Numerical Solution of Stieltjes Differential Equations

Abstract

This work is devoted to the obtaining of a new numerical scheme based on quadrature formulae for the Lebesgue–Stieltjes integral for the approximation of Stieltjes ordinary differential equations. This novel method allows us to numerically approximate models based on Stieltjes ordinary differential equations for which no explicit solution is known. We prove several theoretical results related to the consistency, convergence, and stability of the numerical method. We also obtain the explicit solution of the Stieltjes linear ordinary differential equation and use it to validate the numerical method. Finally, we present some numerical results that we have obtained for a realistic population model based on a Stieltjes differential equation and a system of Stieltjes differential equations with several derivators.