## 3 Numerical Solution of the Problem
Initial and boundary conditions (7)-(16) approximated as  
\[\left.r_{j}=jh_{2}\,,\ \ h_{2}=\frac{a}{N},\ \ j=\overline{1,N},\ \ \ r_{j}=jh_{2}\,,\ \ h_{2}=\frac{b-a}{J-N},\ \ j= \overline{N+1,J}\right\}, \tag{27}\]4  
\[\left.r_{j}=jh_{2}\,,\ \ h_{2}=\frac{a}{N},\ \ j=\overline{1,N},\ \ \ r_{j}=jh_{2}\,,\ \ h_{2}=\frac{b-a}{J-N},\ \ j= \overline{N+1,J}\right\}, \tag{27}\]5  
\[\left.r_{j}=jh_{2}\,,\ \ h_{2}=\frac{a}{N},\ \ j=\overline{1,N},\ \ \ r_{j}=jh_{2}\,,\ \ h_{2}=\frac{b-a}{J-N},\ \ j= \overline{N+1,J}\right\}, \tag{27}\]6  
\[\left.r_{j}=jh_{2}\,,\ \ h_{2}=\frac{a}{N},\ \ j=\overline{1,N},\ \ \ r_{j}=jh_{2}\,,\ \ h_{2}=\frac{b-a}{J-N},\ \ j= \overline{N+1,J}\right\}, \tag{27}\]7  
\[\left.r_{j}=jh_{2}\,,\ \ h_{2}=\frac{a}{N},\ \ j=\overline{1,N},\ \ \ r_{j}=jh_{2}\,,\ \ h_{2}=\frac{b-a}{J-N},\ \ j= \overline{N+1,J}\right\}, \tag{27}\]8  
\[\left.r_{j}=jh_{2}\,,\ \ h_{2}=\frac{a}{N},\ \ j=\overline{1,N},\ \ \ r_{j}=jh_{2}\,,\ \ h_{2}=\frac{b-a}{J-N},\ \ j= \overline{N+1,J}\right\}, \tag{27}\]9  
\[(v_{1x})_{i,j} =-\frac{k_{1}}{\mu}\frac{(p_{1})_{i+1,j}-(p_{1})_{i,j}}{h_{1}},i= \overline{0,I-1},\ j=\overline{0,N}, \tag{28}\]0  
\[(v_{1x})_{i,j} =-\frac{k_{1}}{\mu}\frac{(p_{1})_{i+1,j}-(p_{1})_{i,j}}{h_{1}},i= \overline{0,I-1},\ j=\overline{0,N}, \tag{28}\]1  
\[(v_{1x})_{i,j} =-\frac{k_{1}}{\mu}\frac{(p_{1})_{i+1,j}-(p_{1})_{i,j}}{h_{1}},i= \overline{0,I-1},\ j=\overline{0,N}, \tag{28}\]2  
\[(v_{1x})_{i,j} =-\frac{k_{1}}{\mu}\frac{(p_{1})_{i+1,j}-(p_{1})_{i,j}}{h_{1}},i= \overline{0,I-1},\ j=\overline{0,N}, \tag{28}\]3  
Conditions (17)-(26) we write in finite difference form  
\[(v_{1x})_{i,j} =-\frac{k_{1}}{\mu}\frac{(p_{1})_{i+1,j}-(p_{1})_{i,j}}{h_{1}},i= \overline{0,I-1},\ j=\overline{0,N}, \tag{28}\]4  
\[(v_{1x})_{i,j} =-\frac{k_{1}}{\mu}\frac{(p_{1})_{i+1,j}-(p_{1})_{i,j}}{h_{1}},i= \overline{0,I-1},\ j=\overline{0,N}, \tag{28}\]5  
\[(v_{1x})_{i,j} =-\frac{k_{1}}{\mu}\frac{(p_{1})_{i+1,j}-(p_{1})_{i,j}}{h_{1}},i= \overline{0,I-1},\ j=\overline{0,N}, \tag{28}\]6  
\[(v_{1x})_{i,j} =-\frac{k_{1}}{\mu}\frac{(p_{1})_{i+1,j}-(p_{1})_{i,j}}{h_{1}},i= \overline{0,I-1},\ j=\overline{0,N}, \tag{28}\]7  
\[(v_{1x})_{i,j} =-\frac{k_{1}}{\mu}\frac{(p_{1})_{i+1,j}-(p_{1})_{i,j}}{h_{1}},i= \overline{0,I-1},\ j=\overline{0,N}, \tag{28}\]8  
\[(v_{1x})_{i,j} =-\frac{k_{1}}{\mu}\frac{(p_{1})_{i+1,j}-(p_{1})_{i,j}}{h_{1}},i= \overline{0,I-1},\ j=\overline{0,N}, \tag{28}\]9  
\[(v_{1r})_{i,j} =-\frac{k_{1}}{\mu}\frac{(p_{1})_{i,j+1}-(p_{1})_{i,j}}{h_{2}},i= \overline{0,I},\ j=\overline{0,N-1},\]0  
\[(v_{1r})_{i,j} =-\frac{k_{1}}{\mu}\frac{(p_{1})_{i,j+1}-(p_{1})_{i,j}}{h_{2}},i= \overline{0,I},\ j=\overline{0,N-1},\]1  
Starting values \(\alpha_{i,j},\ \tilde{\chi}_{i,j},\ \alpha_{i,j}^{*},\ \tilde{\xi}_{i,j}^{*}\) in (64), (65) determined from the conditions (37), (40), (43)  
\[\left\{ \begin{array}{ll}\alpha_{i,j}=0,\ \ \tilde{\xi}_{i,j}=p_{c}\ \ \ \ \text{at}\ \ j=1,\ \ i=0,\\ \alpha_{i,j}=\frac{\tau\chi_{1}}{\tau\chi_{1}+h_{2}^{2}},\ \ \ \tilde{\xi}_{i,j}=\frac{(\tilde{\tau}_{3})_{i,j}^{k+1/2}}{B_{3}}\ \ \ \ \text{at}\ \ \ j=1,\ \ i=\overline{1,I}, \end{array} \right. \tag{67}\]  
\[\alpha_{i,j}^{*}=\frac{\tau\chi_{2}}{\tau\chi_{2}+h_{2}^{2}},\ \ \ \tilde{\xi}_{i,j}^{*}=\frac{(\tilde{\tau}_{4})_{i,j}^{k+1/2}}{B_{4}}atj=J-1,\ \ i= \overline{0,I}. \tag{68}\]  
To determine the values \((p_{1})_{i,j}^{k+1}\) and \((p_{2})_{i,j}^{k+1}\) (\(i=\overline{0,I},\ j=S\)) from the boundary conditions (44), (45) we obtain the expression  
\[(p_{1})_{i,j}^{k+1}=(p_{2})_{i,j}^{k+1}=\frac{\tilde{\xi}_{i,j}+k\xi_{i,j}^{*} }{1-\alpha_{i,j}-k\left(\alpha_{i,j}^{*}-1\right)},i=\overline{0,I},\ j=N, \tag{69}\]  
where \(k=k_{2}/k_{1}\). The following computing sequence is set. From (63) taking into account (66), we determine the values of the [PERSON] algorithm coefficients \(\alpha_{i,j}^{m},\ \tilde{\xi}_{i,j}^{m}\) (\(m=1,\ 2,\,i=\overline{1,I},\ j=\overline{0,J}\)).