On a Class of Solutions of Laplace's Two-Dimensional Equation on a Three-Dimensional Manifold

Gladkov, S. O.
Vladikavkaz Mathematical Journal 2024. Vol. 26. Issue 2.

Abstract:
The solution of the two-dimensional Laplace equation on a given set of three independent variables in three-dimensional Euclidean space is found. The problem is solved by converting the two-dimensional Laplace equation into an equation in which the desired function depends on three independent variables. This turns out to be possible by introducing a spherical coordinate system. The proposed method made it possible to find a solution to the two-dimensional Laplace equation in the form of a function of three independent variables. As an example of the application of the obtained solution the problem of an incompressible fluid flow around a three-dimensional body shaped an "iron'' is considered. For this problem detailed reasoning is given that makes it possible to reduce the three-dimensional Laplace equation which describes the distribution of the scalar potential of the flow velocities near the surface of the body and depends on three independent coordinates to the two-dimensional Laplace equation the solution of which was strictly analytically substantiated in the proposed work. It’s also noted that similar problems arise not only in hydrodynamics but also in the theory of elasticity and in the theory of electromagnetism. The described technique, namely, the possibility of moving from two independent variables to three ones using a given transformation enables us to find purely physical solutions for a wide range of problems from different fields of natural sciences.

Keywords: partial differential equations, two-dimensional Laplace equation, spherical coordinates, ordinary differential equations

Language: Russian Download the full text  

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