Impulse response model for a class of distributed parameter systems

A mathematical model is developed here for the impulse response of a class of systems represented by a general linear parabolic partial differential equation. Using a generalized eigenfunction expansion and the concept of a sampled-data system, a canonical set of state equations is obtained in discrete form. The impulse inputs may be any known time-varying functions on the given boundaries of the system or may be assumed to be distributed over a finite boundary region. For this computer model, no spatial discretization is required. Each discrete-time equation represents the dynamic behavior of each eigenvalue of the system and the response variable at any discrete location is given by a spatial linear combination of these state variables. Due to the discrete formulation of the system equations, numerical stability is assured. The equations are stable for any sampling time selected and the solution at any particular time desired may be simply obtained by incorporating this value directly into the model equations. The application of this method is demonstrated by solving a two-dimensional heat transfer problem subject to a single impulse input and comparisons where made to the finite element method of solution.