Liouville theorems for the solution of a second-order linear parabolic equation with discontinuous coefficients

1969 ◽  
Vol 5 (5) ◽  
pp. 359-363
Author(s):  
R. Ya. Glagoleva
Keyword(s):  
Parabolic Equation ◽  
Discontinuous Coefficients ◽  
Second Order ◽  
Linear Parabolic Equation ◽  
Liouville Theorems ◽  
Order Linear

CAUCHY–DIRICHLET PROBLEM ASSOCIATED TO DIVERGENCE FORM PARABOLIC EQUATIONS

2004 ◽  
Vol 06 (03) ◽  
pp. 377-393 ◽  
Author(s):  
MARIA ALESSANDRA RAGUSA
Keyword(s):  
Parabolic Equation ◽  
Dirichlet Problem ◽  
Parabolic Equations ◽  
Discontinuous Coefficients ◽  
Divergence Form ◽  
Second Order ◽  
Linear Parabolic Equation

In this note we study the Cauchy–Dirichlet problem related to a linear parabolic equation of second order in divergence form with discontinuous coefficients. Moreover we prove estimates in the space [Formula: see text], for every 1<p<∞.

High Accuracy Compact Operator Methods for Two-Dimensional Fourth Order Nonlinear Parabolic Partial Differential Equations

2017 ◽  
Vol 17 (4) ◽  
pp. 617-641 ◽  
Author(s):  
Ranjan Kumar Mohanty ◽  
Deepti Kaur
Keyword(s):  
Parabolic Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Kolmogorov Equation ◽  
Linear Parabolic Equation ◽  
Two Dimensional ◽  
Partial Differential ◽  
Order Linear

AbstractIn this study, we develop and implement numerical schemes to solve classes of two-dimensional fourth-order partial differential equations. These methods are fourth-order accurate in space and second-order accurate in time and require only nine spatial grid points of a single compact cell. The proposed discretizations allow the use of Dirichlet boundary conditions only without the need to discretize the derivative boundary conditions and thus avoids the use of ghost points. No transformation or linearization technique is used to handle nonlinearity and the obtained block tri-diagonal nonlinear system has been solved by Newton’s block iteration method. It is discussed how our formulation is able to tackle linear singular problems and it is ensured that the methods retain their orders and accuracy everywhere in the solution region. The proposed two-level method is shown to be unconditionally stable for a class of two-dimensional fourth-order linear parabolic equation. We also discuss the alternating direction implicit (ADI) method for solving two-dimensional fourth-order linear parabolic equation. The proposed difference methods has been successfully tested on the two-dimensional vibration problem, Boussinesq equation, extended Fisher–Kolmogorov equation and Kuramoto–Sivashinsky equation. Numerical results demonstrate that the schemes are highly accurate in solving a large class of physical problems.

Time and space meshes adaptivity for the resolution of a semi-linear heat equation

2021 ◽  
pp. 1-10
Author(s):  
Nejmeddine Chorfi
Keyword(s):  
Parabolic Equation ◽  
Heat Equation ◽  
Order Of Convergence ◽  
Linear Parabolic Equation ◽  
Spectral Elements ◽  
Euler Scheme ◽  
Time Step ◽  
Implicit Euler Scheme ◽  
Linear Heat ◽  
Error Indicators

The aim of this work is to highlight that the adaptivity of the time step when combined with the adaptivity of the spectral mesh is optimal for a semi-linear parabolic equation discretized by an implicit Euler scheme in time and spectral elements method in space. The numerical results confirm the optimality of the order of convergence. The later is similar to the order of the error indicators.

Bases for Second Order Linear ODEs

2020 ◽  
Vol 127 (9) ◽  
pp. 849-849
Author(s):  
Peter McGrath
Keyword(s):  
Second Order ◽  
Order Linear ◽  
Linear Odes
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