Remarks on positive solutions to nonlinear problems and numerical methods

TL;DR

This paper investigates positive solutions to nonlinear differential systems related to chemical reactions, establishing existence results and proposing numerical methods that preserve positivity and stability.

Contribution

It introduces a method for proving positive solutions and develops explicit difference schemes with operator splitting to ensure positivity and stability in numerical simulations.

Findings

Existence of positive solutions for the system is proven.

Numerical schemes maintain positivity and stability.

Operator splitting methods are effective for discretized systems.

Abstract

The existence of positive solutions to the system of ordinary differential equations related to the Belousov-Zhabotinsky reaction is established. The key idea is to use successive approximation of solutions, ensuring its positivity. To obtain the positivity and invariant region for numerical solutions, the system is discretized as difference equations of explicit form, employing operator splitting methods with linear stability conditions.