A class of solutions of the n-dimensional generalized Helmholtz equation which describes generalized Weingarten hypersurfaces

  1. Corro, Armando 3
  2. Riveros, Carlos 2
  3. Carretero, José 1
  1. 1 Universitat d'Alacant
    info

    Universitat d'Alacant

    Alicante, España

    ROR https://ror.org/05t8bcz72

  2. 2 Universidade de Brasília
    info

    Universidade de Brasília

    Brasilia, Brasil

    ROR https://ror.org/02xfp8v59

  3. 3 Universidade Federal de Goiás
    info

    Universidade Federal de Goiás

    Goiânia, Brasil

    ROR https://ror.org/0039d5757

Montrer des affiliations +
Revue:
Serdica Mathematical Journal

ISSN: 2815-5297,  1310-6600

Année de publication: 2024

Volumen: 50

Número: 1

Type: Article

DOI: 10.55630/SERDICA.2024.50.1-34 GOOGLE SCHOLAR lock_openAccès ouvert editor

D'autres publications dans: Serdica Mathematical Journal

Résumé

In this paper, we introduce the -dimensional generalized Helmholtz equation and present explicit solutions to this equation in terms of biharmonic functions, in particular, we get solutions that depend on holomorphic functions. Also, we present explicit radial solutions for this equation and we provide explicit solutions to the -dimensional Helmholtz equation. In addition, as an application we introduced two classes of generalized Weingarten hypersurfaces, namely, the RSHGW-hypersurfaces and the RSGW-hypersurfaces, associated with solutions of the -dimensional generalized Helmholtz equation and classify the RSHGW-hypersurfaces of rotation. For , we obtain a Weierstrass type representation for these surfaces which depend of three holomorphic functions and we classify the RSHGW-surfaces and the RSGW-surfaces of rotation.

Références bibliographiques

  • [1]P. Appell.Surfaces telles que l’origene se projette sur chaque normale aumilieu des centres de courbure principaux.Amer. J. Math.10,2(1888),175–186
  • [2]L. Bianchi.Lezioni di Geometria Differenziale, 2nd edn., Vol 2. Pisa, E.Spoerri, 1903.
  • [3]L. J. Chu.Electromagnetic waves in elliptic hollow pipes of metal,J. Appl.Phys.9(1938), 583–591,https://doi.org/10.1063/1.1710459.
  • [4]A. V. M. Corro, R. Pina, M. A. Souza.Classes of Weingarten surfacesinS2×R,Houston J. Math.46,3(2020), 651–664[
  • 5]D. G. Dias.Classes de hipersuperficies Weingarten generalizada no espa ̧coEuclidiano, PhD thesis, Universidade Federal de Goias, 2014,http://repositorio.bc.ufg.br/tede/handle/tede/4091.
  • [6]D. G. Dias, A. V. Corro.Classes of generalized Weingarten surfaces inthe Euclidean 3-space.Adv. Geom.16,1(2016), 45–55.
  • [7]B. A. Dubrovin, S. P. Novikov.Hydrodinamics of weakly deformedsoliton lattices. Diferential geometry and Hamiltonian theory.Uspekhi Mat.Nauk44,6(1989), 29–98, 203 (in Russian); English translation in:RussianMath. Surveys44,6(1989), 35–124.
  • [8]L. P. Eisenhart.Separable systems of stackel.Ann. of Math.(2)35,2(1934), 284–305.
  • [9]E. V. Ferapontov.Reciprocal transformations and their invariants.Dif-ferentsial’nye Uravneniya25,7(1989), 1256–1265, 1286 (in Russian); En-glish translation in:Differential Equations25,7(1989), 898–905.
  • [10]E. V. Ferapontov.Dupin hypersurfaces an integrable Hamiltonian sys-tems of hidrodinamic type, which do not possess Riemann invariants.Dif-ferential Geom. Appl.5,2(1995), 121–152.
  • [11]W. Ferreira, P. Roitman.Area preserving transformations in two-dimensional space forms and classical differential geometry.Israel J. Math.190(2012), 325–348.
  • [12]A. V. Hirtum.Quasi-analytical solution of two-dimensional Helmholtzequation.Appl. Math. Model.47(2017), 96–102.
  • [13]D. Jones.Acoustic and Electromagnetic Waves. Oxford Sci. Publ. NewYork, The Clarendon Press, Oxford University Press, 1986.
  • [14]Z-C. Li, Y. Wei, Y. Chen, H.-T. Huang.The method of fundamentalsolutions for the Helmholtz equation.Appl. Numer. Math.135(2019), 510–536,https://doi.org/10.1016/j.apnum.2018.09.008.
  • [15]R. L ́opez.Special Weingarten surfaces foliated by circles.Monatsh Math.154,4(2008), 289–302.
  • [16]J. Ma, J. Zhu, M. Li.The Galerkin boundary element method for exteriorproblems of 2-D Helmholtz equation with arbitrary wavenumber.Eng. Anal.Bound. Elem.34,12(2010), 1058–1063.
  • [17]A. Martinez, P. Roitman.A class of surfaces related to a problem posedby ́Elie Cartan.Ann. Mat. Pura Appl.(4)195,2(2016), 513–527.
  • [18]M. A. Porter, R. L. Liboff.Vibrating quantum billiards on Riemannianmanifolds.Internat. J. Bifur. Chaos Appl. Sci. Engrg.11,9(2001), 2305–2315.
  • [19]S. Rienstra, W. A. Eversman.Numerical comparison between themultiple-scales and finite-element solution for sound propagation in linedflow ducts,J. Fluid Mech.437 (2001), 367–384,https://doi.org/10.1017/S0022112001004438.
  • [20]C. M. C. Riveros, A. M. V. Corro.Generalized Helmholtz equation.Selecciones Matematicas6,1(2019), 19–25,https://doi.org/10.17268/sel.mat.2019.01.03.
  • [21]C. M. C. Riveros, A. M. V. Corro.Surfaces with constant Chebyshevangle II.Tokio J. Math.36,2(2013), 379–386.[
  • 22]C. M. C. Riveros, A. V. M. Corro, D. G. Dias.A class of generalizedspecial Weingarten surfaces.Internat. J. Math.30,14(2019), 1950075, 26pp.
  • [23]T. K. Sarkar, Y-S. Chung, M. S. Palma.Solution of the generalHelmholtz equation starting from Laplace’s equation.ACES Journal.17,3(2002), 187–197,https://apps.dtic.mil/sti/tr/pdf/ADP013470.pdf.
  • [24]M. A. Sukhorol’s’kyi.Boundary-value problems for the Helmholtz equa-tion in domains of the complex plane,Ukrainian Mathematical Journal.68(2016), 406–421,https://doi.org/10.1007/s11253-016-1230-5.
  • [25]C. Tsai, D. Young, C. Chiu, C. Fan.Numerical analysis of acousticmodes using the linear least squares method of fundamental solutions.J.Sound Vib.324,3–5(2009), 1086–1110,http://msvlab.hre.ntou.edu.tw/JSV2009-TsaiCC.pdf.
  • [26] M. G. Tzitz ́eica. Sur une nouvelle classe de surfaces,Rend. Circ. Mat.Palermo25(1908), 180–187,https://doi.org/10.1007/BF03029121.
  • [27]H. Wilson, R. Scharstein.Computing elliptic membrane high frequenciesby Mathieu and Galerkin methods,J. Engrg. Math.57,1(2007), 41–55
  • [28]Y. S. Wong, G. Li.Exact finite difference schemes for solving Helmholtzequation at any wavenumber.Int. J. Numer. Anal. Model. Ser. B.2,1(2011), 91–108