Chebyshev Alternance when Approximating Initial Conditions of the Inverse Cauchy Problem

Abstract

Purpose of research. The work is devoted to a range of questions related to Cauchy problem on the segment of real axis with the application of the inverse Cauchy problem, in which real constants are initial conditions which are optimally restored according to experimental or tabular values of the solution of the differential equation. The object of the study is an information-measuring system, in which approximate values of initial conditions are calculated from discrete function values of Cauchy problem solving.Methods. The following problems are solved for this purpose: parameters of measuring section placement on the investigated object and approximation grid on measuring section are developed. Characteristics of recovery accuracy of initial conditions of the task are formulated.Results. An experimental-calculated method of determining initial conditions in the inverse Cauchy problem is proposed. It is based on the concept of objective function of regularization of the problem. Task regularization parameter in the form of minimum value by Lebesgue function is proposed.Conclusion. The reaction of uniformly approximating method of the initial conditions of the inverse Cauchy problem to the deviation of the approximation grid coordinates nodes from the coordinates of Chebyshev alternance was described. Graphs of method reaction to deviation of grid pitch from optimal pitch are given.