2018/02/21 by Robert Stephen Jones, Jones, Robert Stephen
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #FOS: Mathematics #Mathematical functions and polynomials #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1802.07768
openalex publication_date 2018/02/21 · openalex created_date 2022/08/29 · openalex updated_date 2026/07/28
In this project, I examine the lowest Dirichlet eigenvalue of the Laplacian\nwithin the ellipse as a function of eccentricity. Two existing analytic\nexpansions of the eigenvalue are extended: Close to the circle (eccentricity\nnear zero) nine terms are added to the Maclaurin series; and near the infinite\nstrip (eccentricity near unity) four terms are added to the asymptotic\nexpansion. In the past, other methods, such as boundary variation techniques,\nhave been used to work on this problem, but I use a different approach -- which\nnot only offers independent confirmation of existing results, but extends them.\nMy starting point is a high precision computation of the eigenvalue for\nselected values of eccentricity. These data are then fit to polynomials in\nappropriate parameters yielding high-precision coefficients that are fed into\nan LLL integer-relation algorithm with forms guided by prior results.\n