scholarly journals A PROBLEM WITH SECOND KIND INTEGRAL CONDITIONS FOR HYPERBOLIC EQUATION

2017 ◽  
Vol 22 (1-2) ◽  
pp. 33-45
Author(s):  
L. S. Pulkina ◽  
A. E. Savenkova
Keyword(s):  
Hyperbolic Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Main Idea ◽  
Integral Condition ◽  
Generalized Solution ◽  
Integral Conditions ◽  
Priori Estimates ◽  
Nonlocal Integral Condition ◽  
Definition Of

In this paper, we consider a problem for one-dimensional hyperbolic equation with second kind integral conditions and prove unique solvability. To prove this statement we suggest a new approach. The main idea of it is that given nonlocal integral condition is equivalent with a different condition, nonlocal as well but this new condition enables us to introduce a definition of a generalized solution bazed on an integral identity and derive a priori estimates of a required solution in Sobolev space. This approach shows that integral conditions are closely connected with dynamical conditions.

scholarly journals On the solvability of parabolic and hyperbolic problems with a boundary integral condition

2002 ◽  
Vol 31 (4) ◽  
pp. 201-213 ◽  
Author(s):  
Abdelfatah Bouziani
Keyword(s):  
A Priori Estimates ◽  
Hyperbolic Equations ◽  
A Priori ◽  
Integral Condition ◽  
Generalized Solution ◽  
Boundary Integral ◽  
Analysis Method ◽  
Hyperbolic Problems ◽  
Priori Estimates ◽  
Abstract Formulation

We prove the existence, uniqueness, and the continuous dependence of a generalized solution upon the data of certain parabolic and hyperbolic equations with a boundary integral condition. The proof uses a functional analysis method based on a priori estimates established in nonclassical function spaces, and on the density of the range of the linear operator associated to the abstract formulation of the studied problem.

scholarly journals PROBLEM WITH DYNAMIC BOUNDARY CONDITIONS FOR A HYPERBOLIC EQUATION

2017 ◽  
Vol 23 (1) ◽  
pp. 21-27
Author(s):  
V. A. Kirichek ◽  
L. S. Pulkina
Keyword(s):  
Boundary Condition ◽  
Hyperbolic Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Generalized Solution ◽  
Dynamic Boundary Condition ◽  
Initial Boundary Problem ◽  
Dynamic Boundary ◽  
Priori Estimates

We consider an initial-boundary problem with dynamic boundary condition for a hyperbolic equation in a rectangle. Dynamic boundary condition represents a relation between values of derivatives with respect of spacial variables of a required solution and first-order derivatives with respect to time variable. The main result lies in substantiation of solvability of this problem. We prove the existence and uniqueness of a generalized solution. The proof is based on the a priori estimates obtained in this paper, Galyorkin’s procedure and the properties of Sobolev spaces.

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 On initial boundary value problem with Dirichlet integral conditions for a hyperbolic equation with the Bessel operator

2003 ◽  
Vol 2003 (10) ◽  
pp. 487-502
Author(s):  
Abdelfatah Bouziani
Keyword(s):  
Hyperbolic Equation ◽  
A Priori ◽  
Weighted Sobolev Spaces ◽  
A Priori Estimate ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Bessel Operator ◽  
Integral Conditions ◽  
Initial Boundary Value

We consider a mixed problem with Dirichlet and integral conditions for a second-order hyperbolic equation with the Bessel operator. The existence, uniqueness, and continuous dependence of a strongly generalized solution are proved. The proof is based on an a priori estimate established in weighted Sobolev spaces and on the density of the range of the operator corresponding to the abstract formulation of the considered problem.

scholarly journals ON CERTAIN NONLOCAL PROBLEM FOR HYPERBOLIC EQUATION WITH INTEGRAL CONDITIONS OF THE FIRST KIND

2017 ◽  
Vol 17 (5) ◽  
pp. 29-36
Author(s):  
A.V. Duzheva
Keyword(s):  
Hyperbolic Equation ◽  
Existence And Uniqueness ◽  
Nonlocal Problem ◽  
Integral Condition ◽  
Generalized Solution ◽  
Integral Conditions

In this article, we consider a nonlocal problem for hyperbolic equation with integral conditions of the first kind. The main goal of this article is to show the method which allows to reduce posed problem to the problem with integral condition of the second kind. Existence and uniqueness of generalized solution is proved.

scholarly journals Problem on vibration of a bar with nonlinearsecond-order boundary damping

2017 ◽  
Vol 21 (3) ◽  
pp. 9-20
Author(s):  
A.B. Beylin ◽  
L.S. Pulkina
Keyword(s):  
A Priori Estimates ◽  
Longitudinal Vibration ◽  
A Priori ◽  
Initial Boundary ◽  
Generalized Solution ◽  
Application Form ◽  
Boundary Damping ◽  
Initial Boundary Problem ◽  
Priori Estimates ◽  
Pseudohyperbolic Equation

In this paper, we study the initial-boundary problem with nonlinear dynam- ical boundary condition for the pseudohyperbolic equation. This problem repre- sents a mathematical model of longitudinal vibration in a thick short bar with dynamic nonlinear second-order boundary damping. The existence and unique- ness of a generalized solution are proved. The proof is based on a priori estimates and Galerkin procedure. This approach allows to construct approximation in the suitable for practical application form.

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 Semi-Fredholm Solvability in the Framework of Singular Solutions for the (3+1)-D Protter-Morawetz Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
Nedyu Popivanov ◽  
Todor Popov ◽  
Allen Tesdall
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Generalized Solution ◽  
Generalized Solutions ◽  
Singular Solutions ◽  
Priori Estimates ◽  
Characteristic Cone ◽  
Equation Boundary ◽  
Necessary And Sufficient

For the four-dimensional nonhomogeneous wave equation boundary value problems that are multidimensional analogues of Darboux problems in the plane are studied. It is known that for smooth right-hand side functions the unique generalized solution may have a strong power-type singularity at only one point. This singularity is isolated at the vertexOof the boundary light characteristic cone and does not propagate along the bicharacteristics. The present paper describes asymptotic expansions of the generalized solutions in negative powers of the distance toO. Some necessary and sufficient conditions for existence of bounded solutions are proven and additionally a priori estimates for the singular solutions are obtained.

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