A Mixed Boundary Value Problem for the Laplace Equation in a semi-infinite Layer

Abstract

The paper offers a solution of the mixed Dirichlet-Neumann and Dirichlet-Neumann-Robin boundary value problems for the Laplace equation in the semi-infinite layer, using the previously obtained solution of the mixed Dirichlet-Neumann boundary value problem for a layer.The functions on the right-hand sides of the boundary conditions are considered to be functions of slow growth, in particular, polynomials. The solution to boundary value problems is also sought in the class of functions of slow growth. Continuing the functions on the right-hand sides of the boundary conditions on the upper and lower sides of the semi-infinite layer from the semi-hyperplane to the entire hyperplane, we obtain the Dirichlet-Neumann problem for the layer, the solution of which is known and written in the form of a convolution. If the right-hand sides of the boundary conditions are polynomials, then the solution is also a polynomial. To the solution obtained it is necessary to add the solution of the problem for a semi-infinite layer with homogeneous boundary conditions on the upper and lower sides and with an inhomogeneous boundary condition of Dirichlet, Neumann or Robin on the lateral side. This solution is written as a series. If we take a finite segment of the series, then we obtain a solution that exactly satisfies the Laplace equation and the boundary conditions on the upper and lower sides of the semi-infinite layer and approximately satisfies the boundary condition on the lateral side.An example of solving the Dirichlet-Neumann and Dirichlet-Neumann-Robin problems is considered, describing the temperature field of a semi-infinite plate the upper side of which is heat-isolated, on the lower side the temperature is set in the form of a polynomial, and the lateral side is either heat-isolated, or holds a zero temperature, or has heat exchange with a zero-temperature environment. For the first two Dirichlet-Neumann problems, the solution is obtained in the form of polynomials. For the third Dirichlet-Neumann-Robin problem, the solution is obtained as a sum of a polynomial and a series. If in this solution the series is replaced by a finite segment, then an approximate solution of the problem will be obtained, which approximately satisfies the Robin condition on the lateral side of the semi-infinite layer.

Keywords

• laplace equation
• harmonic functions
• mixed boundary value problem
• functions of slow growth