# Entropy Optimization in MHD Nanofluid Flow over an Exponential Stretching Sheet
## 4 Method of Solution
### Linearization  
The non-linear terms in Equations (7)-(9) are converted into a recursive sequence with linear components; see [27; 28; 29].  
Applying the QLM technique to Equations (7)-(9) gives the following:  
\[f_{e+1}^{\prime\prime\prime(m)}+a_{1,e}f_{e+1}^{\prime\prime(m)}+a_{2,e}f_{e+1} ^{\prime(m)}+a_{3,e}f_{e+1}^{(m)}+a_{4,e}\frac{\partial f_{e+1}^{\prime(m)}}{ \partial\xi}+a_{5,e}\frac{\partial f_{e+1}^{(m)}}{\partial\xi}+a_{6,e}\theta_ {e+1}^{\prime(m)}+a_{7,e}\theta_{e+1}^{(m)}+a_{8,e}h_{e+1}^{(m)}=R_{1,e},\]  
\[b_{0,e}\theta_{e+1}^{\prime\prime(m)}+b_{1,e}\theta_{e+1}^{\prime(m)}+b_{2,e} \theta_{e+1}^{(m)}+b_{3,e}\frac{\partial\theta_{e+1}^{(m)}}{\partial\xi}+b_{4,e}f_{e+1}^{\prime\prime(m)}+b_{5,e}f_{e+1}^{\prime(m)}+b_{6,e}f_{e+1}^{(m)}+b_ {7,e}\frac{\partial f_{e+1}^{(m)}}{\partial\xi}+b_{8,e}h_{e+1}^{\prime(m)}=R_{ 2,e},\]  
\[h_{e+1}^{\prime\prime(m)}+c_{1,e}h_{e+1}^{\prime(m)}+c_{2,e}h_{e+1}^{(m)}+c_{3,e}\frac{\partial h_{e+1}^{(m)}}{\partial\xi}+c_{4,e}f_{e+1}^{\prime(m)}+c_{5,e}f_{e+1}^{(m)}+c_{6,e}\frac{\partial f_{e+1}^{(m)}}{\partial\xi}+c_{7,e} \theta_{e+1}^{\prime\prime(m)}=R_{3,e}, \tag{21}\]  
where the variable coefficients \(a_{i,e},b_{i,e},c_{i,e}\) and \(d_{i,e}(i=1,2,3,\ldots)\) are presumed to be known from previous calculations and are given by  
\[a_{1,e}=-\beta_{1}\theta_{e}^{\prime}+\frac{1}{2}e^{\beta_{1} \theta_{e}}f_{e}+\xi e^{\beta_{1}\theta_{e}}\frac{\partial f_{e}}{\partial\xi },\quad a_{2,e}=-\xi e^{\beta_{1}\theta_{e}}\left(1+\frac{\partial f_{e}^{ \prime}}{\partial\xi}\right),\quad a_{3,e}=\frac{1}{2}e^{\beta_{1}\theta_{e}} f_{e}^{\prime\prime},\quad a_{4,e}=-\xi e^{\beta_{1}\theta_{e}}f_{e}^{\prime},\] \[a_{5,e}=\xi e^{\beta_{1}\theta_{e}}f_{e}^{\prime\prime},\quad a_ {6,e}=-\beta_{1}e^{\prime\prime},\quad a_{7,e}=\beta_{1}e^{\beta_{1}\theta_{e }}\left[\frac{1}{2}f_{e}f_{e}^{\prime\prime}+\xi(1-f^{\prime})+G_{T}\theta_{e} +G_{T}c_{h}e_{-}-f_{e}^{\prime}\frac{\partial f_{e}^{\prime}}{\partial\xi}+f _{e}^{\prime\prime}\frac{\partial f_{e}}{\partial\xi}+\frac{G_{T}}{\beta_{1}} \right],\] \[a_{8,e}=e^{\beta_{1}\theta_{e}}G_{T},\] \[b_{0,e}=\frac{1+Nr}{Pr},\quad b_{1,e}=\frac{1}{2}f_{e}^{(m)}+N_ {b}h_{e}^{\prime(m)}+2N_{l}\theta_{e}^{\prime(m)}+\xi\frac{\partial f_{e}^{(m )}}{\partial\xi},\quad b_{2,e}=\lambda\xi,\quad b_{3,e}=-\xi f_{e}^{\prime(m)},\quad b_{4,e}=2 Ecf_{e}^{\prime\prime(m)},\] \[b_{5,e}=2\xi eTf_{e}^{\prime(m)}-\xi\frac{\partial\theta_{e}^{(m )}}{\partial\xi},\quad b_{6,e}=\frac{1}{2}\theta_{e}^{\prime(m)},\quad b_{7,e }=\xi\theta_{e}^{\prime(m)},\quad b_{8,e}=N_{b}\theta_{e}^{\prime(m)},\] \[c_{1,e}=\frac{1}{2}Lef_{e}^{(m)}+Le\xi\frac{\partial f_{e}^{(m)} }{\partial\xi},\quad c_{2,e}=-LeR_{1},\quad c_{3,e}=-Le\xi f_{e}^{\prime(m)},\] \[c_{4,e}=-Le\xi\frac{\partial h_{e}^{(m)}}{\partial\xi},\quad c_{5, e}=\frac{1}{2}Leh_{e}^{\prime(m)},\quad c_{6,e}=Le\xi h_{e}^{\prime(m)},\quad c_{7,e}= \frac{N_{l}}{N_{b}}.\]  
Here, the right-hand side terms \(R_{i,e}(i=1,2,3)\) in Equation (21) are given below:  
\[R_{1,e}=f_{e}^{\prime\prime\prime(m)}+a_{1,e}f_{e}^{\prime\prime (m)}+a_{2,e}f_{e}^{\prime(m)}+a_{3,e}f_{e}^{(m)}+a_{4,e}\frac{\partial f_{e}^{ \prime(m)}}{\partial\xi}+a_{5,e}\frac{\partial f_{e}^{(m)}}{\partial\xi}+a_{ 6,e}\theta_{e}^{\prime(m)}+a_{7,e}\theta_{e}^{(m)}+a_{8,e}h_{e}^{(m)}-F_{f},\] \[R_{2,e}=b_{0,e}\theta_{e}^{\prime\prime(m)}+b_{1,e}\theta_{e}^{ \prime(m)}+b_{2,e}\theta_{e}^{(m)}+b_{3,e}\frac{\partial\theta_{e}^{(m)}}{ \partial\xi}+b_{4,e}f_{e}^{\prime\prime(m)}+b_{5,e}f_{e}^{\prime(m)}+b_{6,e}f _{e}^{(m)}+b_{7,e}\frac{\partial f_{e}^{(m)}}{\partial\xi}+b_{8,e}h_{e}^{\prime (m)}-F_{b},\] \[R_{3,e}=h_{e}^{\prime\prime(m)}+c_{1,e}h_{e}^{\prime(m)}+c_{2,e}h _{e}^{(m)}+c_{3,e}\frac{\partial h_{e}^{(m)}}{\partial\xi}+c_{4,e}f_{e}^{\prime (m)}+c_{5,e}f_{e}^{(m)}+c_{6,e}\frac{\partial f_{e}^{(m)}}{\partial\xi}+c_{7,e} \theta_{e}^{\prime\prime(m)}-F_{h}, \tag{22}\]  
and the functions \(F_{f},F_{\theta}\), and \(F_{h}\) are as defined in Equations (7)-(9).