scholarly journals INITIAL-BOUNDARY VALUE PROBLEM FOR ONE-DIMENSION HYPERBOLIC EQUATION WITH INTEGRAL BOUNDARY CONDITION

2017 ◽  
Vol 17 (8) ◽  
pp. 95-101
Author(s):  
M.V. Strigun
Keyword(s):  
Boundary Value Problem ◽  
Hyperbolic Equation ◽  
Boundary Value ◽  
Integral Condition ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Integral Boundary ◽  
Nonlocal Integral Condition ◽  
Initial Boundary Value

In this paper, we study an initial-boundary value problem with nonlocal integral condition for a hyperbolic equation. The existence and uniqueness of a generalized solution of the problem is proved.

scholarly journals On the connection between solutions of initial boundary-value problems for a some class of integro-differential PDE and a linear hyperbolic equation

2019 ◽  
Vol 21 (4) ◽  
pp. 413-429
Author(s):  
Pavel N. Burago ◽  
Albert I. Egamov
Keyword(s):  
Integral Operator ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Hyperbolic Equation ◽  
Boundary Value ◽  
Fourier Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Phase Constraint ◽  
Initial Boundary Value

We consider the second initial boundary-value problem for a certain class of second-order integro-differential PDE with integral operator. The connection of its solution with the solution of the standard second linear initial boundary-value problem for the hyperbolic equation is shown. Thus, the nonlinear problem is reduced to a standard linear problem, whose numerical solution can be obtained, for example, by the Fourier method or Galerkin method. The article provides examples of five integro-differential equations for various integral operators as particular representatives of the class of integro-differential equations for a better understanding of the problem. The application of the main theorem to these examples is shown. Some simple natural requirement is imposed on the integral operator; so, in four cases out of five the problem’s solution satisfies some phase constraint. The form of these constraints is of particular interest for the further research.

scholarly journals INITIAL-BOUNDARY VALUE PROBLEM FOR HYPERBOLIC EQUATIONS WITH AN ARBITRARY ORDER ELLIPTIC OPERATOR

2020 ◽  
pp. 8-19
Author(s):  
Р.Р. Ашуров ◽  
А.Т. Мухиддинова
Keyword(s):  
Boundary Value Problem ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Fourier Method ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Order Elliptic Operator ◽  
Initial Boundary Value

В настоящей работе исследуется начально-краевые задачи для гиперболических уравнений, эллиптическая часть которых имеет наиболее общий вид и определена в произвольной многомерной области (с достаточно гладкой границей). Установливаются требования на правую часть уравнения и начальные функции, при которых к рассматрываемую задачу применим классический метод Фурье. Другими словами, доказывается методом Фурье существование и единственность решения смешанной задачи и показана устойчивость найденного решения от данных задачи: от начальных функций и правой части уравнения. Введено понятие обобщенного решения и доказана теорема о его существования. Аналогичные результаты справедливы и для параболических уравнений. An initial-boundary value problem for a hyperbolic equation with the most general elliptic differential operator, defined on an arbitrary bounded domain, is considered. Uniqueness, existence and stability of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equation are indicated, under which the corresponding Fourier series converge absolutely and uniformly. The notion of a generalized solution is introduced and existence theorem is proved. Similar results are formulated for parabolic equations too.

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