SOLUTION OF THE LAPLACE EQUATION BY THE METHOD OF SEPARATION OF VARIABLES FOR A LENGTH HOLLOW CYLINDER

DOI: 10.47026/1810-1909-2023-2-32-40

УДК: [517.956.225:517.982.43]:621.313-465

ББК: В161.68:К500.131

Aleksandr A. AFANASYEV, Nadezhda N. IVANOVA

Key words

three-dimensional analytical model, electric machine, hollow cylinder of finite length, Laplace equation, Fourier variable separation method, eigenfunctions and eigenvalues of the Sturm–Liouville problem

Abstract

The physical model of many electrical objects is a hollow cylinder of finite length. As a basis for constructing analytical models of electrical machines, linear equations of mathematical physics are used. They are the solution of the Laplace three-dimensional partial differential equation, which is widely used in analytical calculations.

The purpose of the study is to solve the Laplace three-dimensional differential equation for a hollow cylinder of finite length, which can be adapted for the electromagnetic calculation of electromechanical devices with cylindrical active parts.

Materials and methods. To solve the Laplace equation in a cylindrical coordinate system, the Fourier variable separation method was used. To obtain a non-trivial solution of the equation, eigenfunctions of the Sturm–Liouville problem were used. Dirichlet, Neumann boundary value problems of the mixed type for hollow cylinders are adapted to the electromagnetic calculation of electromechanical devices having cylindrical active parts.

The results of the study. The Laplace equation given in a cylindrical coordinate system is considered, on the basis of which the Sturm–Liouville equation with zero initial values is compiled to find eigenfunctions. The complete solution of the Lapalace equation with given boundary conditions is obtained as the sum of the solutions of two separate Dirichlet problems with different boundary conditions.

Findings. The obtained analytical expression can be used as a mathematical basis for constructing three-dimensional analytical models of electrical machines with cylindrical active parts and carrying out electromagnetic calculations of the corresponding electromechanical devices.

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Information about the authors

Aleksandr A. Afanasyev – Doctor of Technical Sciences, Professor, Department of Automation and Control in Technical Systems, Chuvash State University, Russia, Cheboksary (afan39@mail.ru).

Nadezhda N. Ivanova – Candidate of Technical Sciences, Associate Professor, Department of Mathematical and Hardware Support of Information Systems, Chuvash State University, Russia, Cheboksary (niva_mail@mail.ru; ORCID: https://orcid.org/0000-0001-7130-8588).

For citations

Afanasyev A.A., Ivanova N.N. SOLUTION OF THE LAPLACE EQUATION BY THE METHOD OF SEPARATION OF VARIABLES FOR A LENGTH HOLLOW CYLINDER. Vestnik Chuvashskogo universiteta, 2023, no. 2, pp. 32–40. DOI: 10.47026/1810-1909-2023-2-32-40 (in Russian).

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