Solving Inverse ODE Using Bezier Functions

The simplest inverse boundary value problem is to identify the differential equation and the boundary conditions from a given set of discrete data points. For an ordinary differential equation, it would involve finding a function, which when expressed through some function of itself and its derivatives, and integrated using particular boundary conditions would generate the given data. Parametric Bezier functions are excellent candidates for these functions. They allow efficient approximation of data and its derivative content. The Bezier function is smooth and continuous to a high degree. In this paper the best Bezier function to fit the data represents this function which is being sought. This Bezier approximation also determines the boundary conditions. Next, a generic form of the differential equation is assumed. The Bezier function and its derivatives are then used in this generic form to establish the exponents and coefficients of the various terms in the actual differential equation. The paper looks at homogeneous ordinary differential equations and shows it can recover the exact form of both linear and nonlinear differential equations. Two examples are presented. The first example uses data from the Bessel equation, representing a linear equation. The second example uses the data from the Blassius equation which is nonlinear. In both cases the exact form of the equation is identified from the given discrete data.