Reduced order models for diffusion systems

Mathematical models for diffusion processes like heat propagation, dispersi on of pollutants, etc. are normally partial differential equations which in volve certain unknown parameters. To use these mathematical models as subst itutes of the true system, one has to determine these parameters. Partial d ifferential 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 dime nsional 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 comput es 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), norm ally 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 ex isting numerical techniques like Galerkin and Collocation, etc. And, later, depending on their merit one chooses a suitable approximation for estimati ng the unknown parameters. In this paper we concentrate only on the model reduction aspects of the pro blem and not on the parameter estimation part. In particular, we examine th e model order reduction capabilities of the Chebyshev polynomial methods us ed for solving partial diferential equations.