Dirichlet Problem Solution for Poisson Equation in a Multidimensional Infinite Layer

Polynomial Solutions of the Boundary Value Problems for the Poisson Equation in a Layer

“…It was shown in [8], [9] that if φ(x), ψ(x), φ 1 (x), ψ 1 (x) are generalized functions of slow growth (in particular, polynomials), then the solutions of these problems can be written in the form of the convolution of boundary functions with kernels that are fundamental solutions of these boundary value problems. The solution of the Dirichlet problem (15), (16) is written in the form…”

“…This formula is also valid for n = 1. For n = 1, the integral (27) is calculated in an explicit form [9] P 1 (x, y) = P 1 (|x|, y) = 1 2a sin(πy/a) cosh(πx/a) + cos(πy/a) ,…”

“…In the case of arbitrary dimension n, the Poisson kernel in the form (25) was considered in [13]. In [9] a recurrence formula with respect to dimension is received…”

“…for some m ≥ 0 and for each y ∈ (0, a), then the solution is unique. The solutions of boundary value problems for the Laplace equation are written in the form of convolutions of boundary functions with the corresponding kernels, which are fundamental solutions of the corresponding boundary value problems [8], [9]. It is shown that these solutions are polynomials and algorithm for obtaining them is given.…”

Exact solution to the Dirichlet problem for degenerating on the boundary elliptic equation of Tricomi-Keldysh type in the half-space

“…Earlier, using the Fourier transform method we solved in our joint works the Dirichlet and Dirichlet-Neumann problem for the Laplace and Poisson equations in a multidimensional infinite layer [11], [12].…”