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

TL;DR

This paper develops a method for approximating solutions to linear differential equations with discontinuous coefficients using Henstock integrable functions, transforming the problem into a piecewise-constant coefficient problem.

Contribution

It introduces a novel approximation approach for differential equations with discontinuous coefficients by leveraging Henstock integrability and piecewise-constant transformations.

Findings

Provides a degree of approximation based on modulus of continuity.

Establishes a method for solving equations with Henstock integrable coefficients.

Offers explicit bounds for approximation accuracy.

Abstract

In this article we consider the problem of approximative solution of linear differential equations $y'+p(x)y=q(x)$ with discontinuous coefficients $p$ and $q$. We assume that coefficients of such equation are Henstock integrable functions. To find the approximative solution we change the original Cauchy problem to another problem with piecewise-constant coefficients. The sharp solution of this new problems is the approximative solution of the original Cauchy problem. We find the degree approximation in terms of modulus of continuity $\omega_\delta (P),\ \omega_\delta (Q)$, where $P$ and $Q$ are $f$-primitive for coefficients $p$ and $q$.