Exact Solutions of the Helmholtz equation via the Nikiforov-Uvarov Method

Authors: Said Laachir, Aziz Laaribi

Abstract:

The Helmholtz equation often arises in the study of physical problems involving partial differential equation. Many researchers have proposed numerous methods to find the analytic or approximate solutions for the proposed problems. In this work, the exact analytical solutions of the Helmholtz equation in spherical polar coordinates are presented using the Nikiforov-Uvarov (NU) method. It is found that the solution of the angular eigenfunction can be expressed by the associated-Legendre polynomial and radial eigenfunctions are obtained in terms of the Laguerre polynomials. The special case for k=0, which corresponds to the Laplace equation is also presented.

Keywords: Helmholtz equation, Nikiforov-Uvarov method, exact solutions, eigenfunctions.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1089211

References:

[1] M. Kawahara and K. Kashiyama, Boundary type finite element method for surface wave motion based on the trigonometric function interpolation Int. J. Numer. Meth. Eng. 21, pp. 1833-1852, 1985.
[2] R. A. Altenkirch, M Rezayat, R. Eichhorn, and F. J. Rizzo A study of heat conduction forward of flame in solids, by the boundary integral equation method," Trans. ASME Ser. C. J. Heat Transfer 104, pp. 734-740, 1982.
[3] J T Chen, F. C .Wong Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition. J. Sound Vibration 217, 75–95, 1998.
[4] Beskos DE Boundary element method in dynamic analysis: Part II (1986–1996). ASME Appl. Mech. Rev. 50, pp. 149–197, 1997.
[5] W. S Hall, X. Q. Mao A boundary element investigation of irregular frequencies in electromagnetic scattering. Eng. Anal. Bound. Elem. 16, pp. 245–252, 1995.
[6] S.M. El-Sayed and D.Kaya, Comparing numerical methods for Helmholtz equation model problem, Appl. Math. Comp.150, pp. 763-773, 2004.
[7] D.D. Ganji, A. Rajabi, Assessment of homotopy-perturbation and Perturbation methods in heat radiation equations, International. Commun. Heat Mass Transfer 33 (3) pp. 391–400, 2006.
[8] J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng. 178, pp. 257–262, 1999.
[9] M.A. Abdou, A.A. Soliman, Variational iteration method for solving Burger’s and coupled Burger’sequations, J. Comput. Appl. Math. 181 (2) pp. 245–251, 2005.
[10] G. Szego, Orthogonal polynomials, American Mathematical Society, New York, 1959 (revised edition).
[11] D. Gomez-Ullate, N. Kamran and R. Milson, ArXiv:nlin.SI/0401030
[12] W. Evans, Phys.Rev. A 41 pp. 5666- 5676, 1990.
[13] Shi-Hai Dong, G. Sun and M. Lozada-Cassou, Phys. Lett. A 340, pp. 94–103, 2005.
[14] Shi-Hai Dong, Physica Scripta, 65, pp. 289 -295, 2002.
[15] A.K. Chattopadhyay, Phys. Lett. A 357, pp. 108-111, 2006.
[16] C. Berkdemir, A. Berkdemir, J. Han, Chem. Phys. Lett. 417, pp. 326–329, 2006.
[17] Egrifes, H., Sever, R.: Phys. Lett. A 344, pp. 117 –126, 2005.
[18] Yasuk, F., Durmu¸s, A., Boztosun, I.: J. Math. Phys. 47, pp. 082302-9, 2006.
[19] J.-Y. Guo and Z.-Q. Sheng, Phys. Lett. A 338, pp. 90 –96, 2005.
[20] C. S. Jia, P. Gao, Y. F. Diao, L. Z. Yi, and X. J. Xie, Eur. Phys. J. A 34, pp. 41-48, 2007.
[21] Soylu, O. Bayrak, and I. Boztosun, J. Math. Phys. 48, pp. 082302-082309, 2007.
[22] F. Taşkın, Int. J. Theor. Phys. 48, pp. 1142 -1149, 2009.
[23] A. Soylu, O. Bayrak, and I. Boztosun, J. Phys. A: Math. Theor. 41, pp. 065308-15, 2008.
[24] C. S. Jia, P. Gao, and X. L. Peng, J. Phys. A 39, pp. 7737-7744, 2006.
[25] L. H. Zhang, X. P. Li, and C. S. Jia, Phys. Lett. A 372, pp. 2201- 2207. 2008.
[26] A. F. Nikiforov and V. B. Uvarov. Special Function of Mathematical Physics, Birkhauser, Bassel, 1988.