Solution of some partial differential equations using State space techniques

Abstract

The spatial discretization technique for the solution of certain partial differential equations is discussed. After determining a valid mathematical representation for the canonical parabolic partial differential equation two methods of determining the stability of the resulting high order system of ordinary differential equations is presented. Next the solution to these ordinary differential equations is presented using state space techniques. An expression for the state transition matrix of the ordinary differential equations representing the diffusion equation is obtained as a function of n (the order of the approximation). Applications of the developed theory include a comparison of the minimum energy controls for the diffusion equation obtained by spatial discretization and obtained by harmonic truncation.