Solution to the mixed boundary-value problem for Laplace equation in multidimentional infinite layer

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MATHEMATICS
SOLUTION TO THE MIXED BOUNDARY-VALUE PROBLEM
FOR LAPLACE EQUATION IN MULTIDIMENTIONAL INFINITE LAYER
O.D. Algazin, A.V. Kopaev
Bauman Moscow State Technical University, Moscow, Russian Federation
e-mail: ;
The paper considers the solution to the mixed boundary-value problem of finding
a harmonic function of n variables for the domain confined by two parallel hyper-
planes. This function was determined by its values on a hyper-plane and its normal
derivative on another hyper-plane. The obtained solution is presented as a sum of
two integrals which kernels are expressed only in terms of elementary functions in
the case of the even-dimension space. In contrast to the odd-dimension space they
are also expressed through the Bessel functions. If the given boundary values are
tempered distributions, then the solution is written as a convolution of the kernels with
these functions. The opportunity of practical application of the obtained formulas is
illustrated by the example of forming up a filtration flow under the spot dam with a
aquiclude.
Keywords: harmonic functions, Fourier transform, tempered distributions, filtration
theory.
Harmonic functions of two and three variables describe many stationary
processes of underground hydrodynamics, thermal conductivity, etc.
Therefore, the search for solutions to various boundary value problems
for the Laplace equation (and new simpler forms of solutions) is highly
relevant. In case of simply connected planar domains (e.g., bandwidth)
solving these problems by means of conformal transformations is reduced
to solving them for canonical domains — a circle and a half-plane. If there
are more than two variables, it is not possible. A unique solution has to be
seached for each domain. In case of a half-space the primary method of
solving the boundary value problems for linear partial differential equations
with constant coefficients is Fourier transformation for the variables in the
boundary hyperplane [1]. Poisson and Neumann kernels of integrals which
solve both the first (Dirichlet problem) and second (Neumann problem)
boundary value problems for the Laplace equation in a half-space are
well known [2, 3]. In case of the bandwidth and infinite layer in three-
dimensional space the kernels of the integral representing various solutions
of boundary value problems for the Laplace equation can be obtained by
the method of reflections in the form of sums of infinite series [4, 5]. For
an infinite layer in n-dimensional space the first boundary value problem
was solved by one of the authors of [6]. Wherein it was managed to sum
up the above mentioned series, expressing their sum in terms of elementary
functions. In this paper the mixed boundary value Dirichlet — Neumann
ISSN 1812-3368. Herald of the BMSTU. Series “Natural Sciences”. 2015. No. 1 3