Abstract
AbstractThis paper presents a methodology to quantify computationally the uncertainty in a class of differential equations often met in Mathematical Physics, namely random non-autonomous second-order linear differential equations, via adaptive generalized Polynomial Chaos (gPC) and the stochastic Galerkin projection technique. Unlike the random Fröbenius method, which can only deal with particular random linear differential equations and needs the random inputs (coefficients and forcing term) to be analytic, adaptive gPC allows approximating the expectation and covariance of the solution stochastic process to general random second-order linear differential equations. The random inputs are allowed to functionally depend on random variables that may be independent or dependent, both absolutely continuous or discrete with infinitely many point masses. These hypotheses include a wide variety of particular differential equations, which might not be solvable via the random Fröbenius method, in which the random input coefficients may be expressed via a Karhunen-Loève expansion.
Highlights
Introduction and PreliminariesMany laws of Physics are formulated via di erential equations
This paper presents a methodology to quantify computationally the uncertainty in a class of di erential equations often met in Mathematical Physics, namely random non-autonomous second-order linear di erential equations, via adaptive generalized Polynomial Chaos and the stochastic Galerkin projection technique
Adaptive generalized Polynomial Chaos (gPC) allows for random inputs (2) more general than the random Fröbenius method: A(t), B(t) and C(t) may not be analytic, they may be represented via a truncated Karhunen-Loève expansion, etc
Summary
Introduction and PreliminariesMany laws of Physics are formulated via di erential equations. Input parameters are often not exactly known because of insu cient information, limited understanding of some underlying phenomena, inherent uncertainty, etc. All these facts motivate that input parameters of classical di erential equations. This work is licensed under the Creative Commons are treated as random variables or stochastic processes rather than deterministic constants or functions, respectively. This approach leads to random di erential equations (RDEs) [1, 2]. The random behavior of the solution stochastic process can be understood if one obtains its main statistical features, say expectation, variance, covariance, etc
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