A COMPUTATIONAL MODEL FOR A DIFFERENTIAL EQUATION WITH EMPIRICAL FUNCTIONS BASED ON THE INTEGRAL EQUATIONS FREDHOLM OF THE FIRST KIND

Abstract

The work out lines a method of constructing an approximate solution of a differential equation of the second order with input data obtained in the experiment (empirical functions). In such statement the problem belongs to the class of incorrect mathematical problems and often occurs, for example, in mathematical models of physical phenomena using measurement results of field experiments. To obtain the approximate solution of this problem requires construction of appropriate regularization algorithms based on the methods of the theory of functional analysis and ill-posed problems. In the present work is the construction of the approximate solution of odes with specified boundary conditions, are the so-called singular integrals. This allows you to put in the original equation Fredholm integral equation of the first kind and to find its numerical solution, i.e. the solution of the incorrect task. This uses a machine approximation of functions and their derivatives corresponding singular integrals and regularization method convergence of the sequence of approximate solutions, which implemented the so-called generalized inverse operators. Built in the end, a computational model allows to obtain a stable solution of ill-posed problems.