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 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]: Year: Volume: Issue: Page: Find

 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2019, Volume 23, Number 1, Pages 195–203 (Mi vsgtu1665)

Short Communication

Exact analytical solution for the stationary two-dimensional heat conduction problem with a heat source

I. V. Kudinov*, O. Yu. Kurganova, V. K. Tkachev

Samara State Technical University, Samara, 443100, Russian Federation

Abstract: The exact analytic solution for the stationary two-dimensional heat conduction problem with a heat source for an infinite square bar was obtained. It was based on the Bubnov–Galyorkin orthogonal method using trigonometric systems of coordinate functions. The infinite system of ordinary differential equations obtained by the Bubnov–Galyorkin method is divided and reduced by the orthogonality property of trigonometric coordinate functions to the solution of a generalized equation which provides the exact analytical solution in a simple form, i.e. in the form of an infinite series. In view of the symmetry of the problem, only a quarter of the cross-section of the bar is considered for the boundary conditions of the adiabatic wall (the absence of heat transfer) along the cut lines, which allows (in contrast to the well-known classical exact analytical solution) to significantly simplify the process of the solution and the final equation.

Keywords: Poisson equation, two-dimensional boundary value problem, heat source, Bubnov–Galyorkin method, orthogonal system of coordinate functions, exact analytic solution

 Funding Agency Grant Number Russian Foundation for Basic Research 18-38-00029 ìîë_à This work was supported by the Russian Foundation for Basic Research (project no. 18–38–00029 mol_a).

* Author to whom correspondence should be addressed

DOI: https://doi.org/10.14498/vsgtu1665

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Bibliographic databases:

UDC: 517.958:[536.2+539.219.3]
MSC: 35K05, 80A20, 35C10
Revised: January 19, 2019
Accepted: January 27, 2019
First online: February 13, 2019

Citation: I. V. Kudinov, O. Yu. Kurganova, V. K. Tkachev, “Exact analytical solution for the stationary two-dimensional heat conduction problem with a heat source”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 23:1 (2019), 195–203

Citation in format AMSBIB
\Bibitem{KudKurTka19} \by I.~V.~Kudinov, O.~Yu.~Kurganova, V.~K.~Tkachev \paper Exact analytical solution for the stationary two-dimensional heat conduction problem with a heat source \jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.] \yr 2019 \vol 23 \issue 1 \pages 195--203 \mathnet{http://mi.mathnet.ru/vsgtu1665} \crossref{https://doi.org/10.14498/vsgtu1665} \zmath{https://zbmath.org/?q=an:07097289} \elib{https://elibrary.ru/item.asp?id=37248568} 

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