Solution of inverse ODE using Bezier functions

Abstract

The identification of the actual form of the differential equation and the boundary conditions from a set of discrete data points has been a challenging problem for many decades. A unique two-step approach developed in this article can solve this particular inverse boundary value problem. In the first step, the data are fitted using the best Bezier function. Parametric Bezier functions, based on Bernstein polynomials, allow efficient approximation of smooth data and can generate the derivatives. Bezier functions are also good at handling measured data. The Bezier fit is smooth and large orders of the polynomials can be used for approximation without oscillation. In the second step, the known derivatives are introduced in a generic model of the differential equation. This generic form includes two types of unknowns, real numbers and integers. The real numbers are part of the coefficients of the various terms multiplying the derivatives, while the integers are exponents of the derivatives. The unknown exponents and coefficients are identified using an error formulation through discrete programming. This article looks at homogeneous ordinary differential equations (ODEs) and demonstrates it can recover the exact form of both linear and nonlinear differential equations. Two examples are solved. In both cases, the objective is to identify the exact form of the differential equation. The given data are exact, smooth and they represent solutions to known differential equations.