Convergent Power Series for Boundary Value Problems and Eigenproblems with Application to Atmospheric and Oceanic Tides

Abstract

It is usually impossible to apply power series to solve boundary value and eigenvalue problems because the radius of convergence does not extend to the boundaries. Hough functions, which solve a second order eigenproblem, are always entire functions or can be computed wholly in terms of entire functions. The eigenvalues ϵ are approximated by the zeros of the highest computed power series coefficient, which is always a polynomial in ϵ. The rate of convergence with series truncation N is proportional to exp(-μN) for some positive constant A brief table contains the entire Maple code for the algorithm. The scheme can also be extended to eigenproblems on an unbounded domain as here illustrated by the quantum harmonic oscillator and quartic oscillator. Computer algebra is very useful; the power series method can also be implemented in a stable, purely numerical fashion by applying trigonometric interpolation on a disk in the complex x-plane and Chebyshev interpolation on a real interval in e. Power series are briefly compared with Chebyshev, Fourier, and spherical harmonic spectral methods, which are mostly superior. The success of power series raises an interesting history-of-science issue: Why was there so little progress in solving Laplace's tidal equations in the nineteenth century?