Planning an experiment to estimate the parameters of differential equations based on inaccurate observations

Abstract

Previously, the authors of the work constructed an algorithm for estimating the parameters of a system of first-order ordinary differential equations from a large number of inaccurate observations in the vicinity of a chosen point (reference point). However, the development of this topic requires the selection of several reference points for estimating the parameters of a system of differential equations and constructing estimates for higher-order derivatives. In this case, the number of estimated parameters may be greater than the number of equations. In this paper, we construct an algorithm for estimating the second and third derivatives. It is based on the previously constructed estimate of the first derivative with respect to inaccurate observations of the finite difference formula for the first order derivative. The consistency of the estimate of the second derivative under certain conditions imposed on the choice of observation points is proved. The constructed algorithm was applied for ordinary differential equations of the first and second order, for systems of differential equations of the first order, for differential equations of the first and second order in partial derivatives. When solving these problems, reference points were chosen, in the vicinity of which observations were made along different axes, and estimates of the parameters of the equations under consideration were constructed based on estimates of their derivatives at the chosen points. Computational experiments were carried out showing the quality of the constructed estimates. The most difficult was the choice of reference points in the evaluation of matrix elements in the system of ordinary differential equations of the first order. If the reference points form an arithmetic sequence, then it was possible to construct a matrix relation for estimating the matrix of coefficients of the equation under consideration, which has a unique solution, which is proved on the basis of the Vandermonde determinant being equal to zero.