Solution of 1D Poisson Equation with Neumann-Dirichlet and Dirichlet-Neumann Boundary Conditions, Using the Finite Difference Method

Serigne Bira Gueye; Kharouna Talla; Cheikh Mbow

doi:10.4236/jemaa.2014.610031

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JEMAA> Vol.6 No.10, September 2014

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Solution of 1D Poisson Equation with Neumann-Dirichlet and Dirichlet-Neumann Boundary Conditions, Using the Finite Difference Method

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DOI: 10.4236/jemaa.2014.610031 7,367 Downloads 9,940 Views Citations

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Serigne Bira Gueye^*, Kharouna Talla, Cheikh Mbow

Affiliation(s)

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

ABSTRACT

An innovative, extremely fast and accurate method is presented for Neumann-Dirichlet and Dirichlet-Neumann boundary problems for the Poisson equation, and the diffusion and wave equation in quasi-stationary regime; using the finite difference method, in one dimensional case. Two novels matrices are determined allowing a direct and exact formulation of the solution of the Poisson equation. Verification is also done considering an interesting potential problem and the sensibility is determined. This new method has an algorithm complexity of O(N), its truncation error goes like O(h²), and it is more precise and faster than the Thomas algorithm.

KEYWORDS

1D Poisson Equation, Finite Difference Method, Neumann-Dirichlet, Dirichlet-Neumann, Boundary Problem, Tridiagonal Matrix Inversion, Thomas Algorithm

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Gueye, S. , 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. doi: 10.4236/jemaa.2014.610031.

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. Accepted Manuscript (JEMAA, April 2014).

[2]	Engeln-Muellges, G. and Reutter, F. (1991) Formelsammlung zur Numerischen Mathematik mit QuickBasic-Programmen, Dritte Auflage, BI-Wissenchaftsverlag, 472-481.

[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]	LeVeque, R.J. (2007) Finite Difference Method for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems. SIAM, 15-16. http://dx.doi.org/10.1137/1.9780898717839

[5]	Conte, S.D. and de Boor, C. (1981) Elementary Numerical Analysis: An Algorithmic Approach. 3rd Edition, McGrawHill, New York, 153-157.

[6]	Mathews, J.H. and Fink, K.K. (2004) Numerical Methods Using Matlab. 4th Edition, 323-325, 339-342.