Green's functions of the first and second boundary value problems for the Laplace equation in the nonclassical domain
Authors: Nikolaev Oleksii , Holovchenko Oleksandr , Savchenko Nina 
Published: January 2023

The subject of study is the Green's functions of the first and second boundary value problems for the Laplace equation. The study constructs the Green's functions of the first and second boundary value problems for the Laplace equation in space with a spherical segment in analytical form, as well as numerical analysis of these functions. Research task: to formalize the problem of determining Green's functions for the specified domain; using methods of Fourier, pair summation equations and potential theory to reduce mixed boundary value problems for auxiliary harmonic functions to a system of equations that has an analytical solution; investigate the compatibility of the algebraic system for determining constants of integration; formulate and prove a theorem about the jump of the normal derivative of the potential of a simple layer on the surface of a segment, with the help of which to present the Green's function in the form of the potential of a simple layer; conduct a numerical experiment and identify algorithms and areas of changing the parameters of effective calculations; analyze the behavior of Green's functions. Scientific novelty: for the first time, Green's functions of Dirichlet and Neumann boundary value problems for the Laplace equation in three-dimensional space with a spherical segment were constructed in analytical form, the obtained results were substantiated, and a comprehensive numerical experiment was conducted to analyze the behavior of these functions. The obtained results: mixed boundary value problems in the interior and exterior of the spherical surface to which the segment belongs are set for the auxiliary harmonic functions; using the Fourier method, the problem is reduced to systems of paired equations in series by Legendre functions, the solutions of which are found using discontinuous Mehler-Dirichlet sums. The specified functions are obtained in an explicit view in two forms: series based on the basic harmonic functions in spherical coordinates and the potential of a simple layer on the surface of the segment. To substantiate the results, the lemma on the compatibility of the algebraic system for determining the constants of integration and the theorem on the jump of the normal derivative of the potential of a simple layer on a segment are proved. A numerical experiment was conducted to analyze the behavior of the constructed functions. Conclusions: the analysis of numerical values of Green's functions obtained by different algorithms showed that the highest accuracy of results outside the surface of the segment was obtained when using images of Green's functions in the form of series. On the basis of the calculations, the lines of the level of the Green's functions of two boundary value problems in the plane of the singular point, as well as the graphs of the potential density of the simple layer for the Dirichlet problem and the potential jump for the Neumann problem on the segment at different locations of the singular point were constructed. In the partial case of the location of a singular point at the origin of the coordinates, the potential of the electrostatic field of a point charge near a conductive grounded thin shell in the form of a spherical segment is found. The main characteristics of such a field are found in closed form.