IJPAM: Volume 57, No. 4 (2009)


Harijs Kalis$^1$, Ilmars Kangro$^2$, Aigars Gedroics$^3$
$^1$Institute of Mathematics and Informatics
University of Latvia
29, Raina Blvd., Riga, LV 1459, LATVIA
e-mail: kalis@lanet.lv
$^2$Department of Engineering Science
Rezekne Higher Education Institute
90, Atbrivosanas Aleja, Rezekne, LV 4601, LATVIA
e-mail: kangro@cs.ru.lv
$^3$Faculty of Physics and Mathematics
University of Latvia
8, Zelluiela, Riga, LV 1002, LATVIA
e-mail: aigars.gedroics@lu.lv

Abstract.We study the numerical methods for solving the initial-boundary value problems of some nonlinear heat transfer equations in multi-layer domain. The approximation of corresponding initial boundary value problems is based on the finite volume method (FVM), on the boundary element method (BEM), and on the finite-difference scheme (FDS). These methods enable to reduce the nonlinear heat transfer problem described by nonlinear partial differential equations (PDEs) to initial value problem for system of nonlinear ordinary differential equations (ODEs) of the first order. An example of the initial-boundary problem for PDEs (with power functions)

\frac{\partial u(x,t)}{\partial t}=\frac{\partial^2 \lambd...
...1}}{\partial x^2} +a (u(x,t))^\beta,\quad x \in [0,l], t > 0,
\end{displaymath} (1)

by $ \sigma \ge 0, \beta >0, \lambda>0, a \ge 0 $ and with conditions $ u(0,t)=u(l,t)=0, u(x,0)=u_0(x) \ge 0$ is considered.

A large number of papers in the time period of 1970-1990 are devoted to the quasilinear parabolic equations with the blow-up solutions. In papers [#!Gal-81!#], [#!Gal-82!#], [#!Gal-82-1!#], [#!Sam-83!#],[#!Sam-83-1!#], [#!Sam-84!#], [#!Gal-85!#], [#!Gal-85-1!#], [#!Gal-86!#], [#!sam!#], [#!GalP-88!#] the theoretical investigations (self-similar solutions and a'priori estimations for solving Cauchy and boundary-value problems) and numerical experiments (see [#!Sam-83-2!#], [#!Sam-83-3!#], [#!Mak-84!#], [#!Mak-91!#], [#!Ceg-94!#]) by $\lambda=a=1$ were done. In this paper we study the behaviour of the solutions at the time and also when $ t \rightarrow \infty,$ depending on the parameters $\sigma, \beta, \lambda, a.$

Depending on the parameters two type of solutions are obtained:

1) for large value of the time $t$ the solution is stationary or tends to zero,

2) in the fixed time moment the solution has ``blow up" phenomena - the solution is unbounded and tends to infinity in a small interval or in all domains by a fixed time moment.

Received: November 24, 2009

AMS Subject Classification: 65M06, 65N06, 65M08, 65M20, 65M22, 65N22,65N40

Key Words and Phrases: nonlinear heat transfer, multi-layer domain, the finite volume method, finite-difference schemes, ``blow up" solutions

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 57
Issue: 4