An Algorithm for Numerical Integration of ODE with Sampled Unknown Functional Factors

Abstract

The problem of having ordinary differential equations (ODE) whose coefficients are unknown functions is frequent in several fields. Sometimes, it is possible to obtain samples of the values of these functions in different instants or spatial points. The present paper presents a methodology for the numeric solving of these ODE. There are approximations to the problem for specific cases of equations, especially in the case where the parameters correspond to constants. Other studies focus on the case in which the functions under consideration are linear or meet a certain condition. There are two main advantages of the proposed algorithm. First, it does not impose any condition over the data or the subsequent function from where these sample data are derived. Additionally, the methodology used in the functions modeling can control the possibility of overfitting in the function modeling. This is a crucial point in order to limit the influence of model biases in the numerical solution of the ordinary differential equation under study.