2019/09/10 by Marc Jornet, Jornet, Marc, Julia Calatayud +4
Decision Sciences · Environmental Science · Mathematics · #34F05 #37H10 #60H10 #60H35 #65C05 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design #Probability (math.PR) #Statistical Distribution Estimation and Applications #Wind and Air Flow Studies
paper · pdf · doi:10.48550/arxiv.1909.05907
openalex publication_date 2019/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper concerns the analysis of random second order linear differential\nequations. Usually, solving these equations consists of computing the first\nstatistics of the response process, and that task has been an essential goal in\nthe literature. A more ambitious objective is the computation of the solution\nprobability density function. We present advances on these two aspects in the\ncase of general random non-autonomous second order linear differential\nequations with analytic data processes. The Fr "obenius method is employed to\nobtain the stochastic solution in the form of a mean square convergent power\nseries. We demonstrate that the convergence requires the boundedness of the\nrandom input coefficients. Further, the mean square error of the Fr "obenius\nmethod is proved to decrease exponentially with the number of terms in the\nseries, although not uniformly in time. Regarding the probability density\nfunction of the solution at a given time, we rely on the law of total\nprobability to express it in closed-form as an expectation. For the computation\nof this expectation, a sequence of approximating density functions is\nconstructed by reducing the dimensionality of the problem using the truncated\npower series of the fundamental set. We prove several theoretical results\nregarding the pointwise convergence of the sequence of density functions and\nthe convergence in total variation. The pointwise convergence turns out to be\nexponential under a Lipschitz hypothesis. As the density functions are\nexpressed in terms of expectations, we propose a symbolic Monte Carlo sampling\nalgorithm for their estimation. This algorithm is implemented and applied on\nseveral numerical examples designed to illustrate the theoretical findings of\nthe paper.\n