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.

scholarly journals Additive difference scheme for two-dimensional fractional in time diffusion equation

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 217-226 ◽  
Author(s):  
Sandra Hodzic-Ivanovic ◽  
Bosko Jovanovic
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Difference Scheme ◽  
Numerical Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Two Dimensional ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

An additive finite-difference scheme for numerical approximation of initial-boundary value problem for two-dimensional fractional in time diffusion equation is proposed. Its stability is investigated and a convergence rate estimate is 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.

scholarly journals Factorized difference scheme for 2D fractional in time diffusion equation

2015 ◽  
Vol 9 (2) ◽  
pp. 199-208 ◽  
Author(s):  
Sandra Hodzic
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Difference Scheme ◽  
Numerical Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Two Dimensional ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

A factorized finite-difference scheme for numerical approximation of initial-boundary value problem for two-dimensional fractional in time diffusion equation is proposed. Its stability is investigated and a convergence rate estimate is obtained.

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.

AN INEQUALITY BY GIANAZZA, SAVARÉ AND TOSCANI AND ITS APPLICATIONS TO THE VISCOUS QUANTUM EULER MODEL

2013 ◽  
Vol 15 (05) ◽  
pp. 1250067 ◽  
Author(s):  
XIANGSHENG XU
Keyword(s):  
Weak Solution ◽  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Fisher Information ◽  
Gradient Flow ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Drift Diffusion ◽  
Euler Model ◽  
Initial Boundary Value

In this paper we present a simplified version of a coercivity inequality due to Gianazza, Savaré, and Toscani [The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal.194 (2009) 133–220]. Then we use the inequality to construct a weak solution to the initial-boundary value problem for the viscous quantum Euler model.

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