A method of lines solution of a cylindrical problem via radial discretization

The method of lines (MOL) is applied to solve for the electrostatic potential in a coaxial trough by discretizing in both the longitudinal and radial directions. The coupled equations are diagonalized by eigendecomposition with the aid of MATLAB software. The results agree quite well. An attempt is made to do the same exercise in the case of a cylindrical region without the inner conductor. While it is straight forward when we discretize along the longitudinal dimension, it is difficult to do the same along the radial dimension. We give an approximation to the first two terms of the P matrix. In the analytical solution truncating the infinite summation gives rise to the Gibb's phenomenon in the form of ripples at the discontinuities. A smoothing window gives a potential that is free of overshoots at the discontinuities.