Application of Gegenbauer polynomial expansions to mitigate Gibbs phenomenon in Fourier-Bessel series solutions of a dynamic sphere problem

摘要

We utilize the inverse polynomial reconstruction (IPR) method to mitigate the Gibbs phenomenon observed in Fourier-Bessel (FB) series. Gibbs phenomenon is the oscillatory behavior that occurs near discontinuities when evaluating series solutions for Sturm-Liouville eigenvalue problems. We employ an approach that uses expansions of the solution in terms of Gegenbauer polynomials on each side of solution discontinuities, the location of which must be known in advance. The IPR solutions provide pointwise values that are more accurate than the truncated series solution, which are polluted by Gibbs phenomenon. We apply this method to discontinuous solutions of a time dependent, linear elastic spherical shell problem, for which a series solution is derived in terms of a FB expansion. For the loading conditions and material properties, we consider the Gibbs phenomenon in the FB solution for a perfectly elastic shell renders the numerically evaluated results unusable as an 'exact' solution for code verification analysis. We quantify the degree to which the IPR method eliminates the Gibbs phenomenon in the computed solution.