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

Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, 46022 Valencia, Spain
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Math. Comput. Appl. 2018, 23(4), 76; https://doi.org/10.3390/mca23040076
Received: 11 November 2018 / Revised: 24 November 2018 / Accepted: 24 November 2018 / Published: 27 November 2018
(This article belongs to the Special Issue Mathematical Modelling in Engineering & Human Behaviour 2018)
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. View Full-Text
Keywords: random non-autonomous second order linear differential equation; mean square analytic solution; random power series; uncertainty quantification random non-autonomous second order linear differential equation; mean square analytic solution; random power series; uncertainty quantification
MDPI and ACS Style

Calatayud Gregori, J.; Cortés López, J.C.; Jornet Sanz, M. Some Notes to Extend the Study on Random Non-Autonomous Second Order Linear Differential Equations Appearing in Mathematical Modeling. Math. Comput. Appl. 2018, 23, 76.

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