Identifying differential equations by Galerkin’s method

A numerical technique based on Galerkin's method is presented for computing unknown parameters or functions occurring in a differential equation whose solution is known. Under certain conditions a solution can be shown to exist to the integral equa- tion formulation of this problem. It is also shown that the resulting nonlinear system is nonsingular. 1. Introduction. In most mathematical modeling problems differential equations of specific forms are derived which describe a system. Values of the coefficients, which can be constants or functions, of the differential equations are usually specified, and the solutions are calculated or presented in closed form, with little, if any, indication of how the coefficients can be estimated from observations. Clearly, this inverse problem is very interesting and important but somewhat difficult. The purpose of this paper is to describe a numerical technique for calculating unknown functions in a differential equation (or system) supposing its solution to be known. That is, we assume a function >» is given whose derivative y is continuous on (0, T) and such that y satisfies the differential equa- tion