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International Journal for Uncertainty Quantification
IF: 4.911 5-Year IF: 3.179 SJR: 1.008 SNIP: 0.983 CiteScore™: 5.2

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.v1.i1.50
pages 77-98

ON A POLYNOMIAL CHAOS METHOD FOR DIFFERENTIAL EQUATIONS WITH SINGULAR SOURCES

Jae-Hun Jung
Department of Mathematics, State University of New York at Buffalo, USA
Yunfei Song
Department of Mathematics, State University of New York at Buffalo, USA

ABSTRACT

Singular source terms in the differential equation represented by the Dirac δ-function play a crucial role in determining the global solution. Due to the singular feature of the δ-function, physical parameters associated with the δ-function are highly sensitive to random and measurement errors, which makes the uncertainty analysis necessary. In this paper we use the generalized polynomial chaos method to derive the general solution of the differential equation under uncertainties associated with the δ-function. For simplicity, we assume the uniform distribution of the random variable and use the Legendre polynomials to expand the solution in the random space. A simple differential equation with the singular source term is considered. The polynomial chaos solution is derived. The Gibbs phenomenon and the convergence of high order moments are discussed. We also consider a direct collocation method which can avoid the Gibbs oscillations on the collocation points and enhance the accuracy accordingly.


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