Mathematical methods in electromagnetics
Spectral theory of open structures. In the 1980s, the specific problem settings and
mathematical models related to boundary value problems (BVP)s arising in
electromagnetics and acoustics that are solved on the basis of the spectral theory of
operator-valued functions (OVF)s have become known under the common name of the
spectral theory of open structures [1]. This theory was constructed and developed
during the last two decades on the basis of the spectral theory of OVFs and specially
developed methods using pseudodifferential equations [2] and has become the major field
of my research interests.
My research results are connected with the studies in various areas of applied and
numerical mathematics, mathematical physics and mathematical modelling. In particular
I developed methods of the spectral theory of OFVs. Papers [3, 4] pioneered application
of this theory in electromagnetics. The results summarized in [1, 5] are applied to solving
various elliptic boundary value problems of mathematical physics, e.g. for the Maxwell
equations and systems of the Helmholtz equations with piecewise constant complexvalued coefficients in unbounded domains with mixed boundary and transmission-type
conditions and irregular multi-connected boundary and interface contours that stretch to
infinity [6—8]. Such problems arise in mathematical models of the wave propagation and
oscillations in open resonators, waveguides, transmission lines, and fibers [9, 10]. Results
are based on the development of the theory of singular integral operators and the spectral
theory of integral OFVs with a logarithmic singularity of the kernel [1, 11].
Nonlinear problems. I also studied semilinear Helmholtz and Schrödinger equations
with variable coefficients [12—14] as well as direct and inverse scattering problems in
domains with noncompact boundaries [15]. These problems are connected with the
modeling of wave propagation in layered structures filled with nonlinear media.
General. Studies were also focused on numerical solution to boundary value and
transmission problems for the Helmholtz equations in unbounded domains by
projectional methods [2, 11]; iteration techniques [16]; and development of analytical and
numerical methods for integral equations with a logarithmic singularity of the kernel
[11].
New phenomena: interaction. Among the new phenomena investigated on the basis of
the spectral theory of open structures it should be noted interaction of oscillations and
waves [17—19].
Developments for elasticity. Based on the fact that the developed techniques are of
universal character, I has recently developed them and applied to the analysis of the
mathematical problems of elasticity and related items that enable one to model paper
layers and surfaces using specifically created mathematical models.
References
1. V. Shestopalov, Y. Shestopalov, Spectral Theory and Excitation of Open Structures,
Peter Peregrinus Ltd, London (1996).
2. A. Il'inski, Y. Smirnov, Electromagnetic Wave Diffraction by Conducting Screens, Y.
Shestopalov, Ed., VSP Int. Science Publishers, Utrecht, Boston, Köln, Tokyo (1998).
3. Y. Shestopalov, Properties of the Spectrum of a Class of Nonselfadjoint Boundary
Value Problems for the Systems of Helmholtz Equations, Dokl. Akad. Nauk SSSR,
Maths., 252, pp. 1108--1111 (1980).
4. Y. Shestopalov, On a Spectrum of the Family of Nonselfadjoint Boundary Value
Problems for the System of Helmholtz Equations, Zh. Vych. Mat. Mat. Fiz., 21, pp.
1459—1470 (1980).
5. A. Il'inski, Y. Shestopalov, Applications of the Methods of Spectral Theory in the
Problems of Wave Propagation, Moscow State Univ. Publ. House, Moscow, 1989.
6. Y. Shestopalov and E. Chernokozhin, Solvability of Boundary Value Problems for the
Helmholtz Equation in an Unbounded Domain with Noncompact Boundary, Diff.
Equations, 34, 4, pp. 546--553 (1998).
7. Y. Podlipenko and Y. Shestopalov, On the Electromagnetic Scattering Problem for an
Infinite Dielectric Cylinder of an Arbitrary Cross Section Located in the Wedge, J. Math.
Phys., 40, 10, pp. 4888--4902 (1999).
8. Y. Shestopalov, Y. Smirnov, The Diffraction in a Class of Unbounded Domains
Connected through a Hole, Math. Methods in the Applied Sciences, 26, pp. 1363-1389
(2003).
9. Y. Shestopalov, On the Theory of Cylindrical Resonators, Math. Methods in the
Applied Sciences, 14, pp. 355--375 (1991).
10. Y. Shestopalov, Y. Okuno, and N. Kotik, Oscillations in Slotted Resonators with
Several Slots: Application of Approximate Semi-Inversion, Progress in Electromagnetics
Research, PIER-39, pp. 193--247 (2003).
11. Y. Shestopalov, Y. Smirnov, E. Chernokozhin, Logarithmic Integral Equations in
Electromagnetics, VSP Int. Science Publishers, Utrecht, Boston, Köln, Tokyo (2000).
12. H.W. Schürmann, V. Serov and Y. Shestopalov, TE-polarized Waves Guided by a
Lossless Nonlinear Three-layer Structure, Phys. Rev. E, 58, pp. 1040--1050 (1998).
13. Y. Smirnov, H.W. Schürmann, and Y. Shestopalov, Integral Equation Approach for
the Propagation of TE-Waves in a Nonlinear Dielectric Cylindrical Waveguide, J.
Nonlin. Math. Phys., 11, 2, pp. 256--268 (2004).
14. Y. Smirnov, H.W. Schürmann, and Y. Shestopalov, Propagation of TE-waves in
cylindrical nonlinear dielectric waveguides, Phys. Rev. E, 71, pp. 0166141-10 (2005).
15. Y. Shestopalov and V. Lozhechko, Direct and Inverse Problems of the Wave
Diffraction by Screens with Arbitrary Finite Inhomogeneities, J. Inverse Ill-Posed
Problems, 11, 6, pp. 341-353 (2003).
16. A. Samokhin, Integral Equations and Iteration Methods in Electromagnetic
Scattering, Y. Shestopalov, Ed., VSP Int. Science Publishers, Utrecht, Boston, Köln,
Tokyo (2001).
17. Y. Shestopalov and E. Chernokozhin, Resonant and Nonresonant Diffraction by Open
Image-type Slotted Structures, IEEE Trans. Antennas Propag., 49, 5, pp. 793--801
(2001).
18. Y. Shestopalov and N. Kotik, Interaction and Propagation of Waves in Slotted
Waveguides, New J. Phys., 4, 40, pp. 40.1--40.16 (2002).
19. Y. Shestopalov and O. Kotik, Interaction of Oscillations in Slotted Resonators and its
Applications to Microwave Imaging, J. Electromagn. Waves Appl., 17, 2, pp. 291-311
(2003).
.