SOLUTION OF THE DIRICHLET PROBLEM ON A SPHERE FOR THE LAPLACE EQUATION
Keywords:
Laplace's equation, mathematical physics, Dirichlet problem, spherical geometry, electrostatics, gravity, heat transfer, temperature distribution, distribution of electric charges, gravitational potentials, modeling of physical phenomena, practical application, solution methods, properties of solutions.Abstract
This work considers the formulation and solution of the Dirichlet problem on a sphere. The domain of the problem is a sphere, and the boundary conditions are given on its surface. The solution is presented in spherical coordinates using the method of separation of variables. A general solution is obtained in the form of a series of spherical functions, and the coefficients of the series are determined from the boundary conditions.
The properties of the solution are proven: smoothness inside and on the surface of the sphere, uniqueness of the solution of the Dirichlet problem on the sphere. The physical interpretation of the obtained solution is given, which can describe various physical processes, such as the distribution of potential or temperature.
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