Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties Article Swipe
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· 2018
· Open Access
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· DOI: https://doi.org/10.1186/s13662-018-1848-8
· OA: W2898499626
In this paper we study random non-autonomous second order linear differential \nequations by taking advantage of the powerful theory of random difference \nequations. The coefficients are assumed to be stochastic processes, and the initial \nconditions are random variables both defined in a common underlying complete \nprobability space. Under appropriate assumptions established on the data stochastic \nprocesses and on the random initial conditions, and using key results on difference \nequations, we prove the existence of an analytic stochastic process solution in the \nrandom mean square sense. Truncating the random series that defines the solution \nprocess, we are able to approximate the main statistical properties of the solution, \nsuch as the expectation and the variance. We also obtain error a priori bounds to \nconstruct reliable approximations of both statistical moments. We include a set of \nnumerical examples to illustrate the main theoretical results established throughout \nthe paper. We finish with an example where our findings are combined with Monte \nCarlo simulations to model uncertainty using real data.