Geodesic
The problem of the distribution of heat in the material with a cut on the square
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 3 (2015), pp. 41-54
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The problem of the stationary distribution of the temperature field with a variable coefficient of thermal conductivity in the inner region of a three-dimensional space with a cut on the square, which simulates a heterogeneous material with a crack in the form of a flat square is considered: \begin{gather*} \Delta u(x_1, x_2, x_3)+k\dfrac{\partial u(x_1, x_2, x_3)}{\partial x_3} =0,~\,\,\,x\in {\mathbb R}^3 \backslash\Pi;\\ u(x_1, x_2, +0)-u(x_1, x_2, -0)=q_0(x_1, x_2),~\,\,\,x_1\in [-1;\,\,1],\,~ x_2\in [-1;\,\,1];\\ \dfrac{\partial u(x_1, x_2, +0)}{\partial x_3}+\dfrac k2 u(x_1, x_2, +0)-\dfrac{\partial u(x_1, x_2, -0)}{\partial x_3}-\dfrac k2 u(x_1, x_2, -0)=q_1(x_1, x_2), \end{gather*} where $u(x_1, x_2, x_3)$ is the temperature at the point with coordinates $(x_1, x_2, x_3)$. The article describes a solution of the problem, studies its properties. The main result of this study is to construct asymptotic representations of the temperature field and the heat flux near the boundary. From the formulas for the first derivatives of the solution, we can conclude that these functions at the boundaries of the crack-square are singular terms of higher order than the inside of the cut. Refs 8.
Keywords: thermal potentials, the non-homogeneous material with a square cut, the asymptotic solution. 
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     title = {The problem of the distribution of heat in the material with a cut on the square},
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A. V. Glusgko; E. A. Loginova. The problem of the distribution of heat in the material with a cut on the square. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 3 (2015), pp. 41-54. http://geodesic.mathdoc.fr/item/VSPUI_2015_3_a3/

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