Kostenko O. Mathematical models of diffraction by prefractal electrodynamics structures

1. The objects of investigation: the processes of electromagnetic waves diffraction and scattering; the process of interaction electromagnetic waves and material half spaces. 2. The aim of investigation: to develop the mathematical models of diffraction plane monochromatic electromagnetic waves on ideally thin infinity multielement impedance bounded or periodic lattice; to obtain the second kind boundary integral equations (hypersingular and with logarithmic singularity) of the diffraction and scattering impedance problem; to investigate the diffraction and scattering characteristics of the multielement impedance prefractal lattices; to analyze the second kind hypersingular integral equation and the system of similar equations (a numerical method for solving this equation or system, the existence and uniqueness theorem, a convergence rate of the sequence of approximate solutions to the exact ones); to develop the mathematical models of the interaction of electromagnetic waves and material half spaces in the different states (normal, ideally superconducting or superconducting). 3. The research methods: the analytical methods of mathematical physics; an analytical technique of integral transforms (the parametric representations of hyper singular; singular and with logarithmic singularity improve integrals); the numerical method of discrete singularities; the functional analyze methods; the linear algebra methods; the methods of mathematical theory of diffraction and scattering. 4. The theoretical results: the mathematical diffraction theory development; the hypersingular integral equation qualitative theory extension; the progress of numerical methods for solving hypersingular integral equation. The practice results: the quantitative characteristics of wave diffraction on prefractal lattices (up to the fifth order inclusive) were obtained and can be applied for solving the problems of the radio technology and related industry. 5. The scientific novelty of the research: the two dimensions mathematical models of plane monochromatic electromagnetic wave diffraction on infinity multielement impedance bounded or periodic different form lattice were developed (the pairs of second kind boundary integral equations which are with logarithmic singularities in the kernel and hypersingular respectively); the discrete models and their computer realization were obtained; the numerical method for solving the second kind hypersingular integral equation was developed and justified (the existence and uniqueness theorem, the convergence rate of the sequence of approximate solutions to the exact ones); a justified generalization of this method for solving the system of hypersingular integral equations was obtained; with respect to the refractive index were developed the mathematical models of the interaction of the electromagnetic waves and the material half spaces in the different states (normal, ideally superconducting or superconducting); on the base of this mathematical models the diffraction characteristics of the impedance prefractal and other form lattices were numerical analyzed. 6. The implementation is planned. 7. The radio technology, electronics, the theory of hypersingular integral equations.