QUALITATIVE PARTIAL-DIFFERENTIAL EQUATIONS AND THEIR REALISTIC APPLICATIONS

Partial differential equations are quantitatively well known. However very little has been done in terms of their qualitative analysis. This paper presents two methodologies, a qualitative homogenization, and a qualitative simplification, for qualitative analysis of models which contain partial differential equations. The basic idea of qualitative homogenization is a qualitative version of a conventional numerical method (a partial differential equation is substituted by a set of ordinary differential equations). The simplification method is based on the fact that if a certain submodel of a complex qualitative model is ignored, then the final set of qualitative solutions is a superset of the qualitative solutions of the original complex model. A case study (enzyme immobilization in a porous membrane) is presented in detail.