scholarly journals Smoothing properties in multistep backward difference method and time derivative approximation for linear parabolic equations

2005 ◽  
Vol 2005 (4) ◽  
pp. 523-536
Author(s):  
Yubin Yan
Keyword(s):  
Difference Method ◽  
Parabolic Equations ◽  
Parabolic Problem ◽  
Approximation Scheme ◽  
Time Derivative ◽  
Nonsmooth Data ◽  
Smoothing Property ◽  
Linear Parabolic Equations ◽  
Linear Parabolic Problem ◽  
Data Error

A smoothing property in multistep backward difference method for a linear parabolic problem in Hilbert space has been proved, where the operator is selfadjoint, positive definite with compact inverse. By using the solutions computed by a multistep backward difference method for the parabolic problem, we introduce an approximation scheme for time derivative. The nonsmooth data error estimate for the approximation of time derivative has been obtained.

scholarly journals Asymptotics of the Solution of Bisingular Problem for a System of Linear Parabolic Equations. I

2015 ◽  
Vol 20 (1) ◽  
pp. 5-17
Author(s):  
M. V. Butuzova
Keyword(s):  
Small Parameter ◽  
Asymptotic Expansions ◽  
Parabolic Equations ◽  
Parabolic Problem ◽  
Linear Parabolic Equations

Given a bisingular parabolic problem for a system of linear parabolic equations, we construct an asymptotics for the solution of any order with respect to a small parameter, without using the joining procedure for asymptotic expansions.

scholarly journals A Note on Parabolic Homogenization with a Mismatch between the Spatial Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Liselott Flodén ◽  
Anders Holmbom ◽  
Marianne Olsson Lindberg ◽  
Jens Persson
Keyword(s):  
Spatial Scale ◽  
Parabolic Problem ◽  
Spatial Scales ◽  
Time Derivative ◽  
Local Problem ◽  
Linear Parabolic Problem ◽  
Spatial Oscillations ◽  
Spatial Oscillation

We consider the homogenization of the linear parabolic problem which exhibits a mismatch between the spatial scales in the sense that the coefficient of the elliptic part has one frequency of fast spatial oscillations, whereas the coefficient of the time derivative contains a faster spatial scale. It is shown that the faster spatial microscale does not give rise to any corrector term and that there is only one local problem needed to characterize the homogenized problem. Hence, the problem is not of a reiterated type even though two rapid scales of spatial oscillation appear.

scholarly journals Asymptotic limit of linear parabolic equations with spatio-temporal degenerated potentials

2020 ◽  
Vol 26 ◽  
pp. 50
Author(s):  
Pablo Àlvarez-Caudevilla ◽  
Matthieu Bonnivard ◽  
Antoine Lemenant
Keyword(s):  
Parabolic Equations ◽  
Parabolic Problem ◽  
Decay Estimate ◽  
Convergence Result ◽  
Asymptotic Limit ◽  
Cylindrical Domain ◽  
Noncylindrical Domain ◽  
Linear Parabolic Equations ◽  
Exponential Decay Estimate ◽  
Spatio Temporal

In this paper, we observe how the heat equation in a noncylindrical domain can arise as the asymptotic limit of a parabolic problem in a cylindrical domain, by adding a potential that vanishes outside the limit domain. This can be seen as a parabolic version of a previous work by the first and last authors, concerning the stationary case [Alvarez-Caudevilla and Lemenant, Adv. Differ. Equ. 15 (2010) 649-688]. We provide a strong convergence result for the solution by use of energetic methods and Γ-convergence technics. Then, we establish an exponential decay estimate coming from an adaptation of an argument due to B. Simon.

Sign in / Sign up

Export Citation Format

Share Document