scholarly journals Non-local problem with integral conditions for the three-dimensional high order hyperbolic equations

2020 ◽  
pp. 51-62
Author(s):  
Ш.Ш. Юсубов
Keyword(s):  
Integral Equation ◽  
Hyperbolic Equation ◽  
Hyperbolic Equations ◽  
Three Dimensional ◽  
High Order ◽  
Mixed Derivative ◽  
Local Problem ◽  
Integral Conditions ◽  
Order Hyperbolic Equation ◽  
Non Local

В работе для трехмерного гиперболического уравнения высокого порядка с доминирующей смешанной производной исследуется разрешимость нелокальной задачи с интегральными условиями. Поставленная задача сводится к интегральному уравнению и с помощью априорных оценок доказывается существование единственного решения. In the work the solvability of the non-local problem with integral conditions is investigated for the three-dimensional high order hyperbolic equation with dominated mixed derivative. The problem is reduced to the integral equation and existence of the solution is proved by the help of aprior estimations.

The Equations of Markovian Random Evolution on the Line

1998 ◽  
Vol 35 (1) ◽  
pp. 27-35 ◽  
Author(s):  
Alexander Kolesnik
Keyword(s):  
Poisson Process ◽  
Hyperbolic Equation ◽  
Explicit Form ◽  
General Model ◽  
Hyperbolic Equations ◽  
Random Evolution ◽  
One Dimensional ◽  
Order Hyperbolic Equation

We consider a general model of one-dimensional random evolution with n velocities and rates of a switching Poisson process (n ≥ 2). A governing nth-order hyperbolic equation in a determinant form is given. For two important particular cases it is written in an explicit form. Some known hyperbolic equations are obtained as particular cases of the general model

The Equations of Markovian Random Evolution on the Line

1998 ◽  
Vol 35 (01) ◽  
pp. 27-35 ◽  
Author(s):  
Alexander Kolesnik
Keyword(s):  
Poisson Process ◽  
Hyperbolic Equation ◽  
Explicit Form ◽  
General Model ◽  
Hyperbolic Equations ◽  
Random Evolution ◽  
One Dimensional ◽  
Order Hyperbolic Equation

We consider a general model of one-dimensional random evolution with n velocities and rates of a switching Poisson process (n ≥ 2). A governing nth-order hyperbolic equation in a determinant form is given. For two important particular cases it is written in an explicit form. Some known hyperbolic equations are obtained as particular cases of the general model

scholarly journals Nonlocal problem with the integral condition for a loaded heate equation

2021 ◽  
pp. 47-56
Author(s):  
М.М. Сагдуллаева
Keyword(s):  
Integral Equation ◽  
Heat Equation ◽  
Regular Solution ◽  
Volterra Integral Equation ◽  
Existence And Uniqueness ◽  
Nonlocal Problem ◽  
Integral Condition ◽  
Local Problem ◽  
Non Local ◽  
Loaded Term

В работе рассмотрена нелокальная задача с интегральным условием для нагруженного уравнения теплопроводности, где нагруженное слагаемое представляет собой производную второго порядка от неизвестной функции в начале координат. Доказано существование и единственность регулярного решения. С помощью функции Грина и тепловых потенциалов доказанао существование регулярного решения исследуемой задачи. Доказательство основано на редукции поставленной задачи к интегральному уравнению Вольтерра второго рода со слабой особенностью. Из разрешимости полученных интегральных уравнений Вольтерра следует существование единственного решения поставленной задачи. In this paper, we consider a non-local problem with the integral condition for the loaded heat equation, where the loaded term is a derivative of the second order from an unknown function at the origin. The existence and uniqueness of a regular solution is proven. Using the Green’s functions and thermal potentials, the existence of a regular solution to this problem is proved. The proof is based on the reduction of the formulated problem to the second kind Volterra integral equation with a weak singularity. The solvability of the obtained Volterra integral equations implies the existence of a unique solution to the problem.

scholarly journals Non-local Problems with Integral Displacement for Highorder Parabolic Equations

2021 ◽  
Vol 36 ◽  
pp. 14-28
Author(s):  
A.I. Kozhanov ◽  
◽  
A.V. Dyuzheva ◽  
◽  
Keyword(s):  
Parabolic Equations ◽  
High Order ◽  
Multidimensional Case ◽  
Integral Conditions ◽  
Spatial Variables ◽  
One Dimensional ◽  
Linear Parabolic Equations ◽  
Local Problems ◽  
Non Local ◽  
The One

The aim of this paper is to study the solvability of solutions of non-local problems with integral conditions in spatial variables for high-order linear parabolic equations in the classes of regular solutions (which have all the squared derivatives generalized by S. L. Sobolev that are included in the corresponding equation) . Previously, similar problems were studied for high-order parabolic equations, either in the one-dimensional case, or when certain conditions of smallness on the coefficients are met equations. In this paper, we present new results on the solvability of non-local problems with integral spatial variables for high-order parabolic equations a) in the multidimensional case with respect to spatial variables; b) in the absence of smallness conditions. The research method is based on the transition from a problem with non-local integral conditions to a problem with classical homogeneous conditions of the first or second kind on the side boundary for a loaded integro-differential equation. At the end of the paper, some generalizations of the obtained results will be described.

scholarly journals A NONLOCAL PROBLEM FOR A HYPERBOLIC EQUATION WITH A DOMINANT MIXED DERIVATIVE

2021 ◽  
Vol 26 (4) ◽  
pp. 25-35
Author(s):  
A. V. Gilev
Keyword(s):  
Hyperbolic Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Nonlocal Problem ◽  
Nonlocal Conditions ◽  
Integral Condition ◽  
Goursat Problem ◽  
Mixed Derivative ◽  
Integral Conditions ◽  
Loaded Equation

In this article, we consider the Goursat problem with nonlocal integral conditions for a hyperbolic equation with a dominant mixed derivative. Research methods of solvability of classical boundary value problems for partial differential equations cannot be applied without serious modifications. The choice of a research method of solvability of a nonlocal problem depends on the form of the integral condition. In the process of developing methods that are effective for nonlocal problems, integral conditions of various types were identified [1]. The solvability of the nonlocal Goursat problem with integral conditions of the first kind for a general equation with dominant mixed derivative of the second order was investigated in [2]. In our problem, the integral conditions are nonlocal conditions of the second kind, therefore, to investigate the solvability of the problem, we propose another method, which consists in reducing the stated nonlocal problem to the classical Goursat problem, but for a loaded equation. In this article, we obtain conditions that guarantee the existence of a unique solution of the problem. The main instrument of the proof is the a priori estimates obtained in the paper.

scholarly journals Hyperbolic Equations with Unknown Coefficients

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1539
Author(s):  
Aleksandr I. Kozhanov
Keyword(s):  
Hyperbolic Equation ◽  
Existence And Uniqueness ◽  
Hyperbolic Equations ◽  
Integral Condition ◽  
Uniqueness Theorems ◽  
Regular Solutions ◽  
Final Data ◽  
Order Hyperbolic Equation ◽  
Weak Derivatives ◽  
Second Order Hyperbolic Equation

We study the solvability of nonlinear inverse problems of determining the low order coefficients in the second order hyperbolic equation. The overdetermination condition is specified as an integral condition with final data. Existence and uniqueness theorems for regular solutions are proved (i.e., for solutions having all weak derivatives in the sense of Sobolev, occuring in the equation).

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