## 2 Formulation of the Problem
Using Equation (11), Equations (7)-(13) take the form of  
\[f^{\prime\prime\prime}+ff^{\prime\prime}-{f^{\prime}}^{2}-\lambda\Big{(}f^{ \prime}+\frac{\eta}{2}f^{\prime\prime}\Big{)}-\beta\Big{(}{f^{\prime}}^{2}{f^{ \prime\prime\prime}}-2 ff^{\prime}f^{\prime\prime\prime}\Big{)}-Mf^{\prime}-MEI=0, \tag{16}\]  
\[(1+\frac{4}{3}Rd)\theta^{\prime\prime}+\Pr\Big{[}\big{(}f\theta^{\prime}-f^{ \prime}\theta\big{)}-\lambda\Big{(}\theta+\frac{\eta}{2}\theta^{\prime}\Big{)} \Big{]}+\big{(}Af^{\prime}+B\theta\big{)}=0, \tag{17}\]  
\[q^{\prime\prime\prime}=\frac{\kappa L_{w}(x,t)}{xv}\big{(}A(T_{w}-T_{\infty}) f^{\prime}+(T-T_{\infty})B\big{)}, \tag{9}\]0The relevant boundary conditions are reduced to the form:  
\[ \begin{array}{l}f=f_{w},f^{\prime}=1,\,\theta=1,\,\phi=1\ \ \ at\,\eta=0,\\ f^{\prime}=0,\theta=0,\,\phi=0\ \ at\,\eta\rightarrow\infty.\end{array}  \tag{19}\]  
The dimensional physical parameters after simplification are defined as:  
\[ \begin{array}{l}\lambda=\frac{c}{a},\,\beta=\lambda_{1}\left(\frac{a}{1-ct} \right),M=\frac{\sigma B_{0}{}^{2}}{\rho a}(1-\alpha t),EI=\frac{E_{0}}{axB_{0 }}(1-\alpha t),Nt=-\frac{k_{1}(T_{w}-T_{\infty})}{T_{r}},\\ Sc=\frac{\
u}{D_{m}},Rd=\frac{4T_{w}^{3}\sigma^{*}}{3 kK^{*}}.\end{array}  \tag{20}\]  
where Pr represents the Prandtl number, \(EI\) is the electric parameter, \(Rd\) represents the radiation parameter, \(\lambda\) is the unsteadiness parameter, \(M\) is the magnetic parameter, \(Nt\) is the thermophoresis parameter, \(Sc\) is Schmidt number and \(\beta\) is the Deborah number. **3. Physical Quantities of Interest**  
The physical quantities of interest for the governing Maxwell nanofluid flow problem are Skin friction \(C_{f}\), local Nusselt number \(Nu_{x}\), and local Sherwood number \(Sh_{x}\), which are defined as:  
\[C_{f}=\frac{Nt}{\rho U_{w}^{2}/2},\,\,Nu_{x}=\frac{xq_{w}}{\kappa(T_{w}-T_{ \infty})},\,\,Sh_{x}=\frac{xJ_{w}}{D_{m}(C_{w}-C_{\infty})}, \tag{21}\]  
where \(J_{w}\), \(Nt\) and \(q_{w}\) are the mass flux, surface heat flux and wall shear stress, respectively. Substituting Equation (11) into Equation (21), the dimensionless form of these main design quantities are:  
\[C_{f}=\frac{f^{\prime\prime}(0)}{\text{Re}_{x^{1/2}}},\,\,Nu_{x}=-(1+\frac{4} {3}Rd)\theta^{\prime}(0)\text{Re}_{x^{1/2}},\,\,Sh_{x}=-\phi^{\prime}(0)\text {Re}_{x^{1/2}}, \tag{22}\]  
**4. Solution by HAM**  
[PERSON] [40; 41; 42] worked for the first time on HAM (Homotopy analysis method). He used the idea of Homotopy from in topology and introduced HAM. When one of the functions is not clearly in another, then this is a homotopic function. Consider two continuous functions \(f_{1}\), \(f_{2}\) and two topological spaces \(X\), \(Y\). Wherever the two functions map from \(X\) to \(Y\), \(f_{1}\) is known as homotopic to \(f_{2}\) if it produces continuous function  
\[\psi:X\times[0,\,1]\to Y. \tag{23}\]  
Such that \(\forall,\,x\in\text{X}\)  
\[\psi[x,\,0]=f_{1}(x)\,\&\,\psi[x,\,1]=f_{2}(x). \tag{24}\]  
Then, the mapping is called homotopic. HAM is an alternative method and it is used to solve non-linear differential equation not including linearization and discretization. To solve Equations (16)-(18), the Homotopy Analysis Method (HAM) is used. HAM utilizes the suitable boundary condition in Equation (19) successfully. The given explanations exhibit the supplementary parameters \(\hbar\) which control and normalize the convergence of this very solution.