Higher order Dirichlet type problems for the two-dimensional Helmholtz operator in complex quaternionic analysis
DOI:
https://doi.org/10.21638/11701/spbu01.2019.410Abstract
It is well known that the development of methods for solving Dirichlet problems is important and relevant for various areas of mathematical physics related to the Laplace equation, the Helmholtz equation, the Stokes equation, the Maxwell equation, the Dirac equation, and others. In the previous work, the author studied the solvability of Dirichlet problem of the first order and the second order in the quaternion analysis. In the present paper studies the Dirichlet boundary value problem of high order, associated with the two-dimensional Helmholtz equation with complex potential. In this paper, the existence and uniqueness and a representation formula for the solution of Dirichlet boundary value problems in the two-dimensional case are proved. Most of Dirichlet boundary value problems are focused to the 3D case. Note that the case is not a simple consequence of the three-dimensional case. For solving the problem, we use the method of orthogonal decomposition of the quaternion Sobolev space. This orthogonal decomposition of space is also the basis for the study of many elliptic boundary value problems that are in various areas of mathematics and mathematical physics. Orthogonal decompositions of the quaternionic-valued Sobolev space with respect to the Dirac operator of high order as well as the corresponding orthoprojections onto the subspaces of theses decompositions are obtained.
Keywords:
quaternion analysis, Helmholtz operator, boundary problems of Dirichlet type
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References
Литература
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References
Gürlebeck K., Sprössig W., Quaternionic and Clifford calculus for Physicists and Engineers (Wiley, Chichester, New York, 1997).
Kaehler U., “On a direct decomposition of the space Lp(Ω)”, Z. Anal. Anwend. 18(4), 839–848 (1999). https://doi.org/10.4171/ZAA/917
Le H.T., Hyperholomorphic structures and corresponding explicit orthogonal function systems in 3D and 4D (Ph. D. Thesis, Institute of Applied Analysis, 2014. Freiberg University of Mining and Technology, Germany).
Le H.T., Morais J., “Sprössig W. Orthogonal decompositions of the complex quaternion Hilbert space and their applications”, 9th International Conference on Clifford algebras and theirs Applications in Mathematical Physics, Weimar, Germany 15-20 July, 2011 (ed. K.Gürlebeck, 2011).
Le H.T., Morais J., SprössigW., “Orthogonal decompositions and their applications”, Proceedings of the 19th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering, Weimar, Germany 04–06 July, 2012 (eds. Gürlebeck K., Lahmer T., Werner F., 2012).
Le H.T., Morais J., Sprössig W., “Orthogonal decompositions of the complex quaternion Hilbert space and their applications”, 9th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences: ICNPAA 2012. AIP Conf. Proc. 1493, 595–602 (2012).
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).
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). https://doi.org/10.1002/1099-1476(200008)23:12%3C1057::AID-MMA151%3E3.0.CO;2-6
Li D., Mao J.F., “A Koch-like sided fractal bow-tie dipole antenna”, Antennas and Propagation, IEEE Transactions on 60(5), 2242–2251 (2012). https://doi.org/10.1109/TAP.2012.2189719
Gürlebeck K., Sprössig W., Quaternionic analysis and elliptic boundary value problems (Birkhauser, Basel, 1990). https://doi.org/10.1007/978-3-0348-7295-9
Kravchenko V., Applied quaternionic analysis, in: Research and Exposition in Mathematics 28 (Heldermann Verlag, Germany, 2003).
Kravchenko V., Shapiro M., Integral representations for spatial models of mathematical physics, in: Pitman Res. Notes in Math. Ser. 351 (Longman, Harlow, 1996).
Gerus O.F., Shapiro M., “On boundary properties of metaharmonic simple and double layer potentials on rectifiable curves in R2”, Zb. Pr. Inst. Mat. NAN Ukr. 1(3), 67–76 (2004).
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).
Shapiro M., Tovar L., “Two-dimensional Helmholtz operator and its hyperholomorphic solutions”, J. Natur. Geom. 11, 77–100 (1997).
Shapiro M., Tovar L., “On a class of integral representations related to the two-dimensional Helmholtz operator”, Contemp. Math. 212, 229–244 (Amer. Math. Soc., Providence, RI, 1998). https://doi.org/10.1090/conm/212/02886
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, issue 2, 255–270 (2015). https://doi.org/10.1007/s00006-014-0485-x
Bory Reyes, J., Abreu Blaya, R., Rodríguez-Dagnino R.M., Kats B.A., “On Riemann boundary value problems for null solutions of two dimensional Helmholtz equation”, Anal. Math. Phys. 9, issue 1, 483–496 (2019). https://doi.org/10.1007/s13324-018-0210-3
Bory Reyes J., Abreu Blaya R., Pérez de la Rosa M.A., Schneider B., “On the 2D Quaternionic Metaharmonic Layer Potentials”, Mediterranean Journal of Mathematics 14, issue 4, 195 (2017). https://doi.org/10.1007/s00009-017-0995-6
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). https://doi.org/10.1002/mma.2665
Bory-Reyes J., Abreu-Blaya R., Hernández-Simon L.M., Schneider B., “Dirichlet-Type Problems for the Two-Dimensional Helmholtz Operator in Complex Quaternionic Analysis”, Mediterranean Journal of Mathematics 13, 4901–4916 (2016). https://doi.org/10.1007/s00009-016-0781-x
Wloka J., Partielle Differentialgleichungen [Partial differential equations]. Sobolevräume und Randwertaufgaben [Sobolev spaces and boundary value problems] (Mathematische Leitfäden, B. G. Teubner, Stuttgart, 1982). https://doi.org/10.1007/978-3-322-96662-9_1
Garnir H.G., Les Problèmes aux Limites de la Physique Mathèmatique (Birkhaüser, Basel, 1958).
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Eds. M. Abramowitz, I.A. Stegun, John Wiley & Sons, New York etc., 1972).
Arfken G.B., Weber H. J., Harris F.H., Mathematical Methods for Physicists (7th edn., Academic Press, Elsevier, Waltham, 2013).
Watson G.N., A Treatise on the Theory of Bessel Functions (2nd ed., Cambridge University Press, Cambridge, 1995).
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Published
2019-11-28
How to Cite
Schneider, B. (2019). Higher order Dirichlet type problems for the two-dimensional Helmholtz operator in complex quaternionic analysis. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 6(4), 646–658. https://doi.org/10.21638/11701/spbu01.2019.410
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
