## 3 Numerical Method  
### Formulation of OHAM  
The fundamental concept of optimal homotopy asymptotic method (OHAM) is explained in [43; 52; 53; 54]. The governing differential equation will be of the form:  
\[D(v(\xi))+\omega(\xi)=0,\quad\xi\in\Pi \tag{14}\]\[E(v)=0. \tag{15}\]  
where \(\xi\) is the independent variable, \(\Pi\) is domain, \(v(\xi)\) is unknown function, and \(\omega(\xi)\) is a known function. \[D(v)=P(v)+U(v), \tag{16}\]  
where, \(P\) and \(U\) are the linear and non-linear components, respectively. It is important to note that it does not need to obtain the first solution (initial guess), because we have a great freedom to choose the linear component from the model. Substituting Equation (16) in Equation (14) we obtain,  
\[P(v(\xi))+\omega(\xi)+U(v(\xi)))=0, \tag{17}\]  
By applying OHAM on Equation (17), we obtain  
\[(1-q)[P(\Phi(\xi,q))+\omega(\xi)-H(q)[P(\Phi(\xi,q))-\omega(\xi)+U(\Phi(\xi,q) )]^{\prime}, \tag{18}\]  
\[E(\Phi(\xi,q))=0.\]  
where \(q\in[0,1]\) is embedding parameter and non-zero auxiliary function at \(q\
eq 0\) is \(H(q)\). \(\Phi(\xi,q)\) is an unknown function. Therefore, we have  
\[q=0\to H(\Phi(\xi;0),0)=E(\Phi(\xi;0))+\omega(\xi)=0, \tag{19}\]  
\[q=1\to H(\Phi(\xi;1),1)=H(1)E(\Phi(\xi;1))+\omega(\xi)=0 \tag{20}\]  
when \(q=0\) and \(q=1\), it satisfies that \(\Phi(\xi;0)=v_{0}(\xi))\) and \(\Phi(\xi;1)=v_{n}(\xi)\), respectively. Hence, as q varies from 0 to 1 the \(\Phi(\xi;q)\) varies from \(v_{0}(\xi)\) to \(v_{1}(\xi)\). At \(q=0\), we achieve, using Equations (14) and (15),  
\[P(v_{0}(\xi))+\omega(\xi)=0,\quad E(v_{0})=0 \tag{21}\]  
Auxiliary function will be of the form:  
\[H(q)=qC_{1}+q^{2}C_{2}+q^{3}C_{3}+......, \tag{22}\]  
where \(C_{1},C_{2},C_{3},\dots\) are constants to be evaluated. By applying [PERSON]'s series on \(\Phi(\xi;q)\) with respect to \(q\) for expansion, we obtain  
\[E(v)=0. \tag{15}\]0  
Comparing same exponents of \(q\)-terms from Equations (24) and (17), we obtain the following coefficients:  
\[E(v)=0. \tag{15}\]1  
\[E(v)=0. \tag{15}\]2  
\[E(v)=0. \tag{15}\]3  
where \(U_{n-i}(v_{0}(\xi),v_{1}(\xi),...,v_{n-i}(\xi))\) is the coefficient of \(q^{n-i}\) expansion of \(U(\Phi(\xi;q))\) and it gives that,  
\[E(v)=0. \tag{15}\]4  
Convergence of the Equation (23) depends on the values of auxiliary constants \(C_{1},C_{2},\dots\). For \(q=1\),  
\[E(v)=0. \tag{15}\]5From Equations (27) and (17), we obtain the residual expansion as,  
\[E(v)=0. \tag{15}\]6  
\(\dot{\sigma}(\xi;C_{i})\) obtains the exact solution if \(R(\xi;C_{i})=0\). However, it is not always true in case of non-linear systems. For the determinations of auxiliary constants \(C_{1},C_{2},C_{3},\dots\) Ritz, Galerkin, Least Squares, and Collocation methods can be used. Least squares method is the most commonly used method to evaluate the values of constants. Optimization of error is performed by taking the square of the residuals over the given domain to obtain,  
\[E(v)=0. \tag{15}\]7  
where \(j_{1}\) and \(j_{2}\) are the values that depend on the given system. The optimal values of \(C_{1},C_{2},C_{3},\dots\) can be evaluated with minimum error from:  
\[E(v)=0. \tag{15}\]8  
Solving the above system of algebraic equations obtained in Equation (30), we obtain the values of \(C_{1},C_{2},C_{3},\dots\). Replacing the values of \(C_{1},C_{2},C_{3},\dots,C_{i}\) in Equation (23), we obtain the approximate solution. Figure 2 illustrates the process of modeling and solution of unsteady Hiemenz flow of nanofluid over porous wedge problem using OHAM.