Generalizing ODE modeling structure for multivariate systems with distributed parameters

We propose an approximate model in form of a system of ordinary differential equations (ODEs) for a class of first-order multivariate linear partial differential equations (PDEs) of the hyperbolic type. The resulting scheme utilizes the method of moments and least-square approximations over orthogonal polynomial bases for the factors of PDE depending on the spatial coordinate of the PDE. The class of examined PDEs appears typically in population balance systems with fines removal. The proposed modeling approach is generally of interest for control and optimization of multivariate systems with distributed parameters.