## 3 Problem Formulation
\Biggr{)}\Biggr{\}} \tag{59}\]  
\[f_{2,i}=f_{2,i-1}+\overline{h} \Bigl{\{}a_{1}\Bigl{(}\overline{f}_{1,i}^{2}-\overline{f}_{i} \overline{f}_{2,i}-K_{b}\overline{H}_{1,i}+\lambda\overline{f}_{1,i}+F_{r} \overline{f}_{1,i}^{2}\Bigr{)}+b_{1}\Bigl{(}f_{1,i-1}^{2}-f_{i-1}f_{2,i-1}-K_{ b}H_{1,i-1}+\lambda f_{1,i-1}+F_{r}f_{1,i-1}^{2}\Bigr{)}\Bigr{\}} \tag{60}\] \[+\overline{h}^{2}\{c_{1}\Bigl{(}2\overline{f}_{1,i}\overline{f}_ {2,i}(1+F_{r})-\overline{f}_{1,i}\overline{f}_{2,i}-\overline{f}_{i}\overline {f}_{2,i}^{2}-\overline{f}_{i}\overline{f}_{2,i}-K_{b}\overline{H}_{1,i}+ \lambda\overline{f}_{1,i}+F_{r}\overline{f}_{1,i}^{2}\Bigr{)}-\frac{\kappa_{b}} {\overline{c}}\Bigl{(}2\overline{H}_{i}+\overline{f}_{2,i}\Bigr{)}\] \[+d_{1}\bigl{(}2f_{1,i-1}f_{2,i-1}(1+F_{r})-f_{1,i-1}f_{2,i-1}\] \[-f_{i-1}\Bigl{(}f_{1,i-1}^{2}-f_{i-1}f_{2,i-1}-K_{b}H_{1,i-1}+ \lambda f_{1,i-1}+F_{r}f_{1,i-1}^{2}\Bigr{)}-\frac{\kappa_{b}}{\overline{c}} \bigl{(}2H_{i-1}+f_{2,i-1}\bigr{)}+\lambda f_{2,i-1}\bigr{)}\bigr{)}\}\]  
\[\theta_{i}=\theta_{i-1}+\overline{h}\bigl{(}a_{1}\overline{\theta}_{1,i}+b_{1} \theta_{1,i-1}\bigr{)}+\overline{h}^{2}\biggl{\{}c_{1}\biggl{(}\frac{3P_{r}}{3+4R _{d}}\biggr{)}\Bigl{(}-\overline{f}_{i}\overline{\theta}_{1,i}-Q\overline{\theta }_{i}\Bigr{)}+d_{1}\biggl{(}\frac{3P_{r}}{3+4R_{d}}\biggr{)}\bigl{(}-f_{i-1} \theta_{1,i-1}-Q\theta_{i-1}\Bigr{)}\Biggr{\}} \tag{61}\]\[\theta_{1,j}=\theta_{1,j-1}+\overline{h} \Big{\{}a_{1}\Big{(}\big{(}\tfrac{3\mathcal{P}}{3+4\mathcal{R}_{d}} \big{)}\Big{(}-\overline{f}_{1}^{\prime}\overline{\theta}_{1,i}-Q\overline{ \theta}_{i}\Big{)}+b_{1}\Big{(}\tfrac{3\mathcal{P}}{3+4\mathcal{R}_{d}}\Big{)} \big{(}-f_{i-1}\theta_{1,i-1}-Q\theta_{i-1}\big{)}\Big{\}}\] (62) \[+\overline{h}^{2}\{c_{1}\Big{(}\left(\tfrac{3\mathcal{P}}{3+4 \mathcal{R}_{d}}\right)\Big{(}-\overline{f}_{1,i}\overline{\theta}_{1,i}-Q \overline{\theta}_{1,i}\Big{)}+\Big{(}\tfrac{3\mathcal{P}}{3+4\mathcal{R}_{d}} \Big{)}^{2}\Big{(}\overline{f}_{i}^{2}\overline{\theta}_{1,i}+Q\overline{f}_ {i}\overline{\theta}_{i}\Big{)}\Big{)}\] \[+d_{1}\Big{(}\Big{(}\tfrac{3\mathcal{P}}{3+4\mathcal{R}_{d}} \Big{)}\big{(}-f_{1,i-1}\theta_{1,i-1}-Q\theta_{1,i-1}\big{)}+\Big{(}\tfrac{3 \mathcal{P}}{3+4\mathcal{R}_{d}}\Big{)}^{2}\big{(}\overline{f}_{i-1}^{2} \theta_{1,i-1}+Qf_{i-1}\theta_{i-1}\big{)}\Big{)}\}\] \[g_{i}=g_{i-1}+\overline{h}\Big{(}a_{1}\overline{g}_{1,i}+b_{1} \overline{g}_{1,i-1}\Big{)}+\overline{h}^{2}\Big{\{}c_{1}\Big{(}-S_{c} \overline{f}_{i}\overline{g}_{1,i}+S_{c}K\overline{g}_{i}^{\prime}\overline{ h}_{i}^{2}\Big{)}+d_{1}\Big{(}-S_{c}f_{i-1}g_{1,i-1}+S_{c}Kg_{i-1}h_{i-1}^{2} \Big{)}\Big{\}}\] (63) \[g_{1,i}=g_{1,i-1} +\overline{h}\Big{\{}a_{1}\Big{(}-S_{c}\overline{f}_{i}\overline {g}_{1,i}+S_{c}K\overline{g}_{i}^{\prime}\overline{h}_{i}^{2}\Big{)}+b_{1} \big{(}-S_{c}f_{i-1}g_{1,i-1}+S_{c}Kg_{i-1}h_{i-1}^{2}\big{)}\Big{\}}\] (64) \[+\overline{h}^{2}\Big{\{}c_{1}\Big{(}-S_{c}\overline{f}_{1,i} \overline{g}_{1,i}-S_{c}\overline{f}_{i}\Big{(}-S_{c}\overline{f}_{i} \overline{g}_{1,i}+S_{c}K\overline{g}_{i}^{\prime}\overline{h}_{i}^{2}\Big{)}+ S_{c}K\overline{g}_{i}\overline{h}  
solving equations obtained by applying given boundary conditions. If a second- or higher-order boundary value problem is turned into a system of first-order differential equations, one or more initial conditions will be unknown. Using the shooting strategy, the unknown initial conditions are determined by applying the given boundary condition(s). Thus, in this manner, solutions to ordinary differential equations can be found by employing a particular type of finite difference method that applies only to first-order differential equations. Figures 3-14 were constructed by employing the proposed scheme for solving Equations (31)-(36). The discretizations of Equations (37)-(47) by the proposed numerical scheme with given or assumed initial conditions are provided in Equations (48)-(62). Figures 3 and 4 compare the forward Euler and proposed scheme in terms of the absolute error between these schemes and the Matlab solver bvp4c.