RADIOELECTRONIC AND COMPUTER SYSTEMS

The subject of study is the Green's functions of the first and second boundary value problems for the Laplace equation. The study constructs the Green's functions of the first and second boundary value problems for the Laplace equation in space with a spherical segment in analytical form, as well as numerical analysis of these functions. Research task: to formalize the problem of determining Green's functions for the specified domain; using methods of Fourier, pair summation equations and potential theory to reduce mixed boundary value problems for auxiliary harmonic functions to a system of equations that has an analytical solution; investigate the compatibility of the algebraic system for determining constants of integration; formulate and prove a theorem about the jump of the normal derivative of the potential of a simple layer on the surface of a segment, with the help of which to present the Green's function in the form of the potential of a simple layer; conduct a numerical experiment and identify algorithms and areas of changing the parameters of effective calculations; analyze the behavior of Green's functions. Scientific novelty: for the first time, Green's functions of Dirichlet and Neumann boundary value problems for the Laplace equation in three-dimensional space with a spherical segment were constructed in analytical form, the obtained results were substantiated, and a comprehensive numerical experiment was conducted to analyze the behavior of these functions. The obtained results: mixed boundary value problems in the interior and exterior of the spherical surface to which the segment belongs are set for the auxiliary harmonic functions; using the Fourier method, the problem is reduced to systems of paired equations in series by Legendre functions, the solutions of which are found using discontinuous Mehler-Dirichlet sums. The specified functions are obtained in an explicit view in two forms: series based on the basic harmonic functions in spherical coordinates and the potential of a simple layer on the surface of the segment. To substantiate the results, the lemma on the compatibility of the algebraic system for determining the constants of integration and the theorem on the jump of the normal derivative of the potential of a simple layer on a segment are proved. A numerical experiment was conducted to analyze the behavior of the constructed functions. Conclusions: the analysis of numerical values of Green's functions obtained by different algorithms showed that the highest accuracy of results outside the surface of the segment was obtained when using images of Green's functions in the form of series. On the basis of the calculations, the lines of the level of the Green's functions of two boundary value problems in the plane of the singular point, as well as the graphs of the potential density of the simple layer for the Dirichlet problem and the potential jump for the Neumann problem on the segment at different locations of the singular point were constructed. In the partial case of the location of a singular point at the origin of the coordinates, the potential of the electrostatic field of a point charge near a conductive grounded thin shell in the form of a spherical segment is found. The main characteristics of such a field are found in closed form.

Green's function; Dirichlet boundary value problem; Neumann boundary value problem; Laplace equation; harmonic function; spherical segment; simple layer potential; spherical functions; potential jump

Green, G. An Essay on the Application of mathematical Analysis to the theories of Electricity and Magnetism. – Nottingham: Printed by T. Wheelhouse, 1828. 72 p. Reprinted in three parts in Journal für die reine und angewandte Mathematik, 1850, vol. 39, no. 1, pp. 73-89; 1852, vol. 44, no. 4, pp. 356-74; 1854, vol. 47, no. 3, pp. 161-221. arXiv: 0807.0088v1 [physics.hist-ph] 1Jul 2008.

Bernhard Riemann's gesammelte mathematische Werke, wissenschaftlicher Nachlass. H. Weber and R. Dedekind, Eds. Leipzig: Teubner. Reprint of the second edition of 1892, including the supplement prepared by M. Noether and W. Wirtinger in 1902, NY., Dover Publ., 1953, pp. 3-45.

Morse, P. M., Feshbach, H. Methods of Theoretical Physics. Part I. NY.: McGraw-Hill Publ., 1953. 1060 p.

Morse, P. M., Feshbach, H. Methods of Theoretical Physics. Part II. NY.: McGraw-Hill Publ., 1953. 997 p.

Greenhill, A. G. On Green’s function for a rectangular parallelepiped. Proc. Cambridge Philos. Soc., 1879, vol. 3, pp. 289-293.

Dougall, J. The determination of Green’s function by means of cylindrical and spherical harmonics. Proc. Edinburgh Math. Soc., 1900, vol. 18, ser. 1, pp. 33-83.

Riemann, B., Hattendorff, K. Shwere, Elektrizität und Magnetismus. Hannover, Publ. C. Rümpler, 1876. 358 p.

Helmholtz, H. Theorie der Luftschwingunge in Röhren mit offenen. J. Reine Angew. Math., 1860, vol. 57, pp. 1-72. http://eudml.org/doc/147771

Carslaw, H. S. The use of Green’s functions in the mathematical theory of the conduction of heat. Proc. Edinburgh Math. Soc., 1902, vol. 21, ser. 1, pp. 40-64. DOI: 10.1017/S0013091500034507.

Sommerfeld, A. Über verzweigte Potentiale im Raum. Proc. London Math. Soc., 1897, vol. 28, ser. 1, pp. 395-429.

Hobson, E. W. On Green’s function for a circular disk with applications to electrostatic problems. Trans. Cambridge Philos. Soc., 1900, vol. 18, pp. 277 – 291.

Thomson, W. Note the integration of the equations of equilibrium of an elastic solid. The Cambridge and Dublin Mathematical Journal, 1848, vol. 3, pp. 87-89.

Boussinesq, J. Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques. Paris, France: Gauthier-Villars, 1885. 721 p. http://name.umd1.umich.edu/ABV5032.0001.001

Mindlin, R.D. Force at a point in interior of a semiinfinite solid. J. Appl. Physics, 1936, vol. 7, no. 5, pp. 195-202. DOI: 10.1063/1.1745385.

Lifshits, I.M., Rozentsveyg, L.N. Postroenie tenzora Grina dlya osnovnogo uravneniya teorii uprugosti v sluchae neogranichennoy uprugo-anizotropnoy sredy [Construction of the Green's tensor for the basic equation of the theory of elasticity in the case of an unbounded elastically anisotropic medium]. Zhurnal eksperimental'noy i teoreticheskoy fiziki, 1947, vol. 17, no. 9, pp. 783-791.

Chen, J.-T., Ke, J.-N., Liao, H.-Z. Construction of Green’s function using null-field integral approach for Laplace problems with circular boundaries. CMC, 2009, vol. 9, no. 2, pp. 93-109. DOI: 10.3970/cmc.2009.009.093.

Embree, M., Trefethen, L. N. Green’s functions for multiply connected domains via conformal mapping. SIAM Rev., 1999, vol. 41, no. 4, pp. 745-761. PII. S0036144598349277.

Crowdy, D., Marshall, J., Green’s functions for Laplace’s equation in multiply connected domains. IMA Journal of Applied Mathematics, 2007, no. 9, pp. 1-24. DOI: 10.1093/imamat/hxm007.

Fabricant, V. I. Electrostatic problem of several arbitrarily charged unequal coaxial disks. Journal of Computational and Applied Mathematics, 1987, vol. 18, pp. 129-147. PII:0377-0427(87)90012-4.

Grebenkov, D., Traytak, S. Semi-analitical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries. Journal of Computational Physics, 2019, vol. 379, pp. 91-117. DOI: 10.1016/j.jcp.2018.10.033.

Xue, C., Deng, S. Green’s function and image system for the Laplace operator in the prolate spheroidal geometry. AIP Advances, 2017, vol. 7, article no.015024, pp. 1-17. DOI: 10.1063/1.4974156.

Scharstein, R.W. Green’s Function for the Harmonic Potential of the Three-Dimensional Wedge Transmission Problem. Transactions on Antennas and Propagation, 2004, vol. 52, no. 2, pp. 452-460. DOI: 10.1109/TAP.2004.823949.

Dolgova, I. M., Mel'nikov, Yu. A. Green’s matrix of the plane problem of elasticity theory for an orthotropic strip. Journal of Applied Mathematics and Mechanics, 1989, vol. 53, no. 1, pp. 81-84.

Pan, E. Static Green's functions in multilayered half space. Applied Mathematical Modeling, 1997, vol. 21, iss. 8, pp. 509-521. DOI: 10.1016/S0307-904X(97)00053-X.

Grekov, M. A. Green's functions for periodic problems of elastic half-space. Mechanics of Solids, 1998, vol. 33, no. 3, pp. 142-146. https://www.researchgate.net/publication/314606477.

Chen, W., Shioya, T. Green’s Functions of an External Circular Crack in a Transversely Isotropic Piezoelectric Medium. JSME International Journal. Series A, 1999, vol. 42, no. 1, pp. 73-79. DOI: 10.1299/jsmea.42.73.

Fil’shtinskii, L. A., Matvienko, T. S. Investigation of coupled physical fields in a compound piezoceramic wedge. Mechanics of Composite Materials, 1999, vol. 35, no. 6, pp. 507-512. DOI: 10.1007/BF02259472.

Pan, E., Yuan, F.G. Tree-dimensional Green's functions in anisotropic bimaterials. International Journal of Solid and Structures, 2000, vol. 37, pp. 5329-5351. DOI: 10.1016/S0020-7683(99)00216-4.

Protsyuk, B. V. Funktsiyi Hrina zadach statyky dlya sharuvatykh til iz ploskoparalel'nymy mezhamy podilu [Green's functions of problems of statics for layered bodies with plane-parallel separation boundaries]. Matematychni metody ta fizyko-mekhanichni polya, 2005, vol. 48, iss. 1, pp. 76-87.

Seremet, V. Static equilibrium of a thermoelastic half-plane: Green’s functions and solutions in integrals. Archive of Applied Mechanics, 2014, vol. 84, no. 4, pp. 553-570. DOI: 10.1007/s00419-013-0817-7.

Tadeu, A., Antonio, J., Godinho, L. Applications of Green functions in the study of acoustic problems in open and closed spaces. Journal of Sound and Vibration, 2001, vol. 247, no. 1, pp. 117-130. DOI: 10.1006/jsvi.2001.3717.

Aleixo, E., Capelas de Oliveira, E. Green’s function for the lossy wave equation. Revista Brasileira de Ensino de Fisica, 2008, vol. 30, no. 1, pp. 1302.1-1302.5. DOI: 10.1590/S1806-11172008000100003.

Jun, S., Kim, D. W. Axial Green’s functional method for steady Stokes flow in geometrically complex domains. Journal of Computational Physics, 2011, vol. 230, no. 5, pp. 2095-2124. DOI: 10.1016/j.jcp.2010.12.007.

Newman, J. N. The Green function for potential flow in a rectangular channel. Journal of Numerical Analysis, 1999, vol. 19, no. 3, pp. 357-373. DOI: 10.1007/BF00043225.

Linton, C. M. The Green’s Function for the Two-Dimensional Helmholtz Equation in Periodic Domains. Journal of Engineering Mathematics, 1998, vol. 33, no. 4, pp. 377-401. DOI:10.1023/A:1004377501747.

Melnikov, Yu. A., Powell, J. O. A Green’s function method for detection of cavity from one boundary measurement. Transactions on Modeling and Simulation, 1997, vol. 15, pp. 121-130.

Barbot, S., Fialko, Y. Fourier-domain Green’s function for an elastic semi-infinite solid under gravity, with applications to earthquake and volcano deformation. Geophys. J. Int., 2010, vol. 182, pp. 568–582. DOI: 10.1111/j.1365-246X.2010.04655.x.

Kang, N. Green’s function for Poisson Equation in Layered Nano-Structures including Graphene. A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Applied Math. Waterloo, Ontario, Canada, 2015. http://hdl.handle.net/10012/9312.

Tewary, V. K. Electrostatic Green’s function for advanced materials subject to surface loading. Journal of Engineering Mathematics, 2004, vol. 49, iss. 3, pp. 289-304. DOI: 10.1023/B:ENGI.0000031191.64358.a8.

Hon, Y. C., Li, M., Melnikov, Y. A. Inverse source identification by Green's function. Engineering Analysis with Boundary Elements, 2010, vol. 34, no. 4, pp. 352-358. DOI: 10.1016/j.enganabound.2009.09.009.

Sneddon, I. N. Mixed boundary value problems in potential theory. Amsterdam: North-Holland publ., New York, John Wily&Sons, 1966. 283 p.

Uflyand, Ya.S. Metod parnykh uravneniy v zadachakh matematicheskoy fiziki [Method of pair equations in problems of mathematical physics]. Leningrad, Nauka Publ., 1977. 220 p.

Zyuzin, V.A., Mossakovskiy, V.I. Osesim-metrichnoe nagruzhenie prostranstva so sfericheskim razrezom [Axisymmetric loading of space with a spherical cut]. Prikladnaya matematika i mekhanika, 1970, vol. 34, no. 1, pp. 179-183.

Ulitko, A.F. Metod sobstvennyh vektornyh funkcij v prostranstvennyh zadachah teorii uprugosti [Eigenfunction Method in Spatial Problems of the Theory of Elasticity]. Kiev, Naukova dumka Publ., 1979. 265 p.

Mehler, F.G. Notiz über die Dirichlet’schen Integralausdrücke für die Kugelfunction Pn(cos) und über eine analoge Integralform für die Cylinderfunction Jm(x). Mathematische Annalen, 1872, vol. 5, iss. 1, pp. 141-144.

Nikolayev, O.H., Barakhov, K.P. Deyaki typy fizychnykh poliv u konusi z neodnoridnistyu u vyhlyadi sferychnoho sehmenta [Some types of physical fields in a cone with inhomogeneity in the form of a spherical segment]. Matematychni metody ta fizyko-mekhanichni polya, 2005, vol. 48, no. 4, pp. 191-198.

Nikolayev, O. H., Barakhov, K. P. Zadacha teoriyi potentsialu dlya kruhovoho tsylindru zi sferychnym sehmentom [Potential theory problem for a circular cylinder with a spherical segment]. Visnyk L'vivs'koho universytetu. Seriya prykladna matematyka ta informatyka, 2006, no. 11, pp. 75-81.

Kellogg, O. D. Foundations of potential theory. Berlin, Heidelberg, New York, Springer-Verlag Publ., 1967. 384 p.

Landau, L. D., Lifshic, E. M. Teoreticheskaja fizika. T. VIII. Jelektrodinamika sploshnyh sred [Theoretical physics. V. VIII. Electrodynamics of continuous media]. Moscow, Nauka Publ., 1982. 621 p.

Berdnyk, S. Gomozov, A., Gretskih, D., Kartich, V., Nesterenko, M. Approximate boundary conditions for electromagnetic fields in electrodmagnetics. Radioelectronic and Computer Systems. 2022, no. 3, pp. 141-160. DOI: 10.32620/reks.2022.3.11.