## 3 Numerical Solution of the Problem
Equation (5) is approximated as follows  
\[\theta_{1}\frac{c_{ij}^{k+1/2}-c_{ij}^{k}}{0,5\,\tau}+\frac{(v_{1x})_{ij}^{k} c_{ij}^{k}-(v_{1x})_{i-1,j}^{k}c_{i-1,j}^{k}}{h_{1}}+\frac{(v_{1r})_{i,j}^{k}c_{ ij}^{k}-(v_{1r})_{i,j-1}^{k}c_{i,j-1}^{k}}{h_{2}}=\]  
\[=\,\theta_{1}\left\{D_{1r}\frac{1}{(jh_{2})^{\tau_{1}}}\frac{h_{2}^{\tau_{1}} \left(j+\frac{1}{2}\right)^{\tau_{1}}-\tau_{1}h_{2}^{\tau_{1}}\left(j-\frac{1 }{2}\right)^{\tau_{1}}}{\Gamma\left(2-\tau_{1}\right)h_{2}^{\tau_{1}}}\frac{c_ {ij}^{k}-\tau_{1}c_{i,j-1}^{k}}{\Gamma\left(2-\tau_{1}\right)h_{2}^{\tau_{1}} }+\right.\right.\]  
\[\left.\left.+\frac{D_{1r}}{\Gamma\left(3-2\tau_{1}\right)h_{2}^{2\tau_{1}}} \sum_{l=0}^{j-1}\left(c_{i,j-(l-1)}^{k}-2c_{i,j-l}^{k}+c_{i,j-(l+1)}^{k}\right) \left((l+1)^{2-2\tau_{1}}-l^{2-2\tau_{1}}\right)\right.\right.\]  
\[\left.\left.+\frac{D_{1x}}{\Gamma\left(3-\tau_{1}^{*}\right)h_{1}^{\tau_{1}}} \sum_{l=0}^{i-1}\left(c_{i-(l-1),j}^{k}-2c_{i-l,j}^{k}+c_{i-(l+1),j}^{k}\right) \left((l+1)^{2-\gamma_{1}^{*}}-l^{2-\gamma_{1}^{*}}\right)\right\}, \tag{71}\]  
\[\left.i=\overline{1,I-1},\ \ j=\overline{1,N-1},\ \ k=\overline{0,K-1},\right.\]  
\[\theta_{1}\frac{c_{ij}^{k+1}-c_{ij}^{k+1/2}}{0,5\,\tau}+\frac{(v_{1x})_{i,j}^{ k+1/2}\,c_{ij}^{k+1/2}-(v_{1x})_{i-1,j}^{k+1/2}\,c_{i-1,j}^{k+1/2}}{h_{1}}+ \frac{(v_{1r})_{i,j}^{k+1/2}\,c_{ij}^{k+1/2}-(v_{1r})_{i,j-1}^{k+1/2}\,c_{i,j- 1}^{k+1/2}}{h_{2}}=\]\[= \theta_{1}\left\{D_{1r}\frac{1}{\left(j\mu_{2}\right)^{\gamma_{1}}}\frac{h_{2}^{ \gamma_{1}}\left(j+\frac{1}{2}\right)^{\gamma_{1}}-\gamma_{1}h_{2}^{\gamma_{1}} \left(j-\frac{1}{2}\right)^{\gamma_{1}}}{\Gamma\left(2-\gamma_{1}\right)h_{2}^ {\gamma_{1}}}\frac{c_{ij}^{k+1/2}-\gamma_{1}c_{i,j-1}^{k+1/2}}{\Gamma\left(2- \gamma_{1}\right)h_{2}^{\gamma_{1}}}+\right.\]  
\[(p_{1})_{i,j}^{k+1}=(p_{2})_{i,j}^{k+1}=\frac{\tilde{\xi}_{i,j}+k\xi_{i,j}^{*} }{1-\alpha_{i,j}-k\left(\alpha_{i,j}^{*}-1\right)},i=\overline{0,I},\ j=N, \tag{69}\]0  
\[(p_{1})_{i,j}^{k+1}=(p_{2})_{i,j}^{k+1}=\frac{\tilde{\xi}_{i,j}+k\xi_{i,j}^{*} }{1-\alpha_{i,j}-k\left(\alpha_{i,j}^{*}-1\right)},i=\overline{0,I},\ j=N, \tag{69}\]1  
\[(p_{1})_{i,j}^{k+1}=(p_{2})_{i,j}^{k+1}=\frac{\tilde{\xi}_{i,j}+k\xi_{i,j}^{*} }{1-\alpha_{i,j}-k\left(\alpha_{i,j}^{*}-1\right)},i=\overline{0,I},\ j=N, \tag{69}\]2  
Equation (6) is approximated in the form  
\[(p_{1})_{i,j}^{k+1}=(p_{2})_{i,j}^{k+1}=\frac{\tilde{\xi}_{i,j}+k\xi_{i,j}^{*} }{1-\alpha_{i,j}-k\left(\alpha_{i,j}^{*}-1\right)},i=\overline{0,I},\ j=N, \tag{69}\]3  
\[(p_{1})_{i,j}^{k+1}=(p_{2})_{i,j}^{k+1}=\frac{\tilde{\xi}_{i,j}+k\xi_{i,j}^{*} }{1-\alpha_{i,j}-k\left(\alpha_{i,j}^{*}-1\right)},i=\overline{0,I},\ j=N, \tag{69}\]4  
\[(p_{1})_{i,j}^{k+1}=(p_{2})_{i,j}^{k+1}=\frac{\tilde{\xi}_{i,j}+k\xi_{i,j}^{*} }{1-\alpha_{i,j}-k\left(\alpha_{i,j}^{*}-1\right)},i=\overline{0,I},\ j=N, \tag{69}\]5  
\[(p_{1})_{i,j}^{k+1}=(p_{2})_{i,j}^{k+1}=\frac{\tilde{\xi}_{i,j}+k\xi_{i,j}^{*} }{1-\alpha_{i,j}-k\left(\alpha_{i,j}^{*}-1\right)},i=\overline{0,I},\ j=N, \tag{69}\]6  
\[(p_{1})_{i,j}^{k+1}=(p_{2})_{i,j}^{k+1}=\frac{\tilde{\xi}_{i,j}+k\xi_{i,j}^{*} }{1-\alpha_{i,j}-k\left(\alpha_{i,j}^{*}-1\right)},i=\overline{0,I},\ j=N, \tag{69}\]7  
\[(p_{1})_{i,j}^{k+1}=(p_{2})_{i,j}^{k+1}=\frac{\tilde{\xi}_{i,j}+k\xi_{i,j}^{*} }{1-\alpha_{i,j}-k\left(\alpha_{i,j}^{*}-1\right)},i=\overline{0,I},\ j=N, \tag{69}\]8  
\[(p_{1})_{i,j}^{k+1}=(p_{2})_{i,j}^{k+1}=\frac{\tilde{\xi}_{i,j}+k\xi_{i,j}^{*} }{1-\alpha_{i,j}-k\left(\alpha_{i,j}^{*}-1\right)},i=\overline{0,I},\ j=N, \tag{69}\]9  
\[(p_{m})_{I,j}^{k+1/2}=\xi_{I+1,j}^{m}=\frac{(F_{m})_{I,j}^{k}+A_{m}\xi_{I,j}^{ m}}{B_{m}-A_{m}\xi_{I,j}^{m}},j=\overline{0,J},m=1,\ 2 \tag{70}\]0  
\[(p_{m})_{I,j}^{k+1/2}=\xi_{I+1,j}^{m}=\frac{(F_{m})_{I,j}^{k}+A_{m}\xi_{I,j}^{ m}}{B_{m}-A_{m}\xi_{I,j}^{m}},j=\overline{0,J},m=1,\ 2 \tag{70}\]1  
\[(p_{m})_{I,j}^{k+1/2}=\xi_{I+1,j}^{m}=\frac{(F_{m})_{I,j}^{k}+A_{m}\xi_{I,j}^{ m}}{B_{m}-A_{m}\xi_{I,j}^{m}},j=\overline{0,J},m=1,\ 2 \tag{70}\]2We introduce the following notation  
\[(p_{m})_{I,j}^{k+1/2}=\xi_{I+1,j}^{m}=\frac{(F_{m})_{I,j}^{k}+A_{m}\xi_{I,j}^{ m}}{B_{m}-A_{m}\xi_{I,j}^{m}},j=\overline{0,J},m=1,\ 2 \tag{70}\]3  
\[(p_{m})_{I,j}^{k+1/2}=\xi_{I+1,j}^{m}=\frac{(F_{m})_{I,j}^{k}+A_{m}\xi_{I,j}^{ m}}{B_{m}-A_{m}\xi_{I,j}^{m}},j=\overline{0,J},m=1,\ 2 \tag{70}\]4  
\[(p_{m})_{I,j}^{k+1/2}=\xi_{I+1,j}^{m}=\frac{(F_{m})_{I,j}^{k}+A_{m}\xi_{I,j}^{ m}}{B_{m}-A_{m}\xi_{I,j}^{m}},j=\overline{0,J},m=1,\ 2 \tag{70}\]5  
\[(p_{m})_{I,j}^{k+1/2}=\xi_{I+1,j}^{m}=\frac{(F_{m})_{I,j}^{k}+A_{m}\xi_{I,j}^{ m}}{B_{m}-A_{m}\xi_{I,j}^{m}},j=\overline{0,J},m=1,\ 2 \tag{70}\]6  
\[(p_{m})_{I,j}^{k+1/2}=\xi_{I+1,j}^{m}=\frac{(F_{m})_{I,j}^{k}+A_{m}\xi_{I,j}^{ m}}{B_{m}-A_{m}\xi_{I,j}^{m}},j=\overline{0,J},m=1,\ 2 \tag{70}\]7  
\[(p_{m})_{I,j}^{k+1/2}=\xi_{I+1,j}^{m}=\frac{(F_{m})_{I,j}^{k}+A_{m}\xi_{I,j}^{ m}}{B_{m}-A_{m}\xi_{I,j}^{m}},j=\overline{0,J},m=1,\ 2 \tag{70}\]8  
Given these notation, Equations (71)-(74), have the form respectively  
\[(p_{m})_{I,j}^{k+1/2}=\xi_{I+1,j}^{m}=\frac{(F_{m})_{I,j}^{k}+A_{m}\xi_{I,j}^{ m}}{B_{m}-A_{m}\xi_{I,j}^{m}},j=\overline{0,J},m=1,\ 2 \tag{70}\]9  
\[\theta_{1}\frac{c_{ij}^{k+1/2}-c_{ij}^{k}}{0,5\,\tau}+\frac{(v_{1x})_{ij}^{k} c_{ij}^{k}-(v_{1x})_{i-1,j}^{k}c_{i-1,j}^{k}}{h_{1}}+\frac{(v_{1r})_{i,j}^{k}c_{ ij}^{k}-(v_{1r})_{i,j-1}^{k}c_{i,j-1}^{k}}{h_{2}}=\]0  
\[\theta_{1}\frac{c_{ij}^{k+1/2}-c_{ij}^{k}}{0,5\,\tau}+\frac{(v_{1x})_{ij}^{k} c_{ij}^{k}-(v_{1x})_{i-1,j}^{k}c_{i-1,j}^{k}}{h_{1}}+\frac{(v_{1r})_{i,j}^{k}c_{ ij}^{k}-(v_{1r})_{i,j-1}^{k}c_{i,j-1}^{k}}{h_{2}}=\]1  
\[\theta_{1}\frac{c_{ij}^{k+1/2}-c_{ij}^{k}}{0,5\,\tau}+\frac{(v_{1x})_{ij}^{k} c_{ij}^{k}-(v_{1x})_{i-1,j}^{k}c_{i-1,j}^{k}}{h_{1}}+\frac{(v_{1r})_{i,j}^{k}c_{ ij}^{k}-(v_{1r})_{i,j-1}^{k}c_{i,j-1}^{k}}{h_{2}}=\]2  
\[\theta_{1}\frac{c_{ij}^{k+1/2}-c_{ij}^{k}}{0,5\,\tau}+\frac{(v_{1x})_{ij}^{k} c_{ij}^{k}-(v_{1x})_{i-1,j}^{k}c_{i-1,j}^{k}}{h_{1}}+\frac{(v_{1r})_{i,j}^{k}c_{ ij}^{k}-(v_{1r})_{i,j-1}^{k}c_{i,j-1}^{k}}{h_{2}}=\]3  
\[\theta_{1}\frac{c_{ij}^{k+1/2}-c_{ij}^{k}}{0,5\,\tau}+\frac{(v_{1x})_{ij}^{k} c_{ij}^{k}-(v_{1x})_{i-1,j}^{k}c_{i-1,j}^{k}}{h_{1}}+\frac{(v_{1r})_{i,j}^{k}c_{ ij}^{k}-(v_{1r})_{i,j-1}^{k}c_{i,j-1}^{k}}{h_{2}}=\]4  
\[\theta_{1}\frac{c_{ij}^{k+1/2}-c_{ij}^{k}}{0,5\,\tau}+\frac{(v_{1x})_{ij}^{k} c_{ij}^{k}-(v_{1x})_{i-1,j}^{k}c_{i-1,j}^{k}}{h_{1}}+\frac{(v_{1r})_{i,j}^{k}c_{ ij}^{k}-(v_{1r})_{i,j-1}^{k}c_{i,j-1}^{k}}{h_{2}}=\]5  
\[\theta_{1}\frac{c_{ij}^{k+1/2}-c_{ij}^{k}}{0,5\,\tau}+\frac{(v_{1x})_{ij}^{k} c_{ij}^{k}-(v_{1x})_{i-1,j}^{k}c_{i-1,j}^{k}}{h_{1}}+\frac{(v_{1r})_{i,j}^{k}c_{ ij}^{k}-(v_{1r})_{i,j-1}^{k}c_{i,j-1}^{k}}{h_{2}}=\]6  
From (54) we define \(c^{k+1/2}_{i,j}\) and \(c^{k+1}_{i,j}\) (\(i=\overline{0,\ I-1},\ \ j=N\))  
\[\theta_{1}\frac{c_{ij}^{k+1/2}-c_{ij}^{k}}{0,5\,\tau}+\frac{(v_{1x})_{ij}^{k} c_{ij}^{k}-(v_{1x})_{i-1,j}^{k}c_{i-1,j}^{k}}{h_{1}}+\frac{(v_{1r})_{i,j}^{k}c_{ ij}^{k}-(v_{1r})_{i,j-1}^{k}c_{i,j-1}^{k}}{h_{2}}=\]7  
\[\theta_{1}\frac{c_{ij}^{k+1/2}-c_{ij}^{k}}{0,5\,\tau}+\frac{(v_{1x})_{ij}^{k} c_{ij}^{k}-(v_{1x})_{i-1,j}^{k}c_{i-1,j}^{k}}{h_{1}}+\frac{(v_{1r})_{i,j}^{k}c_{ ij}^{k}-(v_{1r})_{i,j-1}^{k}c_{i,j-1}^{k}}{h_{2}}=\]8  
where \(D=\frac{D_{2r}\Gamma(2-\gamma_{1})h_{2}^{71}}{D_{1r}\Gamma(2-\gamma_{2})h_{2} ^{72}}\).