Explicit determination is considered of the spectrum of normal waves in an inhomogeneous plane-parallel dielectric layer with a parabolic permittivity profile. Exact formulas are obtained for the propagation constants and normal wave field components.

A method for the accurate calculation of the cut-off wavenumbers of a waveguide with an arbitrary cross section and a number of inner conductors is demonstrated. A secure basis for formulating the spectral problem relies upon concepts of integral and infinite-matrix (summation) operator-valued functions depending nonlinearly on the frequency spectral parameter; whilst a variety of methods might be employed in determining the cut-off wavenumbers so specified, the Method of Analytical Regularization provides a route to the construction of an algorithm that is well-conditioned (and hence reliable). The algorithm is based on a mathematically rigorous solution of the homogeneous Dirichlet problem for the Helmholtz equation in the interior of the waveguide, excluding the regions occupied by the inner conductor boundaries; it results in a highly efficient method of calculating the cut-off wavenumbers and the corresponding non-trivial solutions representing the modal distribution. The ability to calculate the cut-off wavenumbers with any prescribed and proven accuracy provides a secure basis for treating the wavenumbers so found as “benchmark solutions”.

A method for the accurate calculation of the cut-off wavenumbers of a waveguide with an arbitrary cross section and a number of inner conductors is demonstrated. A secure basis for formulating the spectral problem relies upon concepts of integral and infinite-matrix (summation) operator-valued functions depending nonlinearly on the frequency spectral parameter; whilst a variety of methods might be employed in determining the cut-off wavenumbers so specified, the Method of Analytical Regularization provides a route to the construction of an algorithm that is well-conditioned (and hence reliable). The algorithm is based on a mathematically rigorous solution of the homogeneous Dirichlet problem for the Helmholtz equation in the interior of the waveguide, excluding the regions occupied by the inner conductor boundaries; it results in a highly efficient method of calculating the cut-off wavenumbers and the corresponding non-trivial solutions representing the modal distribution. The ability to calculate the cut-off wavenumbers with any prescribed and proven accuracy provides a secure basis for treating the wavenumbers so found as "benchmark solutions". 

A method for the accurate calculation of the cut-off wavenumbers of a waveguide with an arbitrary cross section and a number of inner conductors is demonstrated. Concepts of integral and infinite-matrix (summation) operator-valued functions depending nonlinearly on the frequency spectral parameter provide a secure basis for formulating the spectral problem, and the Method of Analytical Regularization is employed to implement an effective algorithm. The algorithm is based on a mathematically rigorous solution of the homogeneous Dirichlet problem for the Helmholtz equation in the interior of the waveguide, excluding the regions occupied by the inner conductor boundaries. A highly efficient method of calculating the cut-off wavenumbers and the corresponding non-trivial solutions representing the modal distribution is developed. The mathematical correctness of the problem statement, the method, and the ability to calculate the cut-off wavenumbers with any prescribed and proven accuracy provide a secure basis for treating these as “benchmark solutions”. In this paper, we use this new approach to validate previously obtained results against our benchmark solutions. Furthermore, we demonstrate its universality in solving some new problems, which are barely accessible by existing methods.

The paper is devoted to the analysis of the TE-wave propagation in a planar waveguide with variable permittivity. The setting is reduced to a transmission eigenvalue problem for an ordinary differential equation where the spectral parameter is the wave propagation constant. For the computation of eigenvalues the shooting method is applied in the form specifically developed for this class of problems. The method allows one to calculate propagating, evanescent, and complex surface and leaky waves. A variety of numerical results illustrating the method is presented.

A brief survey is given of the most valuable contributions of Victor P. Shestopalov to electromagnetics and diffraction theory related to the application of the spectral theory of operator-valued functions. A particular attention is paid to the creation of the theory of intertype interaction of oscillations and waves in open structures on the basis of analysis of critical points of abstract dispersion equations. Contributions to the theory made by other researchers are reported.

Non-stationary flooding of oil-saturated reservoirs has a long-standing and durable place as the main secondary method of oil production and maintenance of reservoir pressure in the development of most oil reservoirs. The water injection into the reservoir creates a delayed problem - the inevitable, often catastrophic flooding of oil production wells, provoked by a sudden and irreversible change in water saturation. The theory of two-phase flow filtration created by Buckley and Leverett does not take into account the loss of stability of the displacement front, which provokes an abrupt change and a triplicity of the water saturation value. Therefore, a mathematically simplified approach was proposed at one time, a repeatedly differentiable approximation to exclude a “jump” in water saturation. Such a simplified solution led to well-known negative consequences of the waterflooding practice, which experts call the “viscous instability of the displacement front”, the “fingering displacement front”. This work has presented a novel approach to formulation decisive rules for the first time allowing timely detection and prevention of the consequences of loss of stability of the displacement front and targeted control of the flooding system by stopping, forcing, limiting operating modes, assigning workover solutions of producing and injection wells. It is possible to quickly solve important short-term practical tasks passing traditional labor- intensive incorrect deterministic tasks and complex methods of solution mobilizing the injected water and controlling the fluid production rate, more precisely water and oil on the basis of the discriminant criterion.

We propose and develop a novel rigorous technique that enables one to obtain the explicit numerical values of parameters at which several lowest-order harmonics of the scattered field are suppressed. This provides partial cloaking of the object, a perfectly conducting cylinder of circular cross section covered by two layers of dielectric separated by an infinitely thin impedance layer, a two-layer impedance Goubau line (GL). The developed approach is a rigorous method that enables one to obtain in the closed form (and without numerical calculations) the values of parameters providing the cloaking effect, achieved particularly in terms of the suppression of several scattered field harmonics and variation of the sheet impedance. This issue constitutes the novelty of the accomplished study. The elaborated technique could be applied to validate the results obtained by commercial solvers with virtually no limitations on the parameter ranges, i.e., use it as a benchmark. The determination of the cloaking parameters is straightforward and does not require computations. We perform comprehensive visualization and analysis of the achieved partial cloaking. The developed parameter-continuation technique enables one to increase the number of the suppressed scattered-field harmonics by appropriate choice of the impedance. The method can be extended to any dielectric-layered impedance structures possessing circular or planar symmetry. 

We consider eigenvalue problems for dielectric cylindrical scatterers of arbitrary cross section with generalized conditions at infinity that enable one to take into account complex eigenvalues. The existence of resonance (scattering) frequencies associated with these eigenvalues is proved. The technique involves the determination of characteristic numbers (CNs) of the Fredholm operator-valued functions of the problems constructed using Green's potentials. Separating principal parts in the form of meromorphic operator pencils, we apply the operator generalization of Rouché's theorem to verify the occurrence of CNs in close proximities of the pencil poles. The results are illustrated in detail using the case of a dielectric cylinder of circular cross section.

The problem of diffraction of a TE-polarized electromagnetic wave by a circular slotted cylinder is investigated. The boundary value problem in question for the Helmholtz equation is reduced to an infinite system of linear algebraic equations of the second kind (SLAE-II) using integral summation identities (ISI). A detailed study of the matrix operator of the problem is performed and its Fredholm property in the weighted Hilbert space of infinite sequences is proven. The convergence of the truncation method constructed in the paper for the numerical solution of SLAE-II is justified and the results of computations are presented and discussed, specifically considering the determination of resonance modes.