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 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 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 A priori bounds of the solution of a one point IBVP for a singular fractional evolution equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Said Mesloub ◽  
Hassan Eltayeb Gadain
Keyword(s):  
Differential Equations ◽  
Evolution Equation ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Initial Boundary ◽  
A Priori Bounds ◽  
Fractional Evolution Equation ◽  
Priori Estimates ◽  
Initial Boundary Value

Abstract A priori bounds constitute a crucial and powerful tool in the investigation of initial boundary value problems for linear and nonlinear fractional and integer order differential equations in bounded domains. We present herein a collection of a priori estimates of the solution for an initial boundary value problem for a singular fractional evolution equation (generalized time-fractional wave equation) with mass absorption. The Riemann–Liouville derivative is employed. Results of uniqueness and dependence of the solution upon the data were obtained in two cases, the damped and the undamped case. The uniqueness and continuous dependence (stability of solution) of the solution follows from the obtained a priori estimates in fractional Sobolev spaces. These spaces give what are called weak solutions to our partial differential equations (they are based on the notion of the weak derivatives). The method of energy inequalities is used to obtain different a priori estimates.

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 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 Accuracy Estimates of Difference Schemes for Quasi-linear Parabolic Equations Taking into Account the Initial-boundary Effect

2003 ◽  
Vol 3 (4) ◽  
pp. 579-595 ◽  
Author(s):  
V. L. Makarov ◽  
L. I. Demkiv
Keyword(s):  
A Priori Estimates ◽  
Initial Conditions ◽  
Boundary Effect ◽  
A Priori ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Difference Schemes ◽  
Linear Parabolic Equation ◽  
Initial Boundary Problem ◽  
Linear Parabolic Equations

AbstractFor difference schemes the initial-boundary problem for quasi-linear parabolic-type equations, ’a priori weight estimates’ of the error have been found. These estimates show how much the accuracy of difference schemes near the boundary of a time rectangle is higher than in the middle of it. Sufficient conditions of smoothness of the coefficients and the right-hand side of the quasi-linear parabolic equation and the initial conditions have been found. These conditions ensure a correctness of these a priori estimates.

On the Dynamic Contact Problem for a Viscoelastic Plate

2010 ◽  
Author(s):  
Igor Bock
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Dynamic Contact ◽  
Viscoelastic Plate ◽  
Weak Formulation ◽  
Priori Estimates ◽  
Elliptic Variational Inequalities ◽  
Initial Boundary Value

We deal with an initial-boundary value problem describing the perpendicular vibrations of an anisotropic viscoelastic plate free on its boundary and with a rigid inner obstacle. A weak formulation of the problem is in the form of the hyperbolic variational inequality. We solve the problem using the discretizing the time variable. The elliptic variational inequalities for every time level are uniquely solved. We derive the a priori estimates and the convergence of the sequence of segment line functions to a variational solution of the considered problem.

scholarly journals Parabolic Nonlinear Second Order Slip Reynolds Equation: Approximation and Existence

2007 ◽  
Vol 12 (1) ◽  
pp. 3-20
Author(s):  
K. Ait Hadi
Keyword(s):  
Reynolds Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Priori Estimates ◽  
Degenerate Parabolic ◽  
Initial Boundary Value ◽  
Model Existence ◽  
Nonlinear Degenerate

This work studies an initial boundary value problem for nonlinear degenerate parabolic equation issued from a lubrication slip model. Existence of solutions is established through a semi discrete scheme approximation combined with some a priori estimates.

scholarly journals A PROBLEM ON VIBRATION OF A BAR WITH UNKNOWN BOUNDARY CONDITION ON A PART OF THE BOUNDARY

2017 ◽  
Vol 23 (2) ◽  
pp. 7-14
Author(s):  
A. B. Beylin ◽  
L. S. Pulkina
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Sobolev Space ◽  
Unique Solvability ◽  
A Priori Estimates ◽  
A Priori ◽  
The Other ◽  
Unknown Boundary ◽  
Priori Estimates ◽  
The Subject

In this paper, we study an inverse problem for hyperbolic equation. This problem arises when we consider vibration of a nonhomogeneous bar if one endpoint is fixed by spring but behavior of the other is unknown and is the subject to find. Overdetermination is given in the form of integral with respect to spacial variable. Unique solvability of this problem is proved under some conditions on data. The proof is based on a priori estimates in Sobolev space.

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