Dirichlet problem for the Laplace equation in a piecewise homogeneous multidimensional layer with fourth-kind conjugation conditions
https://doi.org/10.18384/2310-7251-2022-3-59-87-99
Abstract
Aim. The purpose is to find exact solutions of the Dirichlet problem for the Laplace equation in a piecewise homogeneous multidimensional layer with fourth-kind conjugation conditions.
Methodology. We consider the Dirichlet problem in a piecewise homogeneous layer in a space of arbitrary dimension. Dirichlet conditions are set on the outer boundary hyperplanes, and conjugation conditions of the fourth kind are set on the inner hyperplane dividing the layer into two layers of equal thickness. The functions defined on the boundary are assumed to be generalized functions of slow growth; in particular, they can be polynomials.
Results. Exact solutions of the Dirichlet problem for the Laplace equation in a piecewise homogeneous multidimensional layer with conjugation conditions of the fourth kind are obtained, which are written as convolutions of rapidly decreasing, infinitely differentiable functions (kernels) with boundary functions, which are considered to be generalized functions of slow growth. If the boundary functions are ordinary functions of slow growth, then the solutions are written by integral formulae. In particular, if the boundary functions are polynomials, then the solutions are also polynomials.
Research implications. We have obtained exact solutions of the Dirichlet problem for the Laplace equation in a piecewise homogeneous multidimensional layer with conjugation conditions of the fourth kind.
About the Authors
O. D. Algazin
Bauman Moscow State Technical University
Russian Federation
Russian Federation
Oleg D. Algazin – Cand. Sci. (Phys.-Math.), Assoc. Prof., Department of Computational Mathematics and Mathematical Physics
ul. 2-ya Baumanskaya 5, stroenie 1, Moscow 105005
A. V. Kopaev
Bauman Moscow State Technical University
Russian Federation
Russian Federation
Anatoliy V. Kopaev – Cand. Sci. (Phys.-Math.), Assoc. Prof., Department of Higher Mathematics
ul. 2-ya Baumanskaya 5, stroenie 1, Moscow 105005
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