# New numerical methods for blow-up problems

## TL;DR

This paper introduces two innovative numerical methods for solving ODEs with blow-up solutions by transforming the problem to avoid singularities, enabling standard numerical techniques to be effectively applied.

## Contribution

The paper presents two novel numerical integration methods that transform blow-up problems into regular problems, facilitating more accurate and stable solutions.

## Key findings

- Both methods successfully handle blow-up solutions in test problems.
- The methods outperform traditional approaches in stability and accuracy.
- Numerical experiments demonstrate the effectiveness of the proposed techniques.

## Abstract

Two new methods of numerical integration of Cauchy problems for ODEs with blow-up solutions are described. The first method is based on applying a differential transformation, where the first derivative (given in the original equation) is chosen as a new independent variable. The second method is based on introducing a new non-local variable that reduces ODE to a system of coupled ODEs. Both methods lead to problems whose solutions do not have blowing-up singular points, therefore the standard numerical methods can be applied. The efficiency of the proposed methods is illustrated with several test problems.