Reduced order models for diffusion systems

Mathematical models for diffusion processes like heat propagation, dispersion of pollutants, etc. are normally partial differential equations which involve certain unknown parameters. To use these mathematical models as substitutes of the true system, one has to determine these parameters. Partial differential equations (PDE) of the form partial derivativeu(x, t)/partial derivativet = Lu(x, t) (1) where L is a linear differential (spatial) operator, describe infinite dimensional dynamical systems. To compute a numerical solution for such partial differential equations, one has to approximate the underlying system by a finite order one. By using this finite order approximation, one then computes an approximate numerical solution for the PDE. Here, we consider a simple case of heat propagation in a homogeneous wall. The resulting partial differential equation, which is of the form (1), normally involves some unknown parameters. To estimate these unknown parameters, one has to approximate the infinite order model by a finite order model. For this purpose, we construct some finite order models by using certain existing numerical techniques like Galerkin and Collocation, etc. And, later, depending on their merit one chooses a suitable approximation for estimating the unknown parameters. In this paper we concentrate only on the model reduction aspects of the problem and not on the parameter estimation part. In particular, we examine the model order reduction capabilities of the Chebyshev polynomial methods used for solving partial diferential equations.