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(h2)), 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.
Highlights
The Poisson equation is an elliptic differential equation well known and common to various scientific and technical domains such as physics, mathematics, chemistry, biology, etc
A new method is proposed [1] [2] dealing with this equation in one dimensional case. This new approach, using the finite differences method (FDM), determined the inverse of the matrix obtained from algebraic equations
The present study generalizes the solution of the Poisson equation and determines its solution for boundary conditions of third kind: Robin conditions
Summary
The Poisson equation is an elliptic differential equation well known and common to various scientific and technical domains such as physics, mathematics, chemistry, biology, etc. (2014) Generalization of the Exact Solution of 1D Poisson Equation with Robin Boundary Conditions, Using the Finite Difference Method. A new method is proposed [1] [2] dealing with this equation in one dimensional case This new approach, using the finite differences method (FDM), determined the inverse of the matrix obtained from algebraic equations. The present study generalizes the solution of the Poisson equation and determines its solution for boundary conditions of third kind: Robin conditions. These mixed boundary conditions present a great interest in practice, because of combining two quantities: the function and its derivative. The cases of boundary conditions of type (DR) and (RD) are discussed and solved
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