Read the full paper at: http://www.scirp.org/journal/PaperInformation.aspx?PaperID=50041 DOI: 10.4236/jemaa.2014.610030 Author(s) Serigne Bira Gueye ABSTRACT A new method for solving the 1D Poisson equation is presented using the finite difference method. This method is based on the exact formulation of the inverse of the tridiagonal matrix associated with the Laplacian. This is the first time that the inverse of this remarkable matrix is determined directly and exactly. Thus, solving 1D Poisson equation becomes very accurate and extremely fast. EWW141007GJR This method is a very important tool for physics and engineering where the Poisson equation appears very often in the description of certain phenomena. KEYWORDS 1D Poisson Equation, Finite Difference Method, Tridiagonal Matrix Inversion, Thomas Algorithm, Gaussian Elimination, Potential Problem