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A boundary value problem for Bitsadze equation in the unit disc
Abstract
A boundary value problem for the Bitsadze equation
in the interior of the unit disc is considered. It is proved that the problem is Noetherian and its index is calculated,
and solvability conditions for the non-homogeneous problem are proposed. Some solutions of the homogeneous problem are explicitely
found.
... Vaitekhovich [12] discusses Dirichlet and Schwarz problems for the inhomogeneous Bitsadze equation for a circular ring domain. In the interior of unit disc a boundary value problem for the Bitsadze equation is considered by Babayan [1] and is proved to be Noetherian. In his paper Babayan also proposes solvability conditions for the inhomogeneous Bitsadze equation. ...
This article shows the uniqueness of a solution to a Bitsadze system of equations, with a boundary-value problem that has four additional single point conditions. It also shows how to construct the solution.
... Where f is a complex valued function on B. [10] The Bitsadze equation arises in many areas including structural mechanics, electrostatics, magneto statics, power electromagnetic, conductive media, heat transfer and diffusion ( [11][12][13]). In [1,2], Begehr introduced some boundary value problems for Bitsadze equation under some solvability conditions and presented their solutions as results for applying the Dirichlet and Neumann boundary value problems all those solutions have singular integrals in their context. ...
An algorithm is provided for the fast and accurate computation of the solution of the Bitsadze equation in the complex plane in the interior of the unit disk. The algorithm is based on the representation of the solution in terms of a double integral as it shown by Begehr [1,2], some recursive relations in Fourier space, and Fast Fourier Transforms. The numerical evaluation of integrals at 2 N points on a polar coordinate grid by straightforward summation for the double integral would require 2 O N floating point operation per point. Evaluation of such integrals has been optimized in this paper giving an asymptotic operation count of In O N per point on the average. In actual implementation, the algorithm has even better computational complexity, approximately of the order of 1 O per point. The algorithm has the added advantage of working in place, meaning that no additional memory storage is required beyond that of the initial data. This paper is a result of application of many of the original ideas described in Daripa [3].
Рассматривается задача продолжения n-аналитической функции в область по значениям ее последовательных производных до (n-1) – го порядка на части границы. Построена формула продолжения Карлемана для n – аналитических функций.
We investigate the problem of continuation of an
n-analytic function into a domain by the values of its
successive derivatives up to the th order given on a part
of the boundary. For such functions, we also consider the
problem of finding the conditions under which the Cauchy type
integral becomes the Cauchy integral.
Further mixed boundary value problems are studied for the inhomogeneous polyanalytic equation in the unit disc. As in part I, see [4], the boundary conditions are some combinations of Schwarz, Dirichlet and Neumann conditions. The method applied is based on an iteration process leading from lower order equations to higher ones, see [1-3]. The method can be used to solve all kind of such mixed problems also for other kinds of complex model equations.
The three basic boundary value problems in complex analysis are of Schwarz, of Dirichlet and of Neumann type. When higher order equations are investigated all kind of combinations of these boundary conditions are proper to determine solutions. However, not all of these conditions are leading to well-posed problems. Some are overdetermined so that solvability conditions have to be found. Some of these boundary value problems are treated here for the inhomogeneous polyanalytic equation.
Explicit representations for the solutions of the mixed boundary value problems arising from p-Schwarz, q-Dirichlet and r-Neumann for the inhomogeneous polyanalytic equation have been obtained.
The mixed boundary value problems arising from m-Schwarz, n-Neumann; n-Neumann, m-Dirichlet; m-Dirichlet, n-Schwarz boundary conditions for the inhomogeneous polyanalytic equation have been studied.