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In problems of modeling pollution of water and atmospheric resources of the Earth’s ecosystems, models of potential flow are often used. Recently, considerable attention has also been paid to protecting metropolis from atmospheric pollution with acoustic noise. Very often, without focusing on local flow features, the Laplace equation is used to describe the potential equation motion of a fluid. Acoustic problems are modeled based on the Helmholtz equation.
In the work presented below, the features of the numerical-analytical method for the Helmholtz and Laplace equations are considered. The sound potential is a rapidly oscillating function given at the boundary of the computational domain. In addition, the numerically-analytical method peculiarities are presented for the Laplace equation that uses the expansion of the boundary condition in a Fourier series in eigenvalues of the Sturm-Liouville problem. Despite the fact that the data presented in the work were obtained for canonical domains, the scheme of the method implies the possibility of using it for domains with an arbitrary curvilinear boundary. The numerical-analytical method proposed in the paper allows for low computational costs to solve numerically the problems for the Laplace equation and the acoustic equations. The results of the research conducted can be used as new information technology to address the environmental security challenges of water and air resources.

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