Laplace boundary-value problem in paraboloidal coordinates

Abstract

This paper illustrates both a problem in mathematical physics, whereby the method of separation of variables, while applicable, leads to three ordinary differential equations that remain fully coupled via two separation constants and a five-term recurrence relation for series solutions, and an exactly solvable problem in electrostatics, as a boundary-value problem on a paraboloidal surface. In spite of the complex nature of the former, it is shown that the latter solution can be quite simple. Results are provided for the equipotential surfaces and electric field lines are given near a paraboloidal conductor.

Subspace methods for three‐parameter eigenvalue problems

“…The Helmholtz equation is also separable in paraboloidal coordinates ( ξ 1 , ξ 2 , ξ 3 ), which are related to the Cartesian coordinates by (see, e.g., the works of Moon and Spencer and Duggen et al) where − ∞ < ξ 1 < c < ξ 2 < b < ξ 3 < ∞ and c < b are the parameters of the paraboloidal coordinate system. A constant surface ξ 1 = γ , where γ < c , represents an upward opening elliptic paraboloid that intersects the z ‐axis at z = γ , whereas a constant surface ξ 3 = β , where b < β , represents a downward opening elliptic paraboloid that intersects the z ‐axis at z = β .…”

“…Helmholtz equation ( 5) is also separable in paraboloidal coordinates (ξ 1 , ξ 2 , ξ 3 ), which are related to the Cartesian coordinates by (see, e.g., [5,17])…”

“…Note that we can approximate the solutions X 1 (ξ 1 ), X 2 (ξ 2 ), X 3 (ξ 3 ) of the Baer wave equation by employing (9) subject to the boundary conditions (10), as well as eigenvectors of the discretized algebraic 3-parameter eigenvalue problem. We can combine them in a smooth eigenfunction X(ξ) bounded at the points ξ = 1, ξ = 3 and satisfying 5] subject to X(0) = 0, X(5) = 0. The eigenfunctions corresponding to the six lowest eigenfrequencies computed are displayed in Figure 4.…”