Turning Vector Partial Differential Equations into Multidimensional Transfer Function Models

Abstract

Transfer function models for the description of physical systems have recently been introduced. They provide an alternative to the conventional representation by partial differential equations (PDE) and are suitable for computer implementation. This paper presents transfer function modelling for vector PDEs. They arise from the physical analysis of multidimensional systems in terms of potential and¯ux quantities. Expressing the resulting coupled PDEs in vector form facilitates the direct formulation of boundary and interface conditions in their physical context. It is shown how a carefully constructed transformation for the space variable leads to transfer function models for vector PDEs. They are the starting point for the derivation of discrete models by standard methods for one-dimensional systems. The presented functional transformation approach is suitable for a number of technical applications, like electromagnetics, optics, acoustics and heat and mass transfer. Examples are given for the voltage and current distribution on an electrical transmission line and the velocity and force distribution on longitudinal vibrating strings.

Multidimensional transfer function models

“…Other applications include the propagation of signals on transmission lines [20] and other examples from optics, acoustics or heat and mass transfer.…”

“…For the operator according to (18) the adjoint operator is given by (19) and the Lagrange identity holds in the form (20) An operator is called self-adjoint, if it is equal to its adjoint operator . This property requires that in (18).…”

“…Therefore the eigenvalues can be indexed with integer variable . Replacing the arbitrary functions and in the Lagrange identity (20) with the eigenfunctions and and using (21), (22), and (23) leads to…”

“…Due to the first-order spatial derivative in , the spatial differentation operator is not self-adjoint. Therefore, the adjoint operator must be found to fulfill the Lagrange identity (20). It is shown in this section that even with nonself-adjoint spatial operators it is possible to construct a spatial transformation to obtain a MD TFM for vector PDEs.…”