Generalized Helmholtz equation
DOI:
https://doi.org/10.17268/sel.mat.2019.01.03Keywords:
Helmholtz equation, holomorphic functionsAbstract
In this paper we introduce the generalized Helmholtz equation and present explicit solutions to this generalized Helmholtz equation, these solutions depend on three holomorphic functions. As an application we present explicit solutions to the Helmholtz equation. We note that these solutions are not necessarily limited to certain domains of the complex plane C.
References
Bayliss, A. and Goldstein, C. The numerical solution of the Helmholtz equation for wave propagation problems in underwater acoustics, Comput. Math. Appl. 1985; (11): 655–665.
Chu, L. Electromagnetic waves in elliptic hollow pipes of metal, J. Appl. Phys. 1938; (9): 583–591.
Gladwell, G. and Willms, N. On the mode shapes of the Helmholtz equation, J. Sound Vib. 1995; 188(3): 419–433.
Hirtum, A. V. Quasi-analytical solution of two-dimensional Helmholtz equation, Applied Mathematical Modelling, 2017; (47): 96–102.
Jones, D., Acoustic and Electromagnetic Waves, Clarendon Press, Oxford, UK, 1989.
Li, Zi-Cai, Wei, Y., Chen, Y. and Huang, Hung-Tsai. The method of fundamental solutions for the Helmholtz equation, Applied Numerical Mathematics, 2019; (135): 510–536.
Ma, J., Zhu, J. and Li, M. The Galerkin boundary element method for exterior problems of 2-D Helmholtz equation with arbitrary wavenumber, Engineering Analysis with Boundary Elements, 2010; (34): 1058–1063.
Porter, M. and Liboff, R. Vibrating quantum billiards on Riemannian manifolds, Int. J. Bifurc. Chaos, 2001; 11 (9): 2305–2315.
Rienstra, S. and Eversman, W. A numerical comparison between the multiple-scales and finite-element solution for sound propagation in lined flow ducts, J. Fluid Mech. 2001; (437): 367–384.
Sarkar, T. K., Chung, Y-S and Palma, M. S. Solution of the general Helmholtz equation starting from Laplace’s equation, ACES Journal, 2002, 17(3): 187–197.
Sukhorolskyi, M. A. Boundary- value problems for the Helmholtz equation in domains of the complex plane, Ukrainian Mathematical Journal, 2016; 68(3): 406–421.
Wilson, H. and Scharstein, R. Computing elliptic membrane high frequencies by Mathieu and Galerkin methods, J. Eng. Math. 2007; (57): 41–55.
Tsai, C., Young, D., Chiu, C. and Fan, C. Numerical analysis of acoustic modes using the linear least squares method of fundamental solutions, J. Sound Vib. 2009; (324): 1086–1110.
Wong, Y. and Li, G. Exact finite difference schemes for solving Helmholtz equation at any wavenumber, Int. J. Numer. Anal. Model. 2011; 2 (1): 91–108.
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