Geodesic
On solving problems of heat and mass transfer in piecewise homogeneous regions with a weakly permeable film
Čelâbinskij fiziko-matematičeskij žurnal, Tome 6 (2021) no. 3, pp. 312-320.

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Boundary value problems for the equations of thermal conductivity in a band $D(x\in R,\,0$ divided by a weakly permeable film $x=0$ into two half-bands $D_1(x0,\,0$ and $D_2(x>0,\,0$ with different permeabilities $k_i$ in $D_i$, $i=1,2$, under different types of boundary conditions are considered. A weakly permeable film is modeled as an infinitely thin layer with an infinitesimal permeability. Generalized conjugation conditions on the film are derived for the potentials $u_i(x,y,t)$, $i=1,2$. Problems with a weakly permeable film $x=0$ are considered for steady-state processes in a piecewise homogeneous band $D$ (at $k_1\neq k_2$), for unsteady processes in a homogeneous band $D$ (at $k_1=k_2$), and for unsteady processes in a piecewise homogeneous rod $D(x\in R)=D_1(x0)\cup\{x=0\}\cup D_2(x>0)$ (at $k_1\neq k_2$ for one-dimensional thermal conductivity equations). General formulas are derived that express the solutions of the considered problems through the solutions of similar classical problems in the corresponding homogeneous domain $D$ (without film) in the form of rapidly converging nonconforming integrals. The existence and uniqueness theorem is proved for the considered class of problems.
Keywords: boundary value problems for the heat equation, weakly permeable film. 
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     title = {On solving problems of heat and mass transfer in piecewise homogeneous regions with a weakly permeable film},
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S. E. Kholodovskii. On solving problems of heat and mass transfer in piecewise homogeneous regions with a weakly permeable film. Čelâbinskij fiziko-matematičeskij žurnal, Tome 6 (2021) no. 3, pp. 312-320. http://geodesic.mathdoc.fr/item/CHFMJ_2021_6_3_a4/

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