Generalization of the Exact Solution of 1D Poisson Equation with Robin Boundary Conditions, Using the Finite Difference Method

Abstract

A new and innovative method for solving the 1D Poisson Equation is presented, using the finite differences method, with Robin Boundary conditions. The exact formula of the inverse of the discretization matrix is determined. This is the first time that this famous matrix is inverted explicitly, without using the right hand side. Thus, the solution is determined in a direct, very accurate ( O ( h 2 )), and very fast ( O ( N )) manner. This new approach treats all cases of boundary conditions: Dirichlet, Neumann, and mixed. Therefore, it can serve as a reference for solving the Poisson equation in one dimension.