Fitting Functional Equations to Experimental Data.
An overview of methods for computational solution of system identification problems. In physics, engineering, biology, and economics, a basic problem is that of determining the structure of a system from observations over time. The unknown structure is reflected in unknown parameters that appear in the differential equations or in the initial conditions of a nonlinear boundary-value problem formulation. Functional equations may be reduced exactly or approximately to systems of ordinary differential equations, as the wave equation can be reduced either by taking Laplace transforms or by semidiscretization. Even integral equations may be converted into initial-value problems. If the linearized differential equations are unstable, the method of invariant imbedding provides a reformulation that is stable; by enabling the unknown to be directly determined, invariant imbedding also obviates the need to solve linear algebraic equations. 8 pp. Refs.