Numerical solution of linear differential equations with~discontinuous coefficients and Henstock integral

S. F. Lukomskii ; D. S. Lukomskii

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 2, pp. 151-161

Résumé

We consider the problem of approximate solution of linear differential equations with discontinuous coefficients. We assume that these coefficients have $f$-primitive. It means that these coefficients are Henstock integrable only. Instead of the original Cauchy problem, we consider a different problem with piecewise-constant coefficients. The sharp solution of this new problem is the approximate solution of the original Cauchy problem. We found the degree of approximation in terms of $f$-primitive for Henstock integrable coefficients. Two examples are given. In the first example, the coefficients have an infinite derivative at zero. In the second example, the coefficients have an infinite derivative at interior points.