ON A POLYNOMIAL CHAOS METHOD FOR DIFFERENTIAL EQUATIONS WITH SINGULAR SOURCES

Abstract

Singular source terms in the differential equation represented by the Dirac ‐ -function play a crucial role in determining the global solution. Due to the singular feature of the ‐ -function, physical parameters associated with the ‐ -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 ‐ -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.