On solving problems of heat and mass transfer in piecewise homogeneous regions with a weakly permeable film

Boundary value problems for the equations of thermal conductivity in a band $D(x\in R,\,0<y<a)$ divided by a weakly permeable film $x=0$ into two half-bands $D_1(x<0,\,0<y<a)$ and $D_2(x>0,\,0<y<a)$ 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 $D(x\in R)=D_1(x<0)\cup\{x=0\}\cup D_2(x>0)$ (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.