GREEN'S FUNCTION OF THE LAPLACE OPERATOR FOR A BALL
Keywords:
Green's function, Laplace operator, mathematical physics, potential theory, potential distribution, spherical geometry, Laplace equation, Poisson equation, Dirichlet problem, method of separation of variables, Fourier transform, convolution, boundary conditions, asymptotics, numerical methods.Abstract
This material examines the key concepts of mathematical physics - the Green's function and the Laplace operator, and their application for solving problems related to the distribution of potentials and fields in various physical systems. Particular attention is paid to the study of the properties of the Green's function of the Laplace operator for a sphere and its use in solving practical problems.
The main aspects that will be considered in the work:
- Formulation of the mathematical problem for the Laplace operator in spherical coordinates.
- Finding the analytical expression for the Green's function of the Laplace operator for a sphere.
- Investigation of the basic properties of the Green's function, such as symmetry, boundary conditions, and asymptotics.
- Application of the Green's function to solve various problems of mathematical physics, including the distribution of potentials and the solution of Poisson and Dirichlet equations inside and outside a sphere.
- Analysis of numerical methods for calculating the Green's function and their application in practical problems.
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