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

Serigne Bira Gueye^*, Kharouna Talla, Cheikh Mbow

Département de Physique, Faculté des Sciences et Techniques, Université Cheikh Anta Diop, Dakar-Fann, Sénégal.

DOI: 10.4236/jemaa.2014.612038                

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.

Keywords

Robin Boundary, Poisson Equation, Matrix Inversion

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Gueye, S.B. (2014) The Exact Formulation of the Inverse of the Tridiagonal Matrix for Solving the 1D Poisson Equation with the Finite Difference Method. Journal of Electromagnetic Analysis and Applications, 6, 303-308. http://dx.doi.org/10.4236/jemaa.2014.610030
[2]	Gueye, S.B., Talla, K. and Mbow, C. (2014) Solution of 1D Poisson Equation with Neumann-Dirichlet and Dirichlet-Neumann Boundary Conditions, Using the Finite Difference Method. Journal of Electromagnetic Analysis and Applications, 6, 309-318. http://dx.doi.org/10.4236/jemaa.2014.610031
[3]	Kreiss, H.O. (1972) Difference Approximations for Boundary and Eigenvalue Problems for Ordinary Differential Equations. Mathematics of Computation, 26, 605-624. http://dx.doi.org/10.1090/S0025-5718-1972-0373296-3
[4]	Engeln-Muellges, G. and Reutter, F. (1991) Formelsammlung zur Numerischen Mathematik mit QuickBasic-Programmen, Dritte Auflage. BI-Wissenchaftsverlag, Mannheim, 472-481.
[5]	LeVeque, R.J. (2007) Finite Difference Method for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, SIAM, 15-25. http://dx.doi.org/10.1137/1.9780898717839
[6]	Conte, S.D. and de Boor, C. (1981) Elementary Numerical Analysis: An Algorithmic Approach. 3rd Edition, McGraw-Hill, 153-157.
[7]	Mathews, J.H. and Fink, K.K. (2004) Numerical Methods Using Matlab. 4th Edition, Prentice-Hall Inc., New Jersey, 323-325, 339-342.
[8]	Sadiku Matthew, N.O. (2000) Numerical Techniques in Electromagnetics. 2nd Edition, CRC Press, Boca Raton, 610-626. http://dx.doi.org/10.1201/9781420058277
[9]	Gustafson, K. (1988) Domain Decomposition, Operator Trigonometry, Robin Condition. Contemporary Mathematics, 218, 432-437. http://dx.doi.org/10.1090/conm/218/3039
[10]	Reinhold, P. (2008) Analysis of Electromagnetic Fields and Waves: The Method of Lines. John Wiley, Chichester, 9.