Nonconservation Law Equation in Mathematical Modelling: Aspects of Approximation

In this paper we treat mathematical and computational models of thermodynamical systems in viscous and diiusive media as optimal information control problems. Under assumption of suucient smoothness on the perturbed Hamiltonian of the system we propose reformulations of the classical problems using the concept of informational limit. In the general situation we use Steklov's operator technique to derive generalisations of classical Hamilton-Jacobi-Bellman (HJB) equation in stochastic, nonsmooth and deterministic cases. We show that nonconservation law equation appears in a natural way from such a consideration. For its approximation we use an evolution-associated Markov Chain which permits us to derive stability conditions for our computational model. Computational models of this type give a description of generalised dynamical systems (GDS) which include the modeler (decision maker DM) as an intrinsic part. Associated problems of mathematical modelling in semiconductor technology (including modelling of multilayered structures and superlattices) are discussed. 1. Macroscopic thermodynamical systems and non-conservation law. Many challenging mathematical problems with a wide range of engineering applications arise from the coupled eld theory where interconnection of at least two physical (chemical or biological) elds is essential to obtain a plausible picture of phenomena, processes etc. Presence of a hyperbolic-type operator in many mathematical models of the coupled eld theory precludes an assumption (often made a priori) that subgrid (microscopic) phenomena can be extracted from macroscopic ow (see Oran and Boris 1]). On the other hand approaches based on a parabolisation of the model and attempts to solve the problem of coherent feedback of small scales with respect to large ones by adding a smoothing diiusion-type term into the model in many cases are not adequate to the physics of the process. The mathematical and computational problems which arise from the coupled eld theory (dynamical thermoelasticity, piezoelasticity and semiconductor device modelling, Melnik 2-4]) was the original stimulation for this paper in which we revise the concepts of deterministic mathematical models and conservation law in mathematical modelling. As a rule rigorous mathematical justiications of dynamical system evolution is based on an a priori assumption of system conservativeness and/or ergodicity. Whereas the rst assumption has its roots in Newton-Hamilton classical mechanics, the second one is grounded on the never-proved Gibbs conjecture (Goldstein 5]). Formally a division between Hamiltonian and ergodic systems can be seen as a division between the following two classes of mathematical models: models which are "more hyperbolic than non-linear"; models which are "more …