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.

A Digital Twin of a Compartmental Epidemiological Model based on a Stieltjes Differential Equation

“…The total CPU time required to perform the complete simulation of MDT was 0.18 seconds. Finally we will compare the Main Digital Twin in terms of definition (5) with the solution of the Stieltjes differential equation (11). The numerical approximation of the solution to a system of Stieltjes differential equations was introduced in [5] where a predictorcorrector numerical scheme to approximate the solution to a Stieljes dieferential equation (also for systems) from a quadrature formula for the Lebesgue Stieltjes integral was deduced.…”

On first and second order linear Stieltjes differential equations

“…There has been a recent surge in the study of Stieltjes differential equations focused on obtaining applicable results comparable to those available for classical derivatives [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. These works center their attention in the procuring of solutions of first order differential equations and systems.…”

“…All this work is then illustrated with an application to the Stieltjes harmonic oscillator for which we also analyze the resonance effect. Finally, in order to validate the explicit solutions obtained, we compare them with the numerical approximation of the corresponding first order linear system using the numerical scheme introduced in [2].…”

“…Definition 5.1. We say v ∈ BC 2 g ([0, T ]; F) is a solution of (5.1) if it satisfies the equation v g (t) + P v g (t) + Q v(t) = 0, ∀t ∈ [0, T ] (5.2)…”

“…The numerical approximation of the solution of a system of Stieltjes differential equations was introduced in [2] where a predictor-corrector numerical scheme to approximate the solution of a Stieljes dieferential equation (also for systems) from a quadrature formula for the Lebesgue Stieltjes integral was deduced. For this, a finite set of times {t j } N +1 j=0 ⊂ [0, T ] is considered such that D g ⊂ {t j } N +1 j=0 , t 0 = 0, t N +1 = T and t k+1 − t k = h > 0, for every k = 1, .…”