Mediterranean Journal of Mathematics

, Volume 13, Issue 6, pp 4901–4916 | Cite as

Dirichlet-Type Problems for the Two-Dimensional Helmholtz Operator in Complex Quaternionic Analysis

  • Juan Bory-Reyes
  • Ricardo Abreu-Blaya
  • Luis M. Hernández-Simon
  • Baruch Schneider


This study aims to study a class of Dirichlet-type problems associated with the two-dimensional Helmholtz equation with complex potential. Orthogonal decompositions of the complex quaternionic-valued Sobolev space as well as the corresponding orthoprojections onto the subspaces of theses decompositions are obtained. Analytic representation formulas for the underlying solutions in terms of hypercomplex integral operators are established.


Quaternionic analysis Helmholtz operator Dirichlet-type problems 

Mathematics Subject Classification

30G35 32A25 31B20 


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  1. 1.
    Abreu Blaya R., Ávila-Ávila R., Bory Reyes J., Rodríguez Dagnino R.M.: 2D quaternionic time-harmonic Maxwell system in elliptic coordinates. Adv Appl. Clifford Algebras 25(2), 255–270 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ammari H., Bao G., Wood A.W.: An integral equation method for the electromagnetic scattering from cavities. Math. Methods Appl. Sci. 23(12), 1057–1072 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Garnir H.G.: Les Problèmes aux Limites de la Physique Mathèmatique. Birkhaüser, Basel (1958)MATHGoogle Scholar
  4. 4.
    Gerus O.F., Kutrunov V.N., Shapiro M.: On the spectra of some integral operators related to the potential theory in the plane. Math. Methods Appl. Sci. 33(14), 1685–1691 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Gerus O.F., Shapiro M.: On boundary properties of metaharmonic simple and double layer potentials on rectifiable curves in \({\mathbb{R}^2}\). (English) Zb. Proc. Inst. Mat. NAN Ukr. 1(3), 67–76 (2004)MATHGoogle Scholar
  6. 6.
    Gerus O.F., Shapiro M.: On a Cauchy-type integral related to the Helmholtz operator in the plane. Bol. Soc. Mat. Mex. III Ser. 10(1), 63–82 (2004)MathSciNetMATHGoogle Scholar
  7. 7.
    Gürlebeck K., Sprössig W.: Quaternionic Analysis and Elliptic Boundary Value Problems. Birkhauser, Basel (1990)CrossRefMATHGoogle Scholar
  8. 8.
    Gürlebeck K., Sprössig W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Chichester, New York (1997)MATHGoogle Scholar
  9. 9.
    Kaehler U.: On a direct decomposition of the space \({Lp(\Omega)}\). Z. Anal. Anwend. 18(4), 839–848 (1999)CrossRefGoogle Scholar
  10. 10.
    Kravchenko V.: Applied Quaternionic Analysis. Research and Exposition in Mathematics, vol. 28. Heldermann Verlag, Lemgo (2003)Google Scholar
  11. 11.
    Kravchenko V., Shapiro M.: Integral Representations for Spatial Models of Mathematical Physics. Pitman Research Notes in Mathematical Series, vol. 351. Longman, Harlow (1996)Google Scholar
  12. 12.
    Le, H.T.: Hyperholomorphic structures and corresponding explicit orthogonal function systems in 3D and 4D. Ph.D. Thesis, Institute of Applied Analysis, Freiberg University of Mining and Technology, Germany (2014)Google Scholar
  13. 13.
    Le, H.T., Morais, J., Sprössig, W.: Orthogonal decompositions of the complex quaternion Hilbert space and their applications. In: Gürlebeck, K. (ed.) 9th International Conference on Clifford algebras and theirs Applications in Mathematical Physics, Weimar, Germany, 15–20 July, p. 10 (2011)Google Scholar
  14. 14.
    Le, H.T., Morais, J., Sprössig, W.: Orthogonal decompositions and their applications. In: Gürlebeck, K., Lahmer, T., Werner, F. (eds.) Proceedings of the 19th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering, Weimar, Germany, 04–06 July, p. 10 (2012)Google Scholar
  15. 15.
    Le, H.T., Morais, J., Sprössig, W.: Orthogonal decompositions of the complex quaternion Hilbert space and their applications. In: 9th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences: ICNPAA 2012. AIP Conference Proceedings, vol. 1493, pp. 595–602 (2012)Google Scholar
  16. 16.
    Li D., Mao J.F.: A Koch-like sided fractal bow-tie dipole antenna. Antennas Propag. IEEE Trans. 60(5), 2242–2251 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Luna-Elizarrarás M.E., Pérez-de la Rosa M.A., Rodríguez-Dagnino R.M., Shapiro M.: On quaternionic analysis for the Schrödinger operator with a particular potential and its relation with the Mathieu functions. Math. Methods Appl. Sci. 36(9), 1080–1094 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Magnanini, R., Santosa, F.: Wave propagation in a 2-D optical waveguide. SIAM J. Appl. Math. 61(4), 1237–1252 (2000/2001)Google Scholar
  19. 19.
    Pérez-dela Rosa M.A., Shapiro M.: On the Hilbert operator and the Hilbert formulas on the unit sphere for the time-harmonic Maxwell equations. Appl. Math. Comput. 248, 480–493 (2014)MathSciNetMATHGoogle Scholar
  20. 20.
    Shapiro M., Tovar L.: Two-dimensional Helmholtz operator and its hyperholomorphic solutions. J. Nat. Geom. 11, 77–100 (1997)MathSciNetMATHGoogle Scholar
  21. 21.
    Shapiro M., Tovar L.: On a Class of Integral Representations Related to the Two-Dimensional Helmholtz Operator. Contemporary Mathematics, vol. 212, pp. 229–244. American Mathematical Society, Providence (1998)MATHGoogle Scholar
  22. 22.
    Sprössig W.: On decompositions of the Clifford valued Hilbert space and their applications to boundary value problems. Adv. Appl. Clifford Algebras 5(2), 167–186 (1995)MathSciNetMATHGoogle Scholar
  23. 23.
    Wloka, J.: Partielle Differentialgleichungen. [Partial differential equations] Sobolevräume und Randwertaufgaben. [Sobolev spaces and boundary value problems] Mathematische Leitfäden. [Mathematical Textbooks]. B. G. Teubner, Stuttgart (1982)Google Scholar
  24. 24.
    Wood A.W.: Analysis of electromagnetic scattering from an overfilled cavity in the ground plane. J. Comput. Phys. 215(2), 630–641 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Juan Bory-Reyes
    • 1
  • Ricardo Abreu-Blaya
    • 2
  • Luis M. Hernández-Simon
    • 1
  • Baruch Schneider
    • 3
  1. 1.ESIME-ZacatencoInstituto Politecnico NacionalMexicoMexico
  2. 2.Facultad de Informática y MatemáticaUniversidad de HolguínHolguínCuba
  3. 3.Department of Mathematics, Faculty of Sciences and LiteratureIzmir University of EconomicsIzmirTurkey

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