Laplace Equation Inside a Cylinder: Computational Analysis and Asymptotic Behavior of the Solution

Abstract

The Laplacian in the cylindrical coordinate space has been considered to approximate the solution of a conservative field within a restricted domain.

$$ {{\partial^{2}\psi} \over {\partial\rho^2}} + {{1\delta\psi} \over {\rho\partial\rho}} + {{1} \over {\rho^2}}{ {\partial^{2}\psi} \over {\partial\phi^2}} +{{\partial^{2}\psi} \over {\partial{z}^2} } =0 $$

Solutions of the Laplacian are represented by expansion in series of the appropriate orthonormal functions. By using asymptotic relations of Bessel Series and Fourier Bessel series, we establish some criteria for the solution to properly reflect the nature of the conservative field.