Inverse Solution to the Coupled Ordinary Differential Equation

The identification of the actual form of the constant coefficient coupled differential equations and their boundary conditions, from two sets of discrete data points, is possible through a unique two-step approach developed in this paper. In the first step, the best Bezier function is fitted to the data. This allows an effective approximation of the data and the required number of derivatives for the entire range of the independent variable. In the second step, the known derivatives are introduced in a generic model of the coupled differential equation. This generic form includes two types of unknowns, real numbers and integers. The real numbers are the coefficients of the various terms in the differential equations, while the integers are exponents of the derivatives. The unknown exponents and coefficients are identified using an error formulation. Two examples are solved. The given data is exact, smooth and they represent solutions to coupled linear differential equations. The solution is obtained through discrete programming. Three methods are presented. The first is limited enumeration, which is useful if the coefficients belong to a limited set of discrete values. The second is global search using the genetic algorithm for a larger choice of coefficient values. The third uses a state space integrator driven by the genetic algorithm, to minimize the error between known data and that obtained from numerical integration.