Some notes to extend the study on random non-autonomous second order
linear differential equations appearing in Mathematical Modeling
release_4cc4sqf2ezh67jajcef6h5ca44
by
J. Calatayud and J.-C. Cortés and M. Jornet
2018
Abstract
The objective of this paper is to complete certain issues from our recent
contribution [J. Calatayud, J.-C. Cort\'es, M. Jornet, L. Villafuerte, Random
non-autonomous second order linear differential equations: mean square analytic
solutions and their statistical properties, Advances in Difference Equations,
2018:392, 1--29 (2018)]. 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 an
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, being the latter a reformulation of our random Fr\"obenius method.
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