Opuscula Mathematica

Opuscula Math.
, no. 2
 (), 259-289
Opuscula Mathematica

A finite difference method for nonlinear parabolic-elliptic systems of second order partial differential equations

Abstract. This paper deals with a finite difference method for a wide class of weakly coupled nonlinear second-order partial differential systems with initial condition and weakly coupled nonlinear implicit boundary conditions. One part of each system is of the parabolic type (degenerated parabolic equations) and the other of the elliptic type (equations with a parameter) in a cube in \(\mathbf{R}^{1+n}\). A suitable finite difference scheme is constructed. It is proved that the scheme has a unique solution, and the numerical method is consistent, convergent and stable. The error estimate is given. Moreover, by the method, the differential problem has at most one classical solution. The proof is based on the Banach fixed-point theorem, the maximum principle for difference functional systems of the parabolic type and some new difference inequalities. It is a new technique of studying the mixed-type systems. Examples of physical applications and numerical experiments are presented.
Keywords: partial differential equation, parabolic-elliptic system, finite difference method, finite difference scheme, consistence, convergence, stability, error estimate, uniqueness.
Mathematics Subject Classification: 65M06, 65M12, 65M15, 35M10, 39A10, 39B72.
Cite this article as:
Marian Malec, Lucjan Sapa, A finite difference method for nonlinear parabolic-elliptic systems of second order partial differential equations, Opuscula Math. 27, no. 2 (2007), 259-289
Download this article's citation as:
a .bib file (BibTeX), a .ris file (RefMan), a .enw file (EndNote)
or export to RefWorks.

RSS Feed

horizontal rule

ISSN 1232−9274, e-ISSN 2300−6919, DOI https://doi.org/10.7494/OpMath
Copyright © 2003−2017 OPUSCULA MATHEMATICA
Contact: opuscula@agh.edu.pl
Made by Tomasz Zabawa

horizontal rule

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.