Some notes to extend the study on random non-autonomous second order linear differential equations appearing in Mathematical Modeling
For researchers in random differential equations, this is an incremental refinement that resolves theoretical gaps in the prior study.
This paper completes and clarifies previous work on random non-autonomous second-order linear differential equations, providing clearer hypotheses, existence and uniqueness theorems for mean square analytic solutions, and linking the approach to polynomial chaos expansions and the random differential transform method.
The objective of this paper is to complete certain issues from our recent contribution [J. Calatayud, J.-C. Cortés, 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 Fröbenius method.