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.

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 Convergence Results on the Boundary Conditions for 2D Large-Scale Primitive Equations in Oceanic Dynamics

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Yuanfei Li
Keyword(s):  
Boundary Conditions ◽  
Large Scale ◽  
Weather Prediction ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Convergence Result ◽  
Primitive Equations ◽  
Convergence Results ◽  
Initial Boundary Value

In this paper, the initial boundary value problem for the two-dimensional large-scale primitive equations of large-scale oceanic motion in geophysics is considered, which are fundamental models for weather prediction. By establishing rigorous a priori bounds with coefficients and deriving some useful inequalities, the convergence result for the boundary conditions is obtained.

scholarly journals The Asymptotic Solution of the Initial Boundary Value Problem to a Generalized Boussinesq Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Zheng Yin ◽  
Feng Zhang
Keyword(s):  
Asymptotic Solution ◽  
Boundary Problem ◽  
Existence And Uniqueness ◽  
Large Time ◽  
Boussinesq Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Generalized Boussinesq Equation

TheL2space solution of an initial boundary problem for a generalized damped Boussinesq equation is constructed. Certain assumptions on the coefficients of the equation are found to show the existence and uniqueness of the global solution to the initial boundary problem. The explicit expression for the large time asymptotic solution is obtained.

scholarly journals Rothe method for a mixed problem with an integral condition for the two-dimensional diffusion equation

2003 ◽  
Vol 2003 (16) ◽  
pp. 899-922 ◽  
Author(s):  
Nabil Merazga ◽  
Abdelfatah Bouziani
Keyword(s):  
Diffusion Equation ◽  
Integral Condition ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Two Dimensional ◽  
Dimensional Diffusion ◽  
Rothe Method ◽  
Semidiscrete Approximation ◽  
Initial Boundary Value ◽  
The Given

This paper deals with an initial boundary value problem with an integral condition for the two-dimensional diffusion equation. Thanks to an appropriate transformation, the study of the given problem is reduced to that of a one-dimensional problem. Existence, uniqueness, and continuous dependence upon data of a weak solution of this latter are proved by means of the Rothe method. Besides, convergence and an error estimate for a semidiscrete approximation are obtained.

The Rothe method for the wave equation in several space dimensions

1979 ◽  
Vol 84 (1-2) ◽  
pp. 1-18 ◽  
Author(s):  
Erich Martensen
Keyword(s):  
Wave Equation ◽  
Space Dimension ◽  
Continuous Solution ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
First Order ◽  
Rothe Method ◽  
Initial Boundary Value ◽  
The Given ◽  
Continuous Problem

SynopsisThe interior initial-boundary value problem for the wave equation in m ≧ 1 space dimension is considered for vanishing boundary values. Certain regularity, dependent on m, is required for the solution and additionalboundary conditions, the number of which being also dependent on m, are imposed on the given right hand side. Emphazising the case m = 3, the Rothe method is applied after the problem has been rewritten as a hyperbolic first order evolution problem for m + 1 unknown functions. The sequence of discrete solutions obtained is shown to be discretely convergent to the continuous solution in the sense of uniform convergence if the solution of the continuous problem is assumed to exist. A priori estimates are derived both for the discrete solutions and the continuous solution.

On Dynamical Three-Dimensional Fluid-Solid Interaction Problem

2008 ◽  
Vol 15 (4) ◽  
pp. 601-618
Author(s):  
Gia Avalishvili ◽  
Mariam Avalishvili ◽  
David Gordeziani
Keyword(s):  
Variational Formulation ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Dimensional Problem ◽  
Energy Equality ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Fluid Solid Interaction ◽  
Interaction Problem

Abstract The present paper is devoted to the investigation of one dynamical three-dimensional mathematical model of the fluid-solid interaction. The variational formulation of the corresponding initial boundary problem is considered and a problem for abstract second order evolution equation is formulated, which is a generalization of the three-dimensional initial boundary value problem. For the stated abstract problem the existence and uniqueness of solution, and the energy equality are proved, which yield the corresponding result for the dynamical three-dimensional problem of fluid-solid interaction.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

Sign in / Sign up

Export Citation Format

Share Document