Finite-Difference Schemes for a Nonlinear Parabolic Problem with Nonlocal Boundary Conditions

2004 ◽  
pp. 47-52 ◽  
Author(s):  
Raimondas Čiegis
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Parabolic Problem ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Nonlinear Parabolic Problem ◽  
Nonlinear Parabolic

scholarly journals ON CONSTRUCTION AND ANALYSIS OF FINITE DIFFERENCE SCHEMES FOR PSEUDOPARABOLIC PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS

2014 ◽  
Vol 19 (2) ◽  
pp. 281-297 ◽  
Author(s):  
Raimondas Čiegis ◽  
Natalija Tumanova
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Two Dimensional ◽  
The Stability

In this paper the one- and two-dimensional pseudoparabolic equations with nonlocal boundary conditions are approximated by the Euler finite difference scheme. In the case of classical boundary conditions the stability of all schemes is investigated by the spectral method. Stability regions of finite difference schemes approximating pseudoparabolic problem are compared with the stability regions of the classical discrete parabolic problem. These results are generalized for problems with nonlocal boundary conditions if a matrix of the finite difference scheme can be diagonalized. For the two-dimensional problem an efficient algorithm is constructed, which is based on the combination of the FFT method and the factorization algorithm. General stability results, known for the three level finite difference schemes, are applied to investigate the stability of some explicit approximations of the two-dimensional pseudoparabolic problem with classical boundary conditions. A connection between the energy method stability conditions and the spectrum Hurwitz stability criterion is shown. The obtained results can be applied for pseudoparabolic problems with nonlocal boundary conditions, if a matrix of the finite difference scheme can be diagonalized.

scholarly journals STATIONARY PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS

2001 ◽  
Vol 6 (2) ◽  
pp. 178-191 ◽  
Author(s):  
R. Čiegis ◽  
A. Štikonas ◽  
O. Štikoniene ◽  
O. Suboč
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Difference Schemes ◽  
Maximum Norm ◽  
Finite Difference Schemes ◽  
Stability Estimates ◽  
Stationary Problems ◽  
Theoretical Results

In this article a stationary problems with general nonlocal boundary conditions is considered. The differential problems and finite difference schemes for solving this problem are investigated. Stability estimates are proved in the maximum norm and the non‐negativity of the solution is investigated. All theoretical results are illustrated by representative examples.

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