Eur. Phys. J. Plus (2022) 137: 554
https://doi.org/10.1140/epjp/s13360-022-02691-x

Regular Article

Probabilistic analysis of a foundational class of generalized second-order linear differential equations in classic mechanics

¹ Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Valencia, Spain
² School of Engineering, Polytechnic of Porto and Center for Mathematics, University of Porto, Porto, Portugal

^d cpinto@isep.pt

Received: 30 December 2021
Accepted: 5 April 2022
Published online: 5 May 2022

Abstract

A number of relevant models in Classical Mechanics are formulated by means of the differential equation $y^{\prime \prime }(t)+A{t}^{\beta }y(t)=0$. In this paper, we improve the results recently established for a randomized reformulation of this model that includes a generalized derivative. The stochastic analysis permits solving that generalized model by computing reliable approximations of the probability density function of the solution, which is a stochastic process. The approach avoids constructing these approximations from limited statistical punctual information and the Principle of Maximum Entropy by directly constructing a sequence of approximations using the Probabilistic Transformation Method. We prove that these approximations converge to the exact density under mild conditions on the data. Finally, several numerical examples illustrate our theoretical findings.