## 5 Application of Laplace Transforms to Solve Fractional Differential Systems  
In this section, we consider systems (1)-(3) using the Laplace transform. Suppose the r-level of \(\phi,\psi\) are \([\phi]_{r}=[\underline{\phi}_{r},\overline{\phi}_{r}]\) and \([\psi]_{r}=[\underline{\psi},\overline{\psi}_{r}]\). Now, we write the system and its initial condition with respect to its r-level, and we have  
\[ \begin{cases}\xi^{c}_{H}D^{l}[\underline{\phi}_{r}(\xi),\overline{\phi}_{r}( \xi)]=[\gamma_{1}(\xi,\underline{\phi},(\xi),\underline{\psi}_{r}(\xi))+ \underline{\phi}^{\prime}_{r}(\xi),\gamma_{1}(\xi,\overline{\phi}_{r}(\xi), \overline{\psi}_{r}(\xi))+\overline{\phi}^{\prime}_{r}(\xi)],\\ \xi^{c}_{H}D^{l}[\underline{\psi}_{r}(\xi),\overline{\psi}_{r}(\xi)]=[\gamma_ {2}(t,\underline{\phi}_{r}(\xi),\underline{\psi}_{r}(\xi))+\underline{\psi}^{ \prime}_{r}(\xi),\gamma_{2}(\xi,\overline{\phi}_{r}(\xi),\overline{\psi}_{r}( \xi))+\overline{\psi}^{\prime}_{r}(\xi)],\end{cases} \]  
and  
\[ \begin{cases}[\underline{\phi}_{r}(0),\overline{\phi}_{r}(0)]=[\underline{ \phi}_{0},\overline{\phi}_{0}],\\ \\ [\underline{\psi}_{r}(0),\overline{\psi}_{r}(0)]=[\underline{\psi}_{0}, \overline{\psi}_{0r}],\end{cases} \]  
which are equivalent to the systems  
\[ \begin{cases}\xi^{c}_{H}D^{l}\underline{\phi}_{r}(\xi)=\gamma_{1}(\xi, \underline{\phi}_{r}(\xi),\underline{\psi}_{r}(\xi))+\underline{\phi}^{ \prime}_{r}(\xi),\\ \\ \xi^{c}_{H}D^{l}\underline{\psi}_{r}(\xi)=\gamma_{2}(\xi,\underline{\phi}_{r}( \xi),\underline{\psi}_{r}(\xi))+\underline{\psi}^{\prime}_{r}(\xi),\end{cases} \]  
with the conditions  
\[ \begin{cases}\underline{\phi}_{r}(0)=\underline{\phi}_{0},\\ \\ \underline{\psi}_{r}(0)=\underline{\psi}_{0},\end{cases} \]  
and  
\[ \begin{cases}\xi^{c}_{H}D^{l}\overline{\phi}_{r}(\xi)=\gamma_{1}(\xi, \overline{\phi}_{r}(\xi),\overline{\psi}_{r}(\xi))+\overline{\phi}^{\prime}_{ r}(\xi),\\ \\ \xi^{c}_{H}D^{l}\overline{\psi}_{r}(\xi)=\gamma_{2}(\xi,\overline{\phi}_{r}( \xi),\overline{\psi}_{r}(\xi))+\overline{\psi}^{\prime}_{r}(\xi),\end{cases} \]  
with conditions  
\[ \begin{cases}\overline{\phi}_{r}(0)=\overline{\phi}_{0},\\ \\ \overline{\psi}_{r}(0)=\overline{\psi}_{0}.\end{cases} \]  
Solving these systems using the Laplace transform and then applying the stacking lemma, we have the integral solution of problem (4). **Example 1**. : _Consider the system_  
\[ \begin{cases}\xi^{c}_{H}D^{\frac{1}{2}}\phi(\xi)=\psi(\xi)+\phi^{\prime}(\xi),\\ \\ \xi^{c}_{H}D^{\frac{1}{2}}\psi(\xi)=\psi^{\prime}(\xi),\end{cases} \]with conditions_  
\[ \begin{cases}\phi(0)=1,\\ \psi(0)=1.\end{cases} \]  
_Suppose the r-level of \(\phi,\psi\) are \([\phi]_{r}=[\underline{\phi},\overline{\phi}_{r}]\) and \([\psi]_{r}=[\underline{\psi},\overline{\psi}_{r}]\). Then, we have the systems_  
\[ \begin{cases}\coth_{SH}D^{\frac{1}{2}}\underline{\phi}_{r}(\xi)=\underline{\psi }_{r}(\xi)+\underline{\phi}^{\prime}_{r}(\xi),\\ \\ \coth_{SH}D^{\frac{1}{2}}\underline{\psi}_{r}(\xi)=\underline{\psi}^{\prime}_{r }(\xi),\end{cases} \]  
_with conditions_  
\[ \begin{cases}\coth_{r}(0)=1,\\ \\ \underline{\psi}_{r}(0)=1,\end{cases} \]  
_and_  
\[ \begin{cases}\coth_{SH}D^{\frac{1}{2}}\overline{\phi}_{r}(\xi)=\overline{\psi }_{r}(\xi)+\overline{\phi}^{\prime}_{r}(\xi),\\ \\ \coth_{SH}D^{\frac{1}{2}}\overline{\psi}_{r}(\xi)=\overline{\psi}^{\prime}_{r }(\xi),\end{cases} \]  
_with conditions_  
\[ \begin{cases}\overline{\phi}_{r}(0)=1,\\ \\ \overline{\psi}_{r}(0)=1,\end{cases} \]  
_and note that our initial conditions do not have uncertain parts. We only solve the first system. The solution of the second system is similar. Applying the Laplace transform, we have_  
\[ \begin{cases}\mathds{L}\Big{\{}\coth_{SH}D^{\frac{1}{2}}\underline{\phi}_{r}( \xi)\Big{\}}=\mathds{L}\Big{\{}\underline{\psi}_{r}(\xi)\Big{\}}+\mathds{L} \Big{\{}\underline{\phi}^{\prime}_{r}(\xi)\Big{\}},\\ \\ \mathds{L}\Big{\{}\coth_{SH}D^{\frac{1}{2}}\underline{\psi}_{r}(\xi)\Big{\}}= \mathds{L}\Big{\{}\underline{\psi}^{\prime}_{r}(\xi)\Big{\}}.\end{cases}