ON THE SOLUTION OF A SYSTEM OF DIFFERENTIAL EQUATIONS OCCURRING IN THE THEORY OF HEAT AND MASS TRANSFER

L. A. MARNEVSKAYA

In the present paper we solve a problem in the theory of heat and mass transfer for a two-layer unbounded plate, which, in its mathematical formulation, reduces to finding the solution of the system of equations

\[ \frac{\partial u_1}{\partial t} = a_{11}^{1}\frac{\partial^{2}u_1}{\partial x^{2}} + a_{12}^{1}\frac{\partial^{2}v_1}{\partial x^{2}} + f_1^{1}(x,t), \]

\[ \frac{\partial v_1}{\partial t} = a_{21}^{1}\frac{\partial^{2}u_1}{\partial x^{2}} + a_{22}^{1}\frac{\partial^{2}v_1}{\partial x^{2}} + f_2^{1}(x,t), \]

\[ -\infty < x \le h, \tag{1} \]

\[ \frac{\partial u_2}{\partial t} = a_{11}^{2}\frac{\partial^{2}u_2}{\partial x^{2}} + a_{12}^{2}\frac{\partial^{2}v_2}{\partial x^{2}} + f_1^{2}(x,t), \]

\[ \frac{\partial v_2}{\partial t} = a_{21}^{2}\frac{\partial^{2}u_2}{\partial x^{2}} + a_{22}^{2}\frac{\partial^{2}v_2}{\partial x^{2}} + f_2^{2}(x,t), \]

\[ h \le x < \infty, \tag{2} \]

\[ (a_{ij}^{k}\text{ are constants; } f_j^{k}(x,t)\text{ are given functions})\quad (i,j,k=1,2), \]
satisfying the initial conditions

\[ u_i(x,0)=F_1^{i}(x);\qquad v_i(x,0)=F_2^{i}(x)\quad (i=1,2) \tag{3} \]

and the conjugation conditions at \(x=h\):

\[ u_1-u_2\big|_{x=h}=\varphi_1^{1}(t);\qquad v_1-v_2\big|_{x=h}=\varphi_2^{1}(t); \]

\[ d_{11}^{1}\frac{\partial u_1}{\partial x} + d_{12}^{1}\frac{\partial v_1}{\partial x} - \left( d_{11}^{2}\frac{\partial u_2}{\partial x} + d_{12}^{2}\frac{\partial v_2}{\partial x} \right)\Bigg|_{x=h} = \varphi_1^{2}(t); \]

\[ d_{21}^{1}\frac{\partial u_1}{\partial x} + d_{22}^{1}\frac{\partial v_1}{\partial x} - \left( d_{21}^{2}\frac{\partial u_2}{\partial x} + d_{22}^{2}\frac{\partial v_2}{\partial x} \right)\Bigg|_{x=h} = \varphi_2^{2}(t) \tag{4} \]

\[ (d_{ij}^{k}\text{ are constants, } \varphi_j^{k}(t)\text{ are given functions})\quad (i,j,k=1,2), \]
under the assumption that for \(x=\pm\infty\) the required solution is bounded.

A similar problem for the case of a finite two-layer plate is formulated in [1]. However, it is not solved there. The author confines himself only to stating certain considerations concerning the method of its solution.

There are a number of works in which problems are solved for systems of equations of heat and mass transfer in a homogeneous medium, for example [3—7]. However, we are not aware of any works, with the exception of [2], in which a problem similar to problem (1)—(4) formulated above is solved. Considered

the problem considered by us in [2] corresponds to the case of homogeneous conjugation conditions, as a result of which it is only a special case of the more general problem (1)—(4), which, like the problem in [2], is solved by the method proposed in [1].

1. For convenience in what follows, we pass to the matrix notation of problem (1)—(4), similarly to how this is done in [1, 2]. In this notation, problem (1)—(4) reduces to finding the solution of the system

\[ \frac{\partial U_1}{\partial t} = A_1 \frac{\partial^2 U_1}{\partial x^2} + f_1(x,t) \quad (-\infty < x \leq h), \tag{5} \]

\[ \frac{\partial U_2}{\partial t} = A_2 \frac{\partial^2 U_2}{\partial x^2} + f_2(x,t) \quad (h \leq x < \infty), \]

satisfying the initial conditions

\[ U_i(x,0)=F_i(x), \tag{6} \]

the conjugation conditions on the boundary of the two media

\[ U_1-U_2\big|_{x=h}=\varphi_1(t); \tag{7} \]

\[ D_1\frac{\partial U_1}{\partial x} - D_2\frac{\partial U_2}{\partial x}\bigg|_{x=h} = \varphi_2(t) \]

and the conditions at infinity

\[ U_1\big|_{x\to-\infty}<\infty; \qquad U_2\big|_{x\to\infty}<\infty, \tag{8} \]

where

\[ U_i= \begin{pmatrix} u_i\\ v_i \end{pmatrix}; \qquad f_i= \begin{pmatrix} f_1^i\\ f_2^i \end{pmatrix}; \qquad A_i= \begin{pmatrix} a_{11}^i & a_{12}^i\\ a_{21}^i & a_{22}^i \end{pmatrix}; \tag{9} \]

\[ D_i= \begin{pmatrix} d_{11}^i & d_{12}^i\\ d_{21}^i & d_{22}^i \end{pmatrix}; \qquad \varphi_i= \begin{pmatrix} \varphi_1^i\\ \varphi_2^i \end{pmatrix}; \qquad F_i= \begin{pmatrix} F_1^i\\ F_2^i \end{pmatrix}; \]

\[ (i=1,2). \]

Problem (5)—(8) is solved by us by the method of operational calculus, based on the application of the Laplace transform with respect to the variable \(t\), and the use of the so-called cellular Green matrix [1].

Passing in (5)—(8) to the Laplace transforms, we obtain the system of ordinary differential equations

\[ A_1\frac{d^2\overline{U}_1}{dx^2} - p\overline{U}_1 = -\overline{\Phi}_1(x,p) \quad (-\infty < x \leq h), \]

\[ A_2\frac{d^2\overline{U}_2}{dx^2} - p\overline{U}_2 = -\overline{\Phi}_2(x,p) \quad (h \leq x < \infty), \tag{10} \]

whose solution must satisfy the conjugation conditions:

\[ \overline{U}_1-\overline{U}_2\big|_{x=h} = \overline{\varphi}_1(p); \]

\[ D_1\frac{d\overline{U}_1}{dx} - D_2\frac{d\overline{U}_2}{dx}\bigg|_{x=h} = \overline{\varphi}_2(p) \tag{11} \]

and the conditions at infinity

\[ \overline{U}_1(x,p)\big|_{x\to-\infty}<\infty; \qquad \overline{U}_2(x,p)\big|_{x\to\infty}<\infty. \tag{12} \]

where

\[ \overline{\Phi}_i(x,p)=F_i(x)+\overline{f}_i(x,p)\qquad (i=1,2) \tag{13} \]

and \(\overline{U}_i(x,p)\), \(\overline{f}_i(x,p)\), \(\overline{\varphi}_i(p)\) are the transforms of the functions \(U_i(x,t)\), \(f_i(x,t)\), \(\varphi_i(t)\) \((i=1,2)\).

2. To find the explicit form of the functions \(\overline{U}_i(x,p)\) \((i=1,2)\), we first find the solution \(\overline{W}_i(x,p)\) \((i=1,2)\) of system (10) under the homogeneous conjugation conditions

\[ \overline{W}_1-\overline{W}_2\big|_{x=h}=0, \]

\[ D_1\frac{d\overline{W}_1}{dx} - D_2\frac{d\overline{W}_2}{dx}\bigg|_{x=h}=0 \tag{14} \]

and conditions (12), representable in the form

\[ \overline{W}_1(x,p)= \int_{-\infty}^{h}\overline{G}_{11}(x,\xi,p)\overline{\Phi}_1(\xi,p)\,d\xi + \int_{h}^{\infty}\overline{G}_{12}(x,\xi,p)\overline{\Phi}_2(\xi,p)\,d\xi, \tag{15} \]

\[ \overline{W}_2(x,p)= \int_{-\infty}^{h}\overline{G}_{21}(x,\xi,p)\overline{\Phi}_1(\xi,p)\,d\xi + \int_{h}^{\infty}\overline{G}_{22}(x,\xi,p)\overline{\Phi}_2(\xi,p)\,d\xi, \]

where \(\overline{G}_{11}\), \(\overline{G}_{12}\), \(\overline{G}_{21}\), \(\overline{G}_{22}\) are the so-called cellular Green’s matrices, whose definition and physical meaning are given, for example, in [1].

Domain of definition of the cellular Green’s matrix

Here

\[ \overline{G}_{11}(x,\xi,p)= \begin{cases} \overline{G}_{11}^{*}(x,\xi,p), & -\infty<\xi<x,\\ \overline{G}_{11}^{**}(x,\xi,p), & x<\xi<h; \end{cases} \]

\[ \overline{G}_{22}(x,\xi,p)= \begin{cases} \overline{G}_{22}^{*}(x,\xi,p), & h<\xi<x,\\ \overline{G}_{22}^{**}(x,\xi,p), & x<\xi<\infty; \end{cases} \]

\[ \overline{G}_{ij}= \begin{pmatrix} \overline{G}_{11}^{ij} & \overline{G}_{12}^{ij}\\ \overline{G}_{21}^{ij} & \overline{G}_{22}^{ij} \end{pmatrix} \qquad (i,j=1,2). \]

The elements \(\overline{G}_{kl}^{ij}\) of the matrix \(\overline{G}_{ij}\) \((i,j,k,l=1,2)\) are functions to be defined below. Having found them and substituting into (15), we obtain the solution of system (10) under conditions (14) and (12). The domains of definition of the matrices \(\overline{G}_{ij}\) \((i,j=1,2)\) are the sectors indicated in the figure.

The subdivision of the Green’s matrix on the interval \((-\infty,h)\) for \(\overline{U}_1(x,p)\) and on the interval \((h,+\infty)\) for \(\overline{U}_2(x,p)\) is connected with the possibility of passing to the originals in \(t\) in the elements of this matrix (see [1], pp. 15–20).

Substituting (15) into (10), (14), and (12), we obtain, for determining \(\overline{G}_{ij}(x,\xi,p)\) \((i,j=1,2)\), the differential equations

\[ A_1\frac{\partial^2\overline{G}_{11}^{*}}{\partial x^2} - p\overline{G}_{11}^{*}=0; \qquad A_2\frac{\partial^2\overline{G}_{21}}{\partial x^2} - p\overline{G}_{21}=0; \]

\[ A_1 \frac{\partial^2 \bar{G}_{11}^{**}}{\partial x^2} - p\bar{G}_{11}^{**}=0;\qquad A_2 \frac{\partial^2 \bar{G}_{22}^{*}}{\partial x^2} - p\bar{G}_{22}^{*}=0; \tag{16} \]

\[ A_1 \frac{\partial^2 \bar{G}_{12}}{\partial x^2} - p\bar{G}_{12}=0;\qquad A_2 \frac{\partial^2 \bar{G}_{22}^{**}}{\partial x^2} - p\bar{G}_{22}^{**}=0, \]

the solutions of which must satisfy the conditions

\[ \bar{G}_{11}^{*}-\bar{G}_{11}^{**}\big|_{\xi=x}=0;\qquad \bar{G}_{22}^{*}-\bar{G}_{22}^{**}\big|_{\xi=x}=0; \]

\[ \frac{\partial G_{11}^{*}}{\partial x} - \frac{\partial G_{11}^{**}}{\partial x}\bigg|_{\xi=x} =-A_1^{-1};\qquad \frac{\partial \bar{G}_{22}^{*}}{\partial x} - \frac{\partial \bar{G}_{22}^{**}}{\partial x}\bigg|_{\xi=x} =-A_2^{-1}; \]

\[ \bar{G}_{11}^{*}-\bar{G}_{21}\big|_{x=h}=0;\qquad \bar{G}_{12}-\bar{G}_{22}^{**}\big|_{x=h}=0; \tag{17} \]

\[ D_1\frac{\partial \bar{G}_{11}^{*}}{\partial x} - D_2\frac{\partial \bar{G}_{21}}{\partial x}\bigg|_{x=h}=0;\qquad D_1\frac{\partial \bar{G}_{12}}{\partial x} - D_2\frac{\partial \bar{G}_{22}^{**}}{\partial x}\bigg|_{x=h}=0; \]

\[ \bar{G}_{11}^{**}\big|_{x\to-\infty}=0;\qquad \bar{G}_{12}\big|_{x\to\infty}=0; \]

\[ \bar{G}_{21}\big|_{x\to-\infty}=0;\qquad \bar{G}_{22}^{*}\big|_{x\to\infty}=0. \]

Writing now (16)—(17) in expanded form, we obtain 4 groups of systems of differential equations and conditions for determining the functions \(\bar{G}_{kl}^{i}\), from which, as in [2], we find that

\[ \bar{G}_{11}^{*}(x,\xi,p) = \frac{1}{\sqrt{p}} \sum_{i=1}^{2} \left\{ \left( \begin{array}{cc} A_i & A_i^{*}\\ \beta_j^{1}A_i & \beta_j^{1}A_i^{*} \end{array} \right) e^{-\sqrt{\frac{p}{\lambda_i^{1}}}(x-\xi)} + \right. \]

\[ \left. + \left( \begin{array}{cc} A_{i1}' & A_{i1}^{*}\\ \beta_j^{1}A_{i1}' & \beta_j^{1}A_{i1}^{*} \end{array} \right) e^{-\sqrt{\frac{p}{\lambda_i^{1}}}(2h-x-\xi)} + \right. \]

\[ \left. + \left( \begin{array}{cc} A_{i2}' & A_{i2}^{*}\\ \beta_j^{1}A_{i2}' & \beta_j^{1}A_{i2}^{*} \end{array} \right) e^{-\sqrt{\frac{p}{\lambda_j^{1}}}(h-\xi)-\sqrt{\frac{p}{\lambda_i^{1}}}(h-x)} \right\}; \]

\[ \bar{G}_{11}^{**}(x,\xi,p) = \frac{1}{\sqrt{p}} \sum_{i=1}^{2} \left\{ \left( \begin{array}{cc} A_i & A_i^{*}\\ \beta_j^{1}A_i & \beta_j^{1}A_i^{*} \end{array} \right) e^{-\sqrt{\frac{p}{\lambda_i^{1}}}(\xi-x)} + \right. \]

\[ \left. + \left( \begin{array}{cc} A_{i1}' & A_{i1}^{*}\\ \beta_j^{1}A_{i1}' & \beta_j^{1}A_{i1}^{*} \end{array} \right) e^{-\sqrt{\frac{p}{\lambda_i^{1}}}(2h-x-\xi)} + \right. \]

\[ \left. + \left( \begin{array}{cc} A_{i2}' & A_{i2}^{*}\\ \beta_j^{1}A_{i2}' & \beta_j^{1}A_{i2}^{*} \end{array} \right) e^{-\sqrt{\frac{p}{\lambda_j^{1}}}(h-\xi)-\sqrt{\frac{p}{\lambda_i^{1}}}(h-x)} \right\}; \]

\[ \bar{G}_{12}(x,\xi,p) = \frac{1}{\sqrt{p}} \sum_{i=1}^{2} \left\{ \left( \begin{array}{cc} M_{i1}' & M_{i1}^{*}\\ \beta_j^{1}M_{i1}' & \beta_j^{1}M_{i1}^{*} \end{array} \right) e^{-\sqrt{\frac{p}{\lambda_i^{2}}}(\xi-h)-\sqrt{\frac{p}{\lambda_i^{1}}}(h-x)} + \right. \]

\[ +\left( \begin{array}{cc} M_{i2}' & M_{i2}^{*}\\ \beta_j^1 M_{i2}' & \beta_j^1 M_{i2}^{*} \end{array} \right) e^{-\sqrt{\frac{p}{\lambda_j^2}}(\xi-h)-\sqrt{\frac{p}{\lambda_i^1}}(h-x)} \Biggr\}; \tag{18} \]

\[ \bar G_{21}(x,\xi,p)=\frac{1}{\sqrt p}\sum_{i=1}^{2} \Biggl\{ \left( \begin{array}{cc} B_{i1}' & B_{i1}^{*}\\ \beta_j^2 B_{i1}' & \beta_j^2 B_{i1}^{*} \end{array} \right) e^{-\sqrt{\frac{p}{\lambda_i^2}}(x-h)-\sqrt{\frac{p}{\lambda_i^1}}(h-\xi)} + \]
\[ +\left( \begin{array}{cc} B_{i2}' & B_{i2}^{*}\\ \beta_j^2 B_{i2}' & \beta_j^2 B_{i2}^{*} \end{array} \right) e^{-\sqrt{\frac{p}{\lambda_i^2}}(x-h)-\sqrt{\frac{p}{\lambda_j^1}}(h-\xi)} \Biggr\}; \]

\[ \bar G_{22}^{*}(x,\xi,p)=\frac{1}{\sqrt p}\sum_{i=1}^{2} \Biggl\{ \left( \begin{array}{cc} A_i' & A_i^{*}\\ \beta_j^2 A_i' & \beta_j^2 A_i^{*} \end{array} \right) e^{-\sqrt{\frac{p}{\lambda_i^2}}(x-\xi)} + \]
\[ +\left( \begin{array}{cc} Z_{i1}' & Z_{i1}^{*}\\ \beta_j^2 Z_{i1}' & \beta_j^2 Z_{i1}^{*} \end{array} \right) e^{-\sqrt{\frac{p}{\lambda_i^2}}(x+\xi-2h)} + \]
\[ +\left( \begin{array}{cc} Z_{i2}' & Z_{i2}^{*}\\ \beta_j^2 Z_{i2}' & \beta_j^2 Z_{i2}^{*} \end{array} \right) e^{-\sqrt{\frac{p}{\lambda_i^2}}(x-h)-\sqrt{\frac{p}{\lambda_j^2}}(\xi-h)} \Biggr\}; \]

\[ \bar G_{22}^{**}(x,\xi,p)=\frac{1}{\sqrt p}\sum_{i=1}^{2} \Biggl\{ \left( \begin{array}{cc} A_i' & A_i^{\prime *}\\ \beta_j^2 A_i' & \beta_j^2 A_i^{\prime *} \end{array} \right) e^{-\sqrt{\frac{p}{\lambda_i^2}}(\xi-x)} + \]
\[ +\left( \begin{array}{cc} Z_{i1}' & Z_{i1}^{*}\\ \beta_j^2 Z_{i1}' & \beta_j^2 Z_{i1}^{*} \end{array} \right) e^{-\sqrt{\frac{p}{\lambda_i^2}}(\xi+x-2h)} + \]
\[ +\left( \begin{array}{cc} Z_{i2}' & Z_{i2}^{*}\\ \beta_j^2 Z_{i2}' & \beta_j^2 Z_{i2}^{*} \end{array} \right) e^{-\sqrt{\frac{p}{\lambda_i^2}}(x-h)-\sqrt{\frac{p}{\lambda_j^2}}(\xi-h)} \Biggr\}, \]

where

\[ A_{ik}'=\frac{A_k A_{ik}}{D};\qquad B_{ik}'=\frac{A_k B_{ik}}{D}; \]

\[ A_i=\frac{(\beta_i^1 a_{22}^1+a_{21}^1)\sqrt{\lambda_i^1}} {2(\beta_i^1-\beta_j^1)\Delta_1}; \]

\[ A_{ik}^{*}=\frac{A_k^{*}A_{ik}}{D};\qquad B_{ik}^{*}=\frac{A_k^{*}B_{ik}}{D};\qquad A_i^{*}=\frac{(\beta_i^1 a_{12}^1+a_{11}^1)\sqrt{\lambda_i^1}} {2(\beta_j^1-\beta_i^1)\Delta_1}; \]

\[ A_{i1}=s_{ij}^{i}\frac{\beta_1^j-\beta_2^j}{\sqrt{\lambda_1^i\lambda_2^i}} -b_i\frac{\beta_j^2-\beta_i^j}{\sqrt{\lambda_j^1\lambda_j^2}} -c_i\frac{\beta_j^1-\beta_i^2}{\sqrt{\lambda_j^1\lambda_i^2}}- \]

\[ - s_{ji}^{j}\frac{\beta_{2}^{i}-\beta_{1}^{i}}{\sqrt{\lambda_{1}^{j}\lambda_{2}^{j}}} + b_i\frac{\beta_{i}^{2}-\beta_{i}^{1}}{\sqrt{\lambda_{i}^{1}\lambda_{i}^{2}}} + c_j\frac{\beta_{i}^{1}-\beta_{j}^{2}}{\sqrt{\lambda_{i}^{1}\lambda_{j}^{2}}}; \]

\[ A_{i2}=2\left( \frac{\beta_{i}^{1}-\beta_{j}^{2}}{\sqrt{\lambda_{j}^{1}\lambda_{j}^{2}}}\, b_i + \frac{\beta_{i}^{2}-\beta_{i}^{1}}{\sqrt{\lambda_{j}^{1}\lambda_{i}^{2}}}\, c_i \right); \]

\[ B_{i1}=2\left( \frac{\beta_{i}^{1}-\beta_{j}^{1}}{\sqrt{\lambda_{i}^{1}\lambda_{j}^{2}}}\, c_j + \frac{\beta_{i}^{2}-\beta_{i}^{1}}{\sqrt{\lambda_{1}^{1}\lambda_{2}^{1}}}\, s_{ij}^{1} \right); \]

\[ B_{i2}=2\left( \frac{\beta_{i}^{1}-\beta_{j}^{1}}{\sqrt{\lambda_{j}^{1}\lambda_{j}^{2}}} + \frac{\beta_{i}^{2}-\beta_{i}^{1}}{\sqrt{\lambda_{1}^{1}\lambda_{2}^{2}}}\, s_{ij}^{1} \right); \]

\[ D=\sum_{i=1}^{2}\left( \frac{b_i(\beta_{i}^{2}-\beta_{i}^{1})}{\sqrt{\lambda_{j}^{1}\lambda_{j}^{2}}} + \frac{c_i(\beta_{i}^{1}-\beta_{i}^{2})}{\sqrt{\lambda_{j}^{1}\lambda_{i}^{2}}} + s_{ij}^{i}\frac{\beta_{i}^{j}-\beta_{j}^{i}}{\sqrt{\lambda_{1}^{i}\lambda_{2}^{i}}} \right); \]

\[ s_{ij}^{k}=\gamma_{i2}^{k}\gamma_{j1}^{k}-\gamma_{i1}^{k}\gamma_{j2}^{k};\qquad b_i=\gamma_{i1}^{1}\gamma_{i2}^{2}-\gamma_{i2}^{1}\gamma_{i1}^{2}; \]

\[ c_i=\gamma_{i1}^{1}\gamma_{j2}^{2}-\gamma_{i2}^{1}\gamma_{j1}^{2};\qquad \gamma_{lm}^{k}=d_{m1}^{k}+\beta_{l}^{k}d_{m2}^{k}; \]

\[ \beta_{k}^{i}=\frac{a_{22}^{i}-\lambda_{k}^{i}}{a_{12}^{i}};\qquad \Delta_i=a_{11}^{i}a_{22}^{i}-(a_{12}^{i})^{2}; \]

\[ \lambda_{1}^{i},\ \lambda_{2}^{i}\text{ are the eigenvalues of the matrix }A_i\text{, and where} \]

\[ Z_{ik}^{\prime}=\frac{A_{k}^{\prime}Z_{ik}}{D};\qquad M_{ik}^{\prime}=\frac{A_{k}^{\prime}M_{ik}}{D};\qquad A_{i}^{\prime}=\frac{(a_{22}^{2}\beta_{i}^{2}+a_{21}^{2})\sqrt{\lambda_{i}^{2}}}{2\Delta_{2}(\beta_{i}^{2}-\beta_{j}^{2})}; \]

\[ Z_{ik}^{*}=\frac{A_{k}^{*}Z_{ik}}{D};\qquad M_{ik}^{*}=\frac{A_{k}^{*}M_{ik}}{D};\qquad A_{i}^{*}=-\frac{(\beta_{i}^{2}a_{12}^{2}+a_{11}^{2})\sqrt{\lambda_{i}^{2}}}{2\Delta_{2}(\beta_{j}^{2}-\beta_{i}^{2})}; \]

\[ Z_{i1}= - s_{ij}^{i}\frac{\beta_{1}^{j}-\beta_{2}^{j}}{\sqrt{\lambda_{1}^{i}\lambda_{2}^{i}}} - b_i\frac{\beta_{j}^{2}-\beta_{j}^{1}}{\sqrt{\lambda_{j}^{1}\lambda_{j}^{2}}} - c_j\frac{\beta_{i}^{1}-\beta_{j}^{2}}{\sqrt{\lambda_{i}^{1}\lambda_{j}^{2}}} + \]

\[ + c_i\frac{\beta_{j}^{1}-\beta_{i}^{2}}{\sqrt{\lambda_{j}^{1}\lambda_{i}^{2}}} + b_j\frac{\beta_{i}^{2}-\beta_{i}^{1}}{\sqrt{\lambda_{i}^{1}\lambda_{i}^{2}}} + s_{ji}^{j}\frac{\beta_{2}^{i}-\beta_{1}^{i}}{\sqrt{\lambda_{1}^{j}\lambda_{2}^{j}}}; \]

\[ Z_{i2}=2\left( b_i\frac{\beta_{j}^{1}-\beta_{i}^{2}}{\sqrt{\lambda_{j}^{1}\lambda_{j}^{2}}} + c_j\frac{\beta_{i}^{2}-\beta_{i}^{1}}{\sqrt{\lambda_{i}^{1}\lambda_{j}^{2}}} \right); \]

\[ M_{i1}=2\left( c_i\frac{\beta_{j}^{2}-\beta_{i}^{2}}{\sqrt{\lambda_{i}^{2}\lambda_{j}^{1}}} + s_{ij}^{2}\frac{\beta_{j}^{2}-\beta_{i}^{1}}{\sqrt{\lambda_{1}^{2}\lambda_{2}^{2}}} \right); \]

\[ M_{i2}=2\left( b_i\frac{\beta_{j}^{2}-\beta_{i}^{2}}{\sqrt{\lambda_{j}^{1}\lambda_{j}^{2}}} + s_{ij}^{2}\frac{\beta_{i}^{2}-\beta_{i}^{1}}{\sqrt{\lambda_{1}^{2}\lambda_{2}^{2}}} \right) \]

\[ (i,\ k,\ l,\ m=1,2;\quad j=3-i). \]

Substitution of the expressions obtained into (15) yields the explicit form of the solution of the auxiliary problem (10), (12), (14).

1. Using the operational relation

\[ \frac{1}{\sqrt{p}}\,e^{-\alpha \sqrt{p}} \;\doteq\; \frac{1}{\sqrt{\pi t}}\,e^{-\frac{\alpha^{2}}{4t}}, \]

from (18) we find the explicit form of the matrices \(G_{ij}(x,\xi,t)\) \((i,j=1,2)\):

\[ \begin{aligned} G_{11}(x,\xi,t) &= \frac{1}{\sqrt{\pi t}} \sum_{i=1}^{2} \Biggl\{ \left( \begin{array}{cc} A_i' & A_i^{*}\\ \beta_j^{1} A_i' & \beta_j^{1} A_i^{*} \end{array} \right) e^{-\frac{(x-\xi)^2}{4\lambda_i^{1}t}} \\ &\qquad\qquad + \left( \begin{array}{cc} A_{i1}' & A_{i1}^{*}\\ \beta_j^{1} A_{i1}' & \beta_j^{1} A_{i1}^{*} \end{array} \right) e^{-\frac{(2h-x-\xi)^2}{4\lambda_i^{1}t}} \\ &\qquad\qquad + \left( \begin{array}{cc} A_{i2}' & A_{i2}^{*}\\ \beta_j^{1} A_{i2}' & \beta_j^{1} A_{i2}^{*} \end{array} \right) e^{-\frac{\left[\frac{h-\xi}{\sqrt{\lambda_j^{1}}}+\frac{h-x}{\sqrt{\lambda_i^{1}}}\right]^2}{4t}} \Biggr\}; \\[1.2em] G_{12}(x,\xi,t) &= \frac{1}{\sqrt{\pi t}} \sum_{i=1}^{2} \Biggl\{ \left( \begin{array}{cc} M_{i1}' & M_{i1}^{*}\\ \beta_j^{1} M_{i1}' & \beta_j^{1} M_{i1}^{*} \end{array} \right) e^{-\frac{\left[\frac{\xi-h}{\sqrt{\lambda_i^{2}}}+\frac{h-x}{\sqrt{\lambda_i^{1}}}\right]^2}{4t}} \\ &\qquad\qquad + \left( \begin{array}{cc} M_{i1}' & M_{i2}^{*}\\ \beta_j^{1} M_{i2}' & \beta_j^{1} M_{i2}^{*} \end{array} \right) e^{-\frac{\left[\frac{\xi-h}{\sqrt{\lambda_j^{2}}}+\frac{h-x}{\sqrt{\lambda_i^{1}}}\right]^2}{4t}} \Biggr\}; \\[1.2em] G_{21}(x,\xi,t) &= \frac{1}{\sqrt{\pi t}} \sum_{i=1}^{2} \Biggl\{ \left( \begin{array}{cc} B_{i1}' & B_{i1}^{*}\\ \beta_j^{2} B_{i1}' & \beta_j^{2} B_{i1}^{*} \end{array} \right) e^{-\frac{\left[\frac{x-h}{\sqrt{\lambda_i^{2}}}+\frac{h-\xi}{\sqrt{\lambda_i^{1}}}\right]^2}{4t}} \\ &\qquad\qquad + \left( \begin{array}{cc} B_{i2}' & B_{i2}^{*}\\ \beta_j^{2} B_{i2}' & \beta_j^{2} B_{i2}^{*} \end{array} \right) e^{-\frac{\left[\frac{x-h}{\sqrt{\lambda_i^{2}}}+\frac{h-\xi}{\sqrt{\lambda_j^{1}}}\right]^2}{4t}} \Biggr\}; \\[1.2em] G_{22}(x,\xi,t) &= \frac{1}{\sqrt{\pi t}} \sum_{i=1}^{2} \Biggl\{ \left( \begin{array}{cc} A_i' & A_i^{*}\\ \beta_j^{2} A_i' & \beta_j^{2} A_i^{*} \end{array} \right) e^{-\frac{(x-\xi)^2}{4\lambda_i^{2}t}} \\ &\qquad\qquad + \left( \begin{array}{cc} Z_{i1}' & Z_{i1}^{*}\\ \beta_j^{2} Z_{i1}' & \beta_j^{2} Z_{i1}^{*} \end{array} \right) e^{-\frac{(x+\xi-2h)^2}{4\lambda_i^{2}t}} \\ &\qquad\qquad + \left( \begin{array}{cc} Z_{i2}' & Z_{i2}^{*}\\ \beta_j^{2} Z_{i2}' & \beta_j^{2} Z_{i1}^{*} \end{array} \right) e^{-\frac{\left[\frac{x-h}{\sqrt{\lambda_i^{2}}}+\frac{\xi-h}{\sqrt{\lambda_j^{2}}}\right]^2}{4t}} \Biggr\}. \end{aligned} \tag{19} \]

1. We now proceed to the solution of the system of equations (10) with nonhomogeneous conjugation conditions (11). To this end, in the right-hand sides of (15), instead of \(\bar\Phi_i(\xi,p)\) \((i=1,2)\), we substitute their expressions from (10). Then, as in [1], integrating the expressions obtained by parts and using the homogeneous conjugation conditions satisfied by the solution (15), we obtain the solution of problem (10)—(12) in the form

\[ \begin{aligned} \bar U_1(x,p)={}& \int_{-\infty}^{h}\bar G_{11}\bar\Phi_1\,d\xi + \int_{h}^{\infty}\bar G_{12}\bar\Phi_2\,d\xi + \\[4pt] &+\left\{ \begin{array}{c} -\left.\dfrac{\partial \bar G_{11}^{**}}{\partial \xi}\right|_{\xi=h} A_1\bar\varphi_1(p)\\[8pt] -\left.\dfrac{\partial \bar G_{12}}{\partial \xi}\right|_{\xi=h} A_2\bar\varphi_1(p) \end{array} \right\} + \left\{ \begin{array}{l} \left.\bar G_{11}^{**}\right|_{\xi=h} A_1D_1^{-1}\bar\varphi_2(p)\quad (\text{if } |D_1|\ne 0),\\[4pt] \left.\bar G_{12}\right|_{\xi=h} A_2D_2^{-1}\bar\varphi_2(p)\quad (\text{if } |D_2|\ne 0); \end{array} \right. \end{aligned} \tag{20} \]

\[ \begin{aligned} \bar U_2(x,p)={}& \int_{-\infty}^{h}\bar G_{21}\bar\Phi_1\,d\xi + \int_{h}^{\infty}\bar G_{22}\bar\Phi_2\,d\xi + \\[4pt] &+\left\{ \begin{array}{c} -\left.\dfrac{\partial \bar G_{21}}{\partial \xi}\right|_{\xi=h} A_1\bar\varphi_1(p)\\[8pt] -\left.\dfrac{\partial \bar G_{22}^{*}}{\partial \xi}\right|_{\xi=h} A_2\bar\varphi_1(p) \end{array} \right\} + \left\{ \begin{array}{l} \left.\bar G_{21}\right|_{\xi=h} A_1D_1^{-1}\bar\varphi_2(p)\quad (\text{if } |D_1|\ne 0),\\[4pt] \left.\bar G_{22}^{*}\right|_{\xi=h} A_2D_2^{-1}\bar\varphi_2(p)\quad (\text{if } |D_2|\ne 0). \end{array} \right. \end{aligned} \]

Thus, we have obtained the general form of the solution of problem (10)—(12) in the transform domain. Substituting in (20) the values of \(\bar\Phi_i(\xi,p)\) from (13) \((i=1,2)\) and passing, with the aid of the convolution theorem, to the original for each term, we obtain the solution \(U_i(x,t)\) \((i=1,2)\) of problem (5)—(8) in matrix form. It will have the form

\[ \begin{aligned} U_1(x,t)={}& \int_{-\infty}^{h}G_{11}(x,\xi,t)F_1(\xi)\,d\xi + \int_{h}^{\infty}G_{12}(x,\xi,t)F_2(\xi)\,d\xi + \\[4pt] &+ \int_{0}^{t}d\tau\int_{-\infty}^{h} G_{11}(x,\xi,t-\tau)f_1(\xi,\tau)\,d\xi + \int_{0}^{t}d\tau\int_{h}^{\infty} G_{12}(x,\xi,t-\tau)f_2(\xi,\tau)\,d\xi + \\[6pt] &+ \left\{ \begin{array}{c} -\displaystyle\int_{0}^{t} \dfrac{\partial G_{11}(x,h,t-\tau)}{\partial \xi} A_1\varphi_1(\tau)\,d\tau\\[10pt] -\displaystyle\int_{0}^{t} \dfrac{\partial G_{12}(x,h,t-\tau)}{\partial \xi} A_2\varphi_1(\tau)\,d\tau \end{array} \right\} + \\[6pt] &+ \left\{ \begin{array}{l} \displaystyle\int_{0}^{t} G_{11}(x,h,t-\tau)A_1D_1^{-1}\varphi_2(\tau)\,d\tau \quad (\text{if } |D_1|\ne 0),\\[10pt] \displaystyle\int_{0}^{t} G_{12}(x,h,t-\tau)A_2D_2^{-1}\varphi_2(\tau)\,d\tau \quad (\text{if } |D_2|\ne 0); \end{array} \right. \end{aligned} \]

\[ \begin{aligned} U_2(x,t)={}& \int_{-\infty}^{h} G_{21}(x,\xi,t) F_1(\xi)\,d\xi +\int_{h}^{\infty} G_{22}(x,\xi,t) F_2(\xi)\,d\xi+\\ &+\int_{0}^{t} d\tau \int_{-\infty}^{h} G_{21}(x,\xi,t-\tau) f_1(\xi,\tau)\,d\xi+\\ &+\int_{0}^{t} d\tau \int_{h}^{\infty} G_{22}(x,\xi,t-\tau) f_2(\xi,\tau)\,d\xi+\\ &+\left\{ \begin{array}{l} -\displaystyle\int_{0}^{t} \frac{\partial G_{21}(x,h,t-\tau)}{\partial \xi}\, A_1\varphi_1(\tau)\,d\tau \\[2ex] -\displaystyle\int_{0}^{t} \frac{\partial G_{22}(x,h,t-\tau)}{\partial \xi}\, A_2\varphi_1(\tau)\,d\tau \end{array} \right. +\\ &+\left\{ \begin{array}{l} \displaystyle\int_{0}^{t} G_{21}(x,h,t-\tau) A_1D_1^{-1}\varphi_2(\tau)\,d\tau \quad (\text{when } |D_1|\ne 0),\\[2ex] \displaystyle\int_{0}^{t} G_{22}(x,h,t-\tau) A_2D_2^{-1}\varphi_2(\tau)\,d\tau \quad (\text{when } |D_2|\ne 0), \end{array} \right. \end{aligned} \]

where the matrices \(G_{ij}(x,\xi,t)\) \((i,j=1,2)\) are defined by formulas (19).

Writing out the matrices, we obtain the solution of the original problem (1)—(4). It is not difficult to verify that, under the most general assumptions on the functions \(f_j^i(x,t)\), \(\varphi_j^i(t)\), \(F_j^i(x)\) \((i,j=1,2)\), this solution satisfies all the conditions of the problem.

References

1. A. V. Ivanov, in: Thermophysics in Foundry Production. Minsk, Publishing House of the Academy of Sciences of the BSSR, 1963, pp. 11—52.
2. L. A. Marnevskaya, in: Thermophysics in Foundry Production. Minsk, Publishing House of the Academy of Sciences of the BSSR, 1963, pp. 69—75.
3. A. V. Ivanov, Izvestiya of the Academy of Sciences of the BSSR, Physical-Technical Sciences Series, No. 3, 5—8, 1962.
4. P. V. Tsoi, Izvestiya of the Academy of Sciences of the Tajik SSR, Division of Geological-Chemical and Technical Sciences, No. 3(5), 39—49, 1961.
5. P. V. Tsoi, in: Heat and Mass Transfer, 5. Minsk, Academy of Sciences of the BSSR, 1963, pp. 166—177.
6. A. P. Prudnikov, Doklady of the Academy of Sciences of the USSR, 115, No. 5, 869—871, 1957.
7. A. P. Prudnikov, Izvestiya of the Academy of Sciences of the USSR, Technical Sciences Division, No. 10, 63—67, 1958.

Received by the editors
April 17, 1965

Institute of Mathematics
Academy of Sciences of the BSSR