A Non-Standard Finite Difference Scheme for the Solution of Laplace Equation with Dirichlet Boundary Conditions
Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 33093
A Non-Standard Finite Difference Scheme for the Solution of Laplace Equation with Dirichlet Boundary Conditions

Authors: Khaled Moaddy

Abstract:

In this paper, we present a fast and accurate numerical scheme for the solution of a Laplace equation with Dirichlet boundary conditions. The non-standard finite difference scheme (NSFD) is applied to construct the numerical solutions of a Laplace equation with two different Dirichlet boundary conditions. The solutions obtained using NSFD are compared with the solutions obtained using the standard finite difference scheme (SFD). The NSFD scheme is demonstrated to be reliable and efficient.

Keywords: Standard finite difference schemes, non–standard schemes, Laplace equation, Dirichlet boundary conditions.

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 666

References:


[1] Sadighi, A. & Ganji, D. D. 2007. Exact solutions of Laplace equation by homotopy perturbation and Adomian decomposition methods. Phys. Lett. A 367: 83-87.
[2] J. H. He, Non-perturbative methods for strongly nonlinear problems. Dissertation. Berlin: Verlag im Internet, 2006.
[3] J. H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Phys. Lett. A. vol. 4, no. 363, pp. : 260–262, 2007.
[4] A. Rajabi, D.D. Ganjia and H. Taheriana, “Application of homotopy perturbation method in nonlinear heat conduction and convection equations,” Phys. Lett. A. vol. 360, pp. 570–573, 2007.
[5] A. M. Waswas, “The variational iteration method for exact solutions of Laplace equation,” Chaos, Sol. & Frac. vol. 3, no. 26, pp. 695–700, 2005.
[6] Inc. Mustafa, “On exact solution of Laplace equation with Dirichlet and Neumann boundary conditions by the homotopy analysis method,” Phys. Lett. A. vol. 365, pp. 412-415, 2007.
[7] E. A. Ibijola, J. M. S Lubuma and O. A. Ade-Ibijola, “On nonstandard finite difference schemes for initial value problems in ordinary differential equations,” Int. J. Phys. Sci.. vol. 3, no.2, pp. 59–64, 2008.
[8] R. E. Mickens, Advances in the applications of nonstandard finite difference schemes. Singapore :World Scientific, 2005.
[9] R. E. Mickens. Nonstandard finite difference models of differential equations. Singapore : World Scientific, 1994.
[10] R. E. Mickens, Applications of nonstandard finite difference schemes. Singapore: World Scientific, 2000
[11] R. E. Mickens, “Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition,” Numer. Methods for Partial Diff. Eq. vol. 23, no. 3, pp. 672–-691, 2006.
[12] R. E. Mickens, “Determination of denominator functions for a NSFD scheme for the Fisher PDE with linear advection,” Math. and Comp. in Simul. vol. 74, pp. 190–195, 2007.
[13] K. Moaddy, S. Momani and I. Hashim, “The non-standard finite difference scheme for linear fractional PDEs in fluid mechanic,” Comput. And Math. Appl. vol. 61, no. 4, pp. 1209-1216, 2011.
[14] Q. A. Dang and M. T. Hoang, “Nonstandard finite difference schemes for a general predator–prey system,” Journal of Computational Science. vol. 36, pp. 101015, 2019.