R.M. Gadelshin - Ph. D. (Eng.), Associate Professor, Kazan National Research Technical University named after A.N. Tupolev E-mail: garadman@mail.ru

The traditional solving of the Helmholtz equation for bounded area considers it as a simpler task in contrast with general problem for unbounded area. It determines using sequence of the solving, when sought spatial distribution at this area is selected from ensemble of solutions for unbounded area by meet to terms on its borders. In this case a using the method of eigenfunctions systems allows a capacity of solving this problem only for discrete values of frequency in the form a single component of this system. Moreover, a resulting solution is a arbitrarily selected fragment of the solution for unbounded area and is conflicting the source Helmholtz equation. It is demonstrative reflected in frequency domain, where solution of the source Helmholtz equation in the event of unbounded area is allowed as a alone spectral component, but a solution for a bounded area function excludes such possibility. In this work spectral solving of the Helmholtz equation for bounded area is offered, that allows changing to sequences of a solving. Obviously that spectral presentation for functions, existed on a limited area, turns out to be more general in comparison with a alone spectral component and a task for bounded area is more general in comparison with a task for unbounded area. The sequence of the solving of this problem needs at first a determination a solutions ensemble for bounded area and then a selection the result as a solution of the source Helmholtz equation. Together with that, in this case a transition to frequency domain for considered function requires to be more strictly. The presentation of a function, that differs from zero on limited interval only, in particularly a derived of them, requires exactly estimating, because Fourier transform of them is accompanied the appearance of the garbling. Sources of this inaccuracy are fixed and their manifestation and possibility of their removal from the given spectrum is discussed. Ways to correct results of a simple Fourier transform for that function to get the valid spectral presentation are considered in detail. Fairness of this discourse is confirmed by using the Kotelnikov theorem. After correction of the result of a simple transformation Fourier, it is possible a simple solving of a problem in frequency domain. The finding solution is more general in contrast with the known solution and no limited by only «resonance» cases, but allowing the solution for any value of frequency.

 

1. Vychislitelnye metody v ehlektrodinamike / Pod red. R. Mittry. Per. s angl. pod red. EH.L. Burshtejjna. M: Mir. 1977.
2. Gadelshin R.M. Adaptirovannye sistemy sobstvennykh funkcijj dlja odnomernykh uravnenijj Gelmgolca // EHlektromagnitnye volny i ehlektronnye sistemy. 2013. № 12. S. 4−9.