## 2 Problem Formulation  
The thermal features of a nanofluid in the presence of kerosene oil and water base particles were studied. An inclined surface flow containing a uniform suspension of copper oxide and silver nanoparticles was considered. The magnetic force was normally implemented with an angle of inclination, \(\theta\). The impact of a porous medium was subjected to the flat surface (Figure 1). The suspension of nanoparticles and base materials with a uniform (50-50%) ratio was taken into consideration. The surface was further supported with slip constraints to control the fluid velocity. The flow in the stationary plate was induced after time \(t>0\). The governing model is presented in terms of the following mathematical expressions [26; 27; 28]:  
\[\begin{array}{ll}\rho_{nf}\frac{\partial w(\xi,t)}{\partial t}=\mu_{nf}& \left(1+\alpha_{1}\frac{\partial}{\partial t}\right)\frac{\partial^{2}w(\xi,t )}{\partial\xi^{2}}+g(\rho\beta_{T})_{nf}[T(\xi,t)-T_{\infty}] Cos(\delta)\\ &-\frac{\mu_{nf}\partial}{K}w(\xi,t)-\sigma_{nf}B_{o}^{2}Sin(\theta)w(\xi,t) ;\qquad\xi,t>0\end{array} \tag{1}\]  
\[\begin{array}{ll}\left(\rho C_{p}\right)_{nf}\frac{\partial T(\xi,t)}{ \partial t}=-\frac{\partial q}{\partial\xi}\quad;\qquad\xi,t>0\end{array} \tag{2}\]  
\[\begin{array}{ll}q(\xi,t)=-k_{nf}\frac{\partial T(\xi,t)}{\partial\xi} \end{array} \tag{3}\]  
The boundary conditions for the fractional model are [26; 27; 28]:  
\[w(\xi,0)=0,\qquad T(\xi,0)=T_{\infty}\qquad;\quad\xi>0 \tag{4}\]  
\[\begin{array}{ll}w(0,t)-b\left.\frac{\partial w(\xi,t)}{\partial\xi}\right|_ {\xi=0}&=\frac{\xi(t)}{\mu},\qquad T_{(0,t)}\\ &=\left\{\begin{array}{ll}T_{\infty}+(T_{\infty}-T_{\infty})\frac{t}{t_{o}},&0\leq t\leq t_{o}\\ &\\ T_{\infty},&t>t_{o}\end{array}\right.\\ w(\xi,t)\to 0,\qquad T(\xi,t)\to T_{\infty}\quad;\quad\xi\to\infty,\ t>0 \end{array} \tag{5}\]  
The problem was entertained with following new variables [26; 27; 28]:  
\[\begin{array}{ll}\xi^{*}=\frac{\xi v_{o}}{v_{f}}\quad,\quad w^{*}=\frac{w}{ U_{o}}\quad,\quad t^{*}=\frac{\xi^{2}_{o}t}{v_{f}}\quad,\quad\phi^{*}=\frac{T-T_{ \infty}}{T_{\infty}-T_{\infty}}\\ b^{*}=\frac{h}{k}b,\quad q^{*}=\frac{4}{q_{o}},\quad q_{o}=\frac{k_{nf}(T_{ \infty}-T_{\infty})v_{o}}{v_{f}}\quad,\quad s^{*}(t^{*})=\frac{1}{\mu}\sqrt{ \frac{t_{o}}{v_{f}}}f(t_{o}t^{*})\end{array}\]  
Figure 1: Flow configuration.  
where the AB operator is \({}^{AB}\mathfrak{D}_{t}^{\beta}\). The Mittage-Leffer function, \(E_{\beta}(z)\), is explained as:  
\[E_{\beta}(z)=\sum_{r=0}^{\infty}\frac{z^{\beta}}{\Gamma(r\beta+1)}\qquad\quad; \qquad\quad\quad 0<\beta<1,\;\;z\in\mathbb{C}\]  
**Definition 2**.: _For Equation (13), the definition of the Laplace transform is:_  
\[\mathcal{L}\Big{\{}^{AB}\mathfrak{D}_{t}^{\beta}g(\xi,t)\Big{\}}=\frac{q^{ \beta}\mathcal{L}[g(\xi,t)]-q^{\beta-1}g(\xi,0)}{(1-\beta)q^{\beta}+\beta} \tag{14}\]  
and  
\[\lim_{\beta\to 1}{}^{AB}\mathfrak{D}_{t}^{\beta}g(\xi,t)=\frac{\partial g(\xi,t)}{ \partial t}\]  
**Definition 3**.: _For any function \(g(\xi,t)\), the definition of the Caputo fractional (CF) is associated with the following definition:_  
\[{}^{CF}\mathfrak{D}_{t}^{\alpha}g(\xi,t)=\frac{1}{1-\alpha}\int_{0}^{t}exp \bigg{(}\frac{\alpha(1-\tau)}{1-\alpha}\bigg{)}g^{\prime}(\xi,t)d\tau 0< \alpha<1 \tag{15}\]  
where \({}^{CF}\mathfrak{D}_{t}^{\alpha}\) is the Caputo-Fabrizio operator.  
**Definition 4**.: _The Laplace transformation for CF is:_  
\[\mathcal{L}\Big{\{}^{CF}\mathfrak{D}_{t}^{\alpha}g(\xi,t)\Big{\}}=\frac{s \mathcal{L}[g(\xi,t)]-g(\xi,0)}{(1-\alpha)s+\alpha} \tag{16}\]  
and with the Laplace-transformed variable \(s\) of time \(t\) for CF, we obtain:  
\[\lim_{\alpha\to 1}{}^{CF}\mathfrak{D}_{t}^{\alpha}g(\xi,t)=\frac{\partial g(\xi,t)}{ \partial t}\]  
**4. Solution by Atangana-Baleanu (AB) Operator**  
The implementation of the AB fractional model with operator \({}^{AB}\mathfrak{D}_{t}^{\beta}\) is suggested for Equation (9) as:  
\[q(\xi,t)=-\Lambda_{4}{}^{AB}\mathfrak{D}_{t}^{\beta}\frac{\partial\theta(\xi, t)}{\partial\xi} \tag{17}\]