Finding the Laplace operator's own values for the areas covered by periodic contours with the help of auxiliary means (conduit is a model)

In the bachelor thesis, solving of the Laplace operator's eigenvalue problem in an area with space periodic boundaries is studied by the method of auxiliary sources. Solving of these type problems has great practical importance in acoustics, electrodynamics, quantum mechanics and other areas where quantitative solving is topical. To solve this problem, the model of a resonator is used, which inner area has small conductivity. It is well known from practice that existence of resistance does not change the resonant frequency in the RLC contour, it only affects on Q factor. In general, parametrical study of the resonance system shows that resonance curve becomes wider and the resonant frequency is easier to find. In case of the ideal resonator, the characteristic curve becomes like Dirac delta function. And finding of the resonant frequencies, which have discrete distribution, is practically impossible. The thesis describes the method of auxiliary sources, its application to eigenvalue problem of the Laplace operator, extension for space periodic problems with conductive areas. In the thesis, areas with boundaries shaped like periodic rectangular pulses are considered. 5-6 first eigenvalues are found for the areas and illustrated graphically.