A method for solving the parameter identification problem for ordinary differential equations of the second order
This paper deals with the determination of optimal values for some unknown parameters \(p = (p_1,\dots,p_s)\) entering in a second-order differential equation \(y'' = f(t,y,y',p)\) from a (large) set of experimental data \((t_i, y_i)\) of some solution. Here the optimality of \(p\) must be understood in the sense to minimize the differences between the experimental values and the values given by the differential equation. The above problem falls within the class of parameter identification problems for which several approaches have been proposed. Here the authors combine the ideas of several previous approaches in order to determine its optimal solution. Firstly, they determine a smooth function from the given experimental data by using the so-called moving least squares techniques. Then, they discretize the differential equation by means of the trapezoidal rule and solve (in the least squares sense) the resulting set of algebraic equations in the parameters. Finally, further refinement of the parameters are proposed. To test the efficiency of the method, the authors apply it to a second-order equation which arises in the modelization of an enzyme effusion problem (\textit{Hemker} \& \textit{Kok}, T.R. NM-R9301, Mathematisch Centrum, Amsterdam) which contains four parameters and to a linear second-order equation with two parameters.
scientific article; zbMATH DE number 892426
scientific article