ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

scholarly journals A Reaction-Diffusion System with Nonlinear Nonlocal Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Alexander Gladkov ◽  
Alexandr Nikitin
Keyword(s):  
Boundary Conditions ◽  
Nonlocal Boundary ◽  
Local Existence ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Initial Boundary Value

We consider initial boundary value problem for a reaction-diffusion system with nonlinear and nonlocal boundary conditions and nonnegative initial data. We prove local existence, uniqueness, and nonuniqueness of solutions.

scholarly journals On the solvability of a mixed problem for a fractional partial differential equation with delayed time argument and Laplace operators with nonlocal boundary conditions in Sobolev classes

2020 ◽  
Vol 22 (1) ◽  
pp. 13-23
Author(s):  
Mahkambek M. Babayev
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Sobolev Classes ◽  
Laplace Operators ◽  
Partial Differential ◽  
Initial Boundary Value

In this paper, we study a problem with initial functions and boundary conditions for partial differential equations of fractional order with Laplace operators. The boundary conditions of the problem are nonlocal, and the solution is supposed to belong to one of Sobolev classes. The solution of the initial boundary value problem is constructed as the sum of a series of multidimensional spectral problem’s eigenfunctions. The eigenvalues of the spectral problem are found and the corresponding system of eigenfunctions is constructed. It is shown that this system is complete and forms a Riesz basis in the subspaces of Sobolev spaces. Basing on the completeness of the eigenfunctions’ system, the uniqueness theorem for the solution of the problem is proved. The existence of a regular solution of the initial boundary value problem is proved in Sobolev subspaces.

scholarly journals INVERSE PROBLEM WITH INTEGRAL IN TIME OVERDETERMINATION AND NONLOCAL BOUNDARY CONDITIONS FOR HYPERBOLIC EQUATION

2017 ◽  
Vol 22 (1-2) ◽  
pp. 27-32
Author(s):  
A. V. Duzheva
Keyword(s):  
Inverse Problem ◽  
Boundary Conditions ◽  
Hyperbolic Equation ◽  
Boundary Problem ◽  
Existence Result ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Nonlocal Boundary Conditions ◽  
Time Variable ◽  
Initial Boundary Problem

In this article, we consider a question of sovability of an inverse problem for a linear hyprbolic equation. Properties of the solution of an associated nonlocal initial-boundary problem with displacement in boundary conditions are used to develop an existence result for the identification of the unknown source. Overdetermination is represented as integral with respect to time-variable.

scholarly journals On the stable difference scheme for the time delay telegraph equation

2020 ◽  
Vol 99 (3) ◽  
pp. 105-119
Author(s):  
A. Ashyralyev ◽  
◽  
K. Turk ◽  
D. Agirseven ◽  
◽  
...  
Keyword(s):  
Time Delay ◽  
Difference Scheme ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Telegraph Equation ◽  
Stability Estimates ◽  
The Difference ◽  
Initial Boundary Value

The stable difference scheme for the approximate solution of the initial boundary value problem for the telegraph equation with time delay in a Hilbert space is presented. The main theorem on stability of the difference scheme is established. In applications, stability estimates for the solution of difference schemes for the two type of the time delay telegraph equations are obtained. As a test problem, one-dimensional delay telegraph equation with nonlocal boundary conditions is considered. Numerical results are provided.

scholarly journals An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

2021 ◽  
Vol 5 (3) ◽  
pp. 63
Author(s):  
Emilia Bazhlekova
Keyword(s):  
Boundary Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stability Estimates ◽  
Inverse Source Problem ◽  
Time Data ◽  
Final Time ◽  
Generalized Eigenfunction ◽  
The One ◽  
Initial Boundary Value

An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established.

Initial-boundary value problems for a time-fractional differential equation with involution perturbation

2019 ◽  
Vol 14 (3) ◽  
pp. 312 ◽  
Author(s):  
Nasser Al-Salti ◽  
Sebti Kerbal ◽  
Mokhtar Kirane
Keyword(s):  
Boundary Value Problems ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Nonlocal Boundary Conditions ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value ◽  
Formal Solutions ◽  
The Given

Direct and inverse initial-boundary value problems of a time-fractional heat equation with involution perturbation are considered using both local and nonlocal boundary conditions. Results on existence of formal solutions to these problems are presented. Solutions are expressed in a form of series expansions using appropriate orthogonal basis obtained by separation of variables. Convergence of series solutions are obtained by imposing certain conditions on the given data. Uniqueness of the obtained solutions are also discussed. The obtained general solutions are illustrated by an example using an appropriate choice of the given data.

On convergence and error estimates for the horizontal line method applied to Maxwell's initial boundary problem in anisotropic, inhomogeneous media

1980 ◽  
Vol 86 (1-2) ◽  
pp. 53-59 ◽  
Author(s):  
Rainer Picard
Keyword(s):  
Perturbation Theory ◽  
Error Estimates ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Horizontal Line ◽  
Line Method ◽  
Initial Boundary Problem ◽  
Convergence Results ◽  
Rothe Method ◽  
Initial Boundary Value

SynopsisIn the following paper, the horizontal line method (the Rothe method) is applied to Maxwell's initial boundary value problem. By means of results from abstract perturbation theory, convergence results and error estimates are established.

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