Some Notes to Extend the Study on Random Non-Autonomous Second Order Linear Differential Equations Appearing in Mathematical Modeling

Julia Gregori,Juan López,Marc Sanz

doi:10.3390/mca23040076

Abstract

The objective of this paper is to complete certain issues from our recent contribution (Calatayud, J.; Cortés, J.-C.; Jornet, M.; Villafuerte, L. Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties. Adv. Differ. Equ. 2018, 392, 1–29, doi:10.1186/s13662-018-1848-8). We restate the main theorem therein that deals with the homogeneous case, so that the hypotheses are clearer and also easier to check in applications. Another novelty is that we tackle the non-homogeneous equation with a theorem of existence of mean square analytic solution and a numerical example. We also prove the uniqueness of mean square solution via a habitual Lipschitz condition that extends the classical Picard theorem to mean square calculus. In this manner, the study on general random non-autonomous second order linear differential equations with analytic data processes is completely resolved. Finally, we relate our exposition based on random power series with polynomial chaos expansions and the random differential transform method, the latter being a reformulation of our random Fröbenius method.

Highlights

The important role played by differential equations in dealing with mathematical modeling is beyond discussion
We concentrate on a class of differential equations that have played a distinguished role in a variety of applications in science, in particular in physics and engineering, namely second order linear differential equations
The formulation of second order linear differential equations to describe the aforementioned physical problems appears as direct applications of important laws of physics

Summary

Introduction

The important role played by differential equations in dealing with mathematical modeling is beyond discussion. We concentrate on a class of differential equations that have played a distinguished role in a variety of applications in science, in particular in physics and engineering, namely second order linear differential equations These equations have been successfully applied to describe, for example, vibrations in springs (free undamped or simple harmonic motion, damped vibrations subject to a frictional force, or forced vibrations affected by an external force), the analysis of electric circuits made up of an electromotive force supplied by a battery or a generator, a resistor, an inductor, and a capacitor. The analysis includes a result (Theorem 2) that simplifies the application of a recent finding by the authors that is useful in practical cases This issue is illustrated via several numerical examples where both the random Airy and Hermite differential equations are treated.

Homogeneous Case

Comparison with Other Methods