Solutions of helmholtz equation in complex domain

Authors

  • Михайло Антонович Сухорольський National University “Lviv polytechnic” Bandery Str.,12 , Lviv, Ukraine, 79013, Ukraine
  • Галина Володимирівна Івасик National University “Lviv polytechnic” Bandery Str.,12 , Lviv, Ukraine, 79013, Ukraine
  • Вероніка Володимирівна Достойна National University “Lviv polytechnic” Bandery Str.,12 , Lviv, Ukraine, 79013, Ukraine

DOI:

https://doi.org/10.15587/1729-4061.2014.27680

Keywords:

Helmholtz equation, analytical solution of Helmholtz equation, conformal mapping, boundary value problems

Abstract

A harmonic equation and the Helmholtz equation are elliptic type equations and describe important physical processes (the first – stationary, the second - stationary and dynamic). Effective solutions of boundary value problems for harmonic equation (in different regions in the plane) are constructed by the methods of the theory of analytic functions of a complex variable. These methods can not be applied directly to solving problems for the Helmholtz equation. In the scientific literature, solutions of boundary value problems for this equation are known only in certain areas that are represented by cumbersome formulas.

In the paper, using the solution of the Helmholtz equation in a circle through the functions (not analytical) of complex variable and conformal mapping of a given area on the circle, a general approach to building a solution of the corresponding boundary value problem is formulated. An important prerequisite for presenting this solution as functional series is finding the solution of harmonic equation in a given region that satisfies the given boundary conditions and an analytic function in this region respectively. The solutions of the Helmholtz equation in the plane with an elliptic hole and half-plane are constructed. For effective formulation of boundary value prob­lems and finding analytic functions in these areas, systems of basic functions in the corresponding spaces of analytic functions are found. 

Author Biographies

Михайло Антонович Сухорольський, National University “Lviv polytechnic” Bandery Str.,12 , Lviv, Ukraine, 79013

Proffessor, doctor of phisics and math sciences,

The department of higher mathematics

Галина Володимирівна Івасик, National University “Lviv polytechnic” Bandery Str.,12 , Lviv, Ukraine, 79013

Candidate of phisics and math sciences,

The department of higher mathematics

Вероніка Володимирівна Достойна, National University “Lviv polytechnic” Bandery Str.,12 , Lviv, Ukraine, 79013

PhD student

The department of higher mathematics

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Published

2014-10-21

How to Cite

Сухорольський, М. А., Івасик, Г. В., & Достойна, В. В. (2014). Solutions of helmholtz equation in complex domain. Eastern-European Journal of Enterprise Technologies, 5(4(71), 10–15. https://doi.org/10.15587/1729-4061.2014.27680

Issue

Section

Mathematics and Cybernetics - applied aspects