Calculation of Green’s Function for Poisson’s Equation in Plane Polar Coordinates using Eigenfunction Expansion in the Angular Variable

Abstract

. A new calculation of Green’s function for the problem with Poisson’s equation in plane polar coordinates is presented. The method consists in calculating the solution of a problem that is simpler but that has the same Green’s function – the problem that results from the homogenization of the boundary conditions – and then inferring Green’s function by comparing this calculated solution with Green’s formula for the solution. To describe the method, it is applied to the particular case of a disc sector under mixed Dirichlet-Neumann boundary conditions. The solution of the simplified problem is obtained as an eigenfunction expansion in the angular variable. Green’s function arises from the calculations as an infinite series but is finally presented in closed form because it is possible to compute the sum of this series.