The method of solving a scalar initial value problem with a required tolerance

TL;DR

This paper introduces an integrating numerical method for scalar initial value problems that guarantees solutions within a user-defined tolerance, offering a reliable alternative to traditional methods with undefined error bounds.

Contribution

The paper presents a novel integrating method that ensures solutions within a specified tolerance, differing from traditional methods with approximate error bounds.

Findings

Method guarantees solutions within user-defined tolerance

Numerical experiments validate theoretical accuracy

Advantages and limitations of the method are discussed

Abstract

A new numerical method for solving a scalar ordinary differential equation with a given initial condition is introduced. The method is using a numerical integration procedure for an equivalent integral equation and is called in this paper an integrating method. Bound to specific constraints, the method returns an approximate solution assuredly within a given tolerance provided by a user. This makes it different from a large variety of single- and multi-step methods for solving initial value problems that provide results up to some undefined error in the form O(h^k), where h is a step size and k is concerned with the method's accuracy. Advantages and disadvantages of the method are presented. Some improvements in order to avoid the latter are also made. Numerical experiments support these theoretical results.