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ISSN Print: 2152-5080
ISSN Online: 2152-5099
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ON A POLYNOMIAL CHAOS METHOD FOR DIFFERENTIAL EQUATIONS WITH SINGULAR SOURCES
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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Chakraborty Debananda, Jung Jae-Hun, Efficient determination of the critical parameters and the statistical quantities for Klein–Gordon and sine-Gordon equations with a singular potential using generalized polynomial chaos methods, Journal of Computational Science, 4, 1-2, 2013. Crossref
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Chakraborty Debananda, Jung Jae-Hun, Lorin Emmanuel, An efficient determination of critical parameters of nonlinear Schrödinger equation with a point-like potential using generalized polynomial chaos methods, Applied Numerical Mathematics, 72, 2013. Crossref
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Yi Lijun, Wang Zhongqing, A Legendre–Gauss–Radau spectral collocation method for first order nonlinear delay differential equations, Calcolo, 53, 4, 2016. Crossref