Abstract
This memory deals with the construction of stable numerical solutions of coupled parabolic
and hyperbolic systems. The characteristic stages of this memory are: the construction of
discrete solutions using difference schemes and a discrete separation of variables method,
the study of the stability and the consistency of the calculated solution, and the use of a
method of projections in order to extend the obtained results to a more general class of
initial value functions.
By the application of a discrete separation of variables method, the proposed numerical
solution to the problems, is the exact solution of a coupled difference system, which is obtained
from the discretization in finite differences of the coupled mixed partial differential
system.
The boundary conditions of the problems which have been dealt here are coupled and of
non-Dirichlet type.
Our methodological approach is alternative to the more traditional algebraic treatment
which writes the scheme in matrix form, and offers the advantage of not having to solve
the big-sized algebraic systems with matrix blocks appearing in the standard difference
method, thanks to the use of a discrete separation of variables method.
The uncoupled techniques aren’t applied except for excessively restrictive hypotheses. Even
uncoupling the partial differential equation, the boundary conditions don’t necessarily have
to be so, hence the uncoupled techniques aren’t very suitable.
The treated problems model, among others, diffusion problems, nerve conduction and armament
models (chapter 2), microwave heating processes, optics, cardiology and soil flows
(chapter 3).