пространствах случайных процессов

This paper investigates the solvability of the optimal control problem for solutions of stochastic Sobolev type equations. It is shown that the optimal dynamic measurement problem can be considered as an optimal control problem. To do this, the mathematical model of dynamic measurements is reduced to a stochastic Sobolev type equation of the first order in the spaces of stochastic processes. The article presents theorems on the existence of a unique classical and strong solutions of the Sobolev type equation with initial condition of Showalter–Sidorov in the spaces of stochastic processes. The theorem of the unique solvability of the optimal control problem for such equation is proved. The abstract results obtained for Sobolev type equation are applied to the problem of restoring a dynamically distorted signal as an optimal dynamic measurement.Исследована разрешимость задачи оптимального управления решениями стохастических уравнений соболевского типа. Показано, что задачу оптимального динамического измерения можно рассматривать как задачу оптимального управления. Для этого математическая модель динамических измерений редуцируется к стохастическому уравнению соболевского типа первого порядка в пространствах случайных процессов. В статье приведены теоремы о существовании единственного классического и сильного решений уравнения соболевского типа с начальным условием Шоуолтера–Сидорова в пространствах стохастических процессов. Доказана теорема об однозначной разрешимости задачи оптимального управления для такого уравнения. Полученные абстрактные результаты для уравнения соболевского типа применены для задачи восстановления динамически искаженного сигнала как оптимального динамического измерения

Linear Sobolev Type Equations with Relatively -Sectorial Operators in Space of "Noises"

The concept of "white noise," initially established in finite-dimensional spaces, is transferred to infinite-dimensional case. The goal of this transition is to develop the theory of stochastic Sobolev type equations and to elaborate applications of practical interest. To reach this goal the Nelson-Gliklikh derivative is introduced and the spaces of "noises" are developed. The Sobolev type equations with relatively sectorial operators are considered in the spaces of differentiable "noises." The existence and uniqueness of classical solutions are proved. The stochastic Dzektser equation in a bounded domain with homogeneous boundary condition and the weakened Showalter-Sidorov initial condition is considered as an application

On convergence of difference schemes of high accuracy for one pseudo-parabolic Sobolev type equation

Difference schemes of the finite difference method and the finite element method of high-order accuracy in time and space are proposed and investigated for a pseudo-parabolic Sobolev type equation. The order of accuracy in space is improved in two ways using the finite difference method and the finite element method. The order of accuracy of the scheme in time is improved by a special discretization of the time variable. The corresponding a priori estimates are determined and, on their basis, the accuracy estimates of the proposed difference schemes are obtained with sufficient smoothness of the solution to the original differential problem. Algorithms for the implementation of the constructed difference schemes are proposed

Многоточечная начально-конечная задача для стохастической модели Баренблатта – Желтова – Кочиной

In the paper we observe the multipoint initial-finish problem for the Barenblatt - Zheltov - Kochina equation for the perturbed white noise. We show the reduction of the problem under consideration to the multipoint initial-finish problem for stochastic Sobolev-type equation. We obtain sufficient conditions for the unique solvability for the abstract problem and for the stochastic Barenblatt - Zheltov - Kochina model.Рассматривается многоточечная начально-конечная задача для уравнения Баренблатта – Желтова – Кочиной, возмущенного белым шумом. Показана редукция рассматриваемой задачи к многоточечной начально-конечной задаче для стохастического уравнения соболевского типа. Получены достаточные условия однозначной разрешимости как для абстрактной задачи, так и для стохастической модели Баренблатта – Желтова – Кочиной

Positive solutions to Sobolev type equations with relatively p-sectorial operators

The article describes sufficient conditions for the existence of positive solutions to both the Cauchy problem and the Showalter-Sidorov problem for an abstract linear Sobolev type equation. A distinctive feature of such equations is the phenomenon of non-existence and non-uniqueness of solutions. The research is based on the theory of positive semigroups of operators and the theory of degenerate holomorphic semigroups of operators. The merger of these theories leads to a new theory of degenerate positive holomorphic semigroups of operators. In spaces of sequences, which are analogues of Sobolev function spaces, the constructed abstract theory is used to study a mathematical model. The results can be used to study economic and engineering problems.https://mmp.susu.ru/page/en/greetpm2021Mathematics and Applied Mathematic

A Finite Difference method for the Wide-Angle `Parabolic' equation in a waveguide with downsloping bottom

We consider the third-order wide-angle `parabolic' equation of underwater acoustics in a cylindrically symmetric fluid medium over a bottom of range-dependent bathymetry. It is known that the initial-boundary-value problem for this equation may not be well posed in the case of (smooth) bottom profiles of arbitrary shape if it is just posed e.g. with a homogeneous Dirichlet bottom boundary condition. In this paper we concentrate on downsloping bottom profiles and propose an additional boundary condition that yields a well posed problem, in fact making it $L^2$-conservative in the case of appropriate real parameters. We solve the problem numerically by a Crank-Nicolson-type finite difference scheme, which is proved to be unconditionally stable and second-order accurate, and simulates accurately realistic underwater acoustic problems.Comment: 2 figure