Boundary Value Problems in Ellipsoidal Geometry and Applications

Panayiotis Vafeas

doi:10.61927/igmin263

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Biology Group Review Article 記事ID: igmin263

Boundary Value Problems in Ellipsoidal Geometry and Applications

Mathematics

Panayiotis Vafeas ^*

Affiliation

Department of Chemical Engineering, University of Patras, 26504 Patras, Greece

DOI10.61927/igmin263 全文HTML 全文PDF この記事を引用する

要約

Many applications in science, engineering, and modern technology require the solution of boundary value problems for genuine three-dimensional objects. These objects often are of or can be approximated by, an ellipsoidal shape, where the three ellipsoidal semiaxes correspond to three independent degrees of freedom. The triaxial ellipsoid represents the sphere of any anisotropic space and for this reason, it appears naturally in many scientific disciplines. Consequently, despite the complications of the ellipsoidal geometry and mainly its analysis, based on the theory of ellipsoidal harmonics, a lot of progress has been made in the solution of ellipsoidal boundary value problems, due to its general applicability. In this mini-review, we aim to present to the scientific community the main achievements towards the investigation of three such physical problems of medical, engineering and technological significance, those comprising intense research in (a) electroencephalography (EEG) and magnetoencephalography (MEG), (b) creeping hydrodynamics (Stokes flow) and (c) identification of metallic impenetrable bodies, either embedded within the Earth’s conductive subsurface or located into a lossless air environment. In this context, special expertise and particular skills are needed in solving open boundary value problems that incorporate the ellipsoidal geometry and the related harmonic analysis, revealing the fact that there still exists the necessity of involving with these issues.

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参考文献

Dassios G. Ellipsoidal harmonics: theory and applications. Cambridge: Cambridge University Press; 2012.
Moon P, Spencer E. Field theory handbook. Berlin: Springer-Verlag; 1971.
Morse PM, Feshbach H. Methods of theoretical physics. Vols. I & II. New York: McGraw-Hill; 1953.
Hobson EW. The theory of spherical and ellipsoidal harmonics. New York: Chelsea Publishing Company; 1965.
Dassios G, Satrazemi K. Lamé functions and ellipsoidal harmonics up to degree seven. Int J Spec Funct Appl. 2014;2(1):27-40.
Dassios G, Kariotou F, Vafeas P. Invariant vector harmonics: the ellipsoidal case. J Math Anal Appl. 2013;405:652-60.
Fragoyiannis G, Vafeas P, Dassios G. On the reducibility of the ellipsoidal system. Math Methods Appl Sci. 2022;45:4497-4554.
Kariotou F. On the mathematics of EEG and MEG in spheroidal geometry. Bull Greek Math Soc. 2003;47:117-35.
Dassios G, Fokas AS, Hadjiloizi D. On the complementarity of electroencephalography and magnetoencephalography. Inverse Problems. 2007;23:2541.
Dassios G, Doschoris M, Satrazemi K. Localizing brain activity from multiple distinct sources via EEG. J Appl Math. 2014;2014:232747.
Dassios G, Hadjiloizi D. On the non-uniqueness of the inverse problem associated with electroencephalography. Inverse Problems. 2009;25:115012.
Dassios G, Fragoyiannis G, Satrazemi K. On the inverse EEG problem for a 1D current distribution. J Appl Math. 2014;2014:715785.
Dassios G, Satrazemi K. Inversion of electroencephalography data for a 2D current distribution. Math Methods Appl Sci. 2014;38:1098-1105.
Dassios G, Doschoris M, Satrazemi K. On the resolution of synchronous dipolar excitations via MEG measurements. Q Appl Math. 2018;76:39-45.
Doschoris M, Dassios G, Fragoyiannis G. Sensitivity analysis of the forward electroencephalographic problem depending on head shape variations. Math Probl Eng. 2015;2015:1-14.
Doschoris M, Vafeas P, Fragoyiannis G. The influence of surface deformations on the forward magnetoencephalographic problem. SIAM J Appl Math. 2018;78:963-76.
Papargiri A, Kalantonis VS, Vafeas P, Doschoris M, Kariotou F, Fragoyiannis G. On the geometrical perturbation of a three-shell spherical model in electroencephalography. Math Methods Appl Sci. 2022;45:8876-89.
Papargiri A, Kalantonis VS, Fragoyiannis G. Mathematical modeling of brain swelling in electroencephalography and magnetoencephalography. Mathematics. 2023;11:2582.
Dassios G, Vafeas P. Connection formulae for differential representations in Stokes flow. J Comput Appl Math. 2001;133:283-94.
Dassios G, Vafeas P. The 3D Happel model for complete isotropic Stokes flow. Int J Math Math Sci. 2004;46:2429-41.
Dassios G, Vafeas P. On the spheroidal semiseparation for Stokes flow. Res Lett Phys. 2008;2008:135289:1-4.
Vafeas P, Protopapas E, Hadjinicolaou M. On the analytical solution of the Kuwabara-type particle-in-cell model for the non-axisymmetric spheroidal Stokes flow via the Papkovich - Neuber representation. Symmetry. 2022;14:170:1-21.
Svarnas P, Papadopoulos PK, Vafeas P, Gkelios A, Clément F, Mavon A. Influence of atmospheric pressure guided streamers (plasma bullets) on the working gas pattern in air. IEEE Trans Plasma Sci. 2014;42:2430-1.
Papadimas V, Doudesis C, Svarnas P, Papadopoulos PK, Vafakos GP, Vafeas P. SDBD flexible plasma actuator with Ag-Ink electrodes: experimental assessment. Appl Sci. 2021;11:11930:1-13.
Vafeas P, Bakalis P, Papadopoulos PK. Effect of the magnetic field on the ferrofluid flow in a curved cylindrical annular duct. Phys Fluids. 2019;31:117105:1-15.
Vafeas P, Perrusson G, Lesselier D. Low-frequency solution for a perfectly conducting sphere in a conductive medium with dipolar excitation. Prog Electromagn Res. 2004;49:87-111.
Stefanidou E, Vafeas P, Kariotou F. An analytical method of electromagnetic wave scattering by a highly conductive sphere in a lossless medium with low-frequency dipolar excitation. Mathematics. 2021;9:3290:1-25.
Vafeas P, Papadopoulos PK, Lesselier D. Electromagnetic low-frequency dipolar excitation of two metal spheres in a conductive medium. J Appl Math. 2012;628261:1-37.
Vafeas P, Lesselier D, Kariotou F. Estimates for the low-frequency electromagnetic fields scattered by two adjacent metal spheres in a lossless medium. Math Methods Appl Sci. 2015;38:4210-37.
Vafeas P, Perrusson G, Lesselier D. Low-frequency scattering from perfectly conducting spheroidal bodies in a conductive medium with magnetic dipole excitation. Int J Eng Sci. 2009;47:372-90.
Vafeas P. Dipolar excitation of a perfectly electrically conducting spheroid in a lossless medium at the low-frequency regime. Adv Math Phys. 2018;9587972:1-20.
Vafeas P, Papadopoulos PK, Ding P-P, Lesselier D. Mathematical and numerical analysis of low-frequency scattering from a PEC ring torus in a conductive medium. Appl Math Model. 2016;40:6477-500.
Vafeas P. Low-frequency electromagnetic scattering by a metal torus in a lossless medium with magnetic dipolar illumination. Math Methods Appl Sci. 2016;39:4268-92.
Stratton JA. Electromagnetic theory. New York: McGraw-Hill; 1941.
Maxwell JC. A treatise on electricity and magnetism. Vols. I & II. Oxford: Oxford University Press; 1998.
Dassios G, Fokas AS. Electroencephalography and magnetoencephalography. Boston: De Gruyter; 2020.
Dassios G, Fokas AS. The definite non-uniqueness results for deterministic EEG and MEG data. Inverse Problems. 2013;29:065012.
Dassios G. Electric and magnetic activity of the brain in spherical and ellipsoidal geometry. In: Ammari H, editor. Mathematical modeling in biomedical imaging I. Berlin: Springer. 2009; 133-202.
Fokas AS. Electro-magneto-encephalography for a three-shell model: distributed current in arbitrary, spherical and ellipsoidal geometries. J R Soc Interface. 2009 May 6;6(34):479-88. doi: 10.1098/rsif.2008.0309. Epub 2008 Aug 29. PMID: 18757270; PMCID: PMC2659695.
Dassios G, Fokas AS. Electro-magneto-encephalography for a three-shell model: dipoles and beyond for the spherical geometry. Inverse Problems. 2009;25:035001.
Kariotou F. Electroencephalography in ellipsoidal geometry. J Math Anal Appl. 2004;290:324-42.
Giapalaki S, Kariotou F. The complete ellipsoidal shell model in EEG imaging. Abstr Appl Anal. 2006;2006:57429.
Doschoris M, Kariotou F. Quantifying errors during the source localization process in electroencephalography for confocal systems. IMA J Appl Math. 2018;83:243-60.
Doschoris M, Kariotou F. Error analysis for nonconfocal ellipsoidal systems in the forward problem of electroencephalography. Math Methods Appl Sci. 2018;41:6793-813.
Dassios G, Kariotou F. Magnetoencephalography in ellipsoidal geometry. J Math Phys. 2003;44:220-41.
Dassios G, Giapalaki SN, Kandili AN, Kariotou F. The exterior magnetic field for the multilayer ellipsoidal model of the brain. Q J Mech Appl Math. 2007;60:1-25.
Dassios G, Hadjiloizi D, Kariotou F. The octapolic ellipsoidal term in magnetoencephalography. J Math Phys. 2009;50:013508.
Vafeas P, Dassios G. Stokes flow in ellipsoidal geometry. J Math Phys. 2006;47:093102:1-38.
Hatzikonstantinou PM, Vafeas P. A general theoretical model for the magnetohydrodynamic flow of micropolar magnetic fluids: application to Stokes flow. Math Methods Appl Sci. 2010;33:233-48.
Stratton JA. Electromagnetic theory. New York: McGraw-Hill; 1941.
Dassios G, Kleinman RE. Low-frequency scattering. Oxford: Oxford University Press; 2000.
Ammari H, Kang H. Polarization and moment tensors: with applications to inverse problems and effective medium theory. Applied Mathematical Sciences. Vol. 162. New York: Springer-Verlag; 2007.
Björkberg J, Kristenson G. Three-dimensional subterranean target identification by use of optimization techniques. Prog Electromagn Res. 1997;15:141-64.
Yu T, Carin L. Analysis of the electromagnetic inductive response of a void in a conducting-soil background. IEEE Trans Geosci Remote Sens. 2000;38:1320-7.
Huang H, Won IJ. Detecting metal objects in magnetic environments using a broadband electromagnetic method. Geophysics. 2003;68:1877-87.
Cui TJ, Chew WC, Wright DL, Smith DV. Three-dimensional imaging for buried objects in a very lossy earth by inversion of VETEM data. IEEE Trans Geosci Remote Sens. 2003;41:2197-210.
Tortel H. Electromagnetic imaging of a three-dimensional perfectly conducting object using a boundary integral formulation. Inverse Problems. 2004;20:385-98.
Perrusson G, Vafeas P, Lesselier D. Low-frequency dipolar excitation of a perfect ellipsoidal conductor. Q Appl Math. 2010;68:513-36.
Perrusson G, Vafeas P, Chatjigeorgiou IK, Lesselier D. Low-frequency on-site identification of a highly conductive body buried in Earth from a model ellipsoid. IMA J Appl Math. 2015;80:963-80.
Vafeas P. Low-frequency dipolar electromagnetic scattering by a solid ellipsoid in the lossless environment. Stud Appl Math. 2020;145:217-46.

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[1] Dassios G. Ellipsoidal harmonics: theory and applications. Cambridge: Cambridge University Press; 2012.

[2] Moon P, Spencer E. Field theory handbook. Berlin: Springer-Verlag; 1971.

[3] Morse PM, Feshbach H. Methods of theoretical physics. Vols. I & II. New York: McGraw-Hill; 1953.

[4] Hobson EW. The theory of spherical and ellipsoidal harmonics. New York: Chelsea Publishing Company; 1965.

[5] Dassios G, Satrazemi K. Lamé functions and ellipsoidal harmonics up to degree seven. Int J Spec Funct Appl. 2014;2(1):27-40.

[6] Dassios G, Kariotou F, Vafeas P. Invariant vector harmonics: the ellipsoidal case. J Math Anal Appl. 2013;405:652-60.

[7] Fragoyiannis G, Vafeas P, Dassios G. On the reducibility of the ellipsoidal system. Math Methods Appl Sci. 2022;45:4497-4554.

[8] Kariotou F. On the mathematics of EEG and MEG in spheroidal geometry. Bull Greek Math Soc. 2003;47:117-35.

[9] Dassios G, Fokas AS, Hadjiloizi D. On the complementarity of electroencephalography and magnetoencephalography. Inverse Problems. 2007;23:2541.

[10] Dassios G, Doschoris M, Satrazemi K. Localizing brain activity from multiple distinct sources via EEG. J Appl Math. 2014;2014:232747.

[11] Dassios G, Hadjiloizi D. On the non-uniqueness of the inverse problem associated with electroencephalography. Inverse Problems. 2009;25:115012.

[12] Dassios G, Fragoyiannis G, Satrazemi K. On the inverse EEG problem for a 1D current distribution. J Appl Math. 2014;2014:715785.

[13] Dassios G, Satrazemi K. Inversion of electroencephalography data for a 2D current distribution. Math Methods Appl Sci. 2014;38:1098-1105.

[14] Dassios G, Doschoris M, Satrazemi K. On the resolution of synchronous dipolar excitations via MEG measurements. Q Appl Math. 2018;76:39-45.

[15] Doschoris M, Dassios G, Fragoyiannis G. Sensitivity analysis of the forward electroencephalographic problem depending on head shape variations. Math Probl Eng. 2015;2015:1-14.

[16] Doschoris M, Vafeas P, Fragoyiannis G. The influence of surface deformations on the forward magnetoencephalographic problem. SIAM J Appl Math. 2018;78:963-76.

[17] Papargiri A, Kalantonis VS, Vafeas P, Doschoris M, Kariotou F, Fragoyiannis G. On the geometrical perturbation of a three-shell spherical model in electroencephalography. Math Methods Appl Sci. 2022;45:8876-89.

[18] Papargiri A, Kalantonis VS, Fragoyiannis G. Mathematical modeling of brain swelling in electroencephalography and magnetoencephalography. Mathematics. 2023;11:2582.

[19] Dassios G, Vafeas P. Connection formulae for differential representations in Stokes flow. J Comput Appl Math. 2001;133:283-94.

[20] Dassios G, Vafeas P. The 3D Happel model for complete isotropic Stokes flow. Int J Math Math Sci. 2004;46:2429-41.

[21] Dassios G, Vafeas P. On the spheroidal semiseparation for Stokes flow. Res Lett Phys. 2008;2008:135289:1-4.

[22] Vafeas P, Protopapas E, Hadjinicolaou M. On the analytical solution of the Kuwabara-type particle-in-cell model for the non-axisymmetric spheroidal Stokes flow via the Papkovich - Neuber representation. Symmetry. 2022;14:170:1-21.

[23] Svarnas P, Papadopoulos PK, Vafeas P, Gkelios A, Clément F, Mavon A. Influence of atmospheric pressure guided streamers (plasma bullets) on the working gas pattern in air. IEEE Trans Plasma Sci. 2014;42:2430-1.

[24] Papadimas V, Doudesis C, Svarnas P, Papadopoulos PK, Vafakos GP, Vafeas P. SDBD flexible plasma actuator with Ag-Ink electrodes: experimental assessment. Appl Sci. 2021;11:11930:1-13.

[25] Vafeas P, Bakalis P, Papadopoulos PK. Effect of the magnetic field on the ferrofluid flow in a curved cylindrical annular duct. Phys Fluids. 2019;31:117105:1-15.

[26] Vafeas P, Perrusson G, Lesselier D. Low-frequency solution for a perfectly conducting sphere in a conductive medium with dipolar excitation. Prog Electromagn Res. 2004;49:87-111.

[27] Stefanidou E, Vafeas P, Kariotou F. An analytical method of electromagnetic wave scattering by a highly conductive sphere in a lossless medium with low-frequency dipolar excitation. Mathematics. 2021;9:3290:1-25.

[28] Vafeas P, Papadopoulos PK, Lesselier D. Electromagnetic low-frequency dipolar excitation of two metal spheres in a conductive medium. J Appl Math. 2012;628261:1-37.

[29] Vafeas P, Lesselier D, Kariotou F. Estimates for the low-frequency electromagnetic fields scattered by two adjacent metal spheres in a lossless medium. Math Methods Appl Sci. 2015;38:4210-37.

[30] Vafeas P, Perrusson G, Lesselier D. Low-frequency scattering from perfectly conducting spheroidal bodies in a conductive medium with magnetic dipole excitation. Int J Eng Sci. 2009;47:372-90.

[31] Vafeas P. Dipolar excitation of a perfectly electrically conducting spheroid in a lossless medium at the low-frequency regime. Adv Math Phys. 2018;9587972:1-20.

[32] Vafeas P, Papadopoulos PK, Ding P-P, Lesselier D. Mathematical and numerical analysis of low-frequency scattering from a PEC ring torus in a conductive medium. Appl Math Model. 2016;40:6477-500.

[33] Vafeas P. Low-frequency electromagnetic scattering by a metal torus in a lossless medium with magnetic dipolar illumination. Math Methods Appl Sci. 2016;39:4268-92.

[34] Stratton JA. Electromagnetic theory. New York: McGraw-Hill; 1941.

[35] Maxwell JC. A treatise on electricity and magnetism. Vols. I & II. Oxford: Oxford University Press; 1998.

[36] Dassios G, Fokas AS. Electroencephalography and magnetoencephalography. Boston: De Gruyter; 2020.

[37] Dassios G, Fokas AS. The definite non-uniqueness results for deterministic EEG and MEG data. Inverse Problems. 2013;29:065012.

[38] Dassios G. Electric and magnetic activity of the brain in spherical and ellipsoidal geometry. In: Ammari H, editor. Mathematical modeling in biomedical imaging I. Berlin: Springer. 2009; 133-202.

[39] Fokas AS. Electro-magneto-encephalography for a three-shell model: distributed current in arbitrary, spherical and ellipsoidal geometries. J R Soc Interface. 2009 May 6;6(34):479-88. doi: 10.1098/rsif.2008.0309. Epub 2008 Aug 29. PMID: 18757270; PMCID: PMC2659695.

[40] Dassios G, Fokas AS. Electro-magneto-encephalography for a three-shell model: dipoles and beyond for the spherical geometry. Inverse Problems. 2009;25:035001.

[41] Kariotou F. Electroencephalography in ellipsoidal geometry. J Math Anal Appl. 2004;290:324-42.

[42] Giapalaki S, Kariotou F. The complete ellipsoidal shell model in EEG imaging. Abstr Appl Anal. 2006;2006:57429.

[43] Doschoris M, Kariotou F. Quantifying errors during the source localization process in electroencephalography for confocal systems. IMA J Appl Math. 2018;83:243-60.

[44] Doschoris M, Kariotou F. Error analysis for nonconfocal ellipsoidal systems in the forward problem of electroencephalography. Math Methods Appl Sci. 2018;41:6793-813.

[45] Dassios G, Kariotou F. Magnetoencephalography in ellipsoidal geometry. J Math Phys. 2003;44:220-41.

[46] Dassios G, Giapalaki SN, Kandili AN, Kariotou F. The exterior magnetic field for the multilayer ellipsoidal model of the brain. Q J Mech Appl Math. 2007;60:1-25.

[47] Dassios G, Hadjiloizi D, Kariotou F. The octapolic ellipsoidal term in magnetoencephalography. J Math Phys. 2009;50:013508.

[48] Vafeas P, Dassios G. Stokes flow in ellipsoidal geometry. J Math Phys. 2006;47:093102:1-38.

[49] Hatzikonstantinou PM, Vafeas P. A general theoretical model for the magnetohydrodynamic flow of micropolar magnetic fluids: application to Stokes flow. Math Methods Appl Sci. 2010;33:233-48.

[50] Stratton JA. Electromagnetic theory. New York: McGraw-Hill; 1941.

[51] Dassios G, Kleinman RE. Low-frequency scattering. Oxford: Oxford University Press; 2000.

[52] Ammari H, Kang H. Polarization and moment tensors: with applications to inverse problems and effective medium theory. Applied Mathematical Sciences. Vol. 162. New York: Springer-Verlag; 2007.

[53] Björkberg J, Kristenson G. Three-dimensional subterranean target identification by use of optimization techniques. Prog Electromagn Res. 1997;15:141-64.

[54] Yu T, Carin L. Analysis of the electromagnetic inductive response of a void in a conducting-soil background. IEEE Trans Geosci Remote Sens. 2000;38:1320-7.

[55] Huang H, Won IJ. Detecting metal objects in magnetic environments using a broadband electromagnetic method. Geophysics. 2003;68:1877-87.

[56] Cui TJ, Chew WC, Wright DL, Smith DV. Three-dimensional imaging for buried objects in a very lossy earth by inversion of VETEM data. IEEE Trans Geosci Remote Sens. 2003;41:2197-210.

[57] Tortel H. Electromagnetic imaging of a three-dimensional perfectly conducting object using a boundary integral formulation. Inverse Problems. 2004;20:385-98.

[58] Perrusson G, Vafeas P, Lesselier D. Low-frequency dipolar excitation of a perfect ellipsoidal conductor. Q Appl Math. 2010;68:513-36.

[59] Perrusson G, Vafeas P, Chatjigeorgiou IK, Lesselier D. Low-frequency on-site identification of a highly conductive body buried in Earth from a model ellipsoid. IMA J Appl Math. 2015;80:963-80.

[60] Vafeas P. Low-frequency dipolar electromagnetic scattering by a solid ellipsoid in the lossless environment. Stud Appl Math. 2020;145:217-46.

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