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

A number of relevant models in Classical Mechanics are formulated by means of the differential equation y ''(t) + At-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.

Automated summary

This paper improves a randomized reformulation of a model in Classical Mechanics that includes a generalized derivative. By utilizing stochastic analysis, reliable approximations of the probability density function of the solution are computed, avoiding the limitations of limited statistical punctual information and the Principle of Maximum Entropy. It is proven that these approximations converge to the exact density under mild conditions on the data. Numerical examples are provided to illustrate the theoretical findings.