## 6. Adomian Decomposition Method for Solving Fuzzy Fractional Equations
\]  
_Putting \(n=0,1\cdots\), solving both systems, and finally, summing all functions, we have_  
\[\underline{\phi}(\xi,r)=\underline{\psi}(\xi,r) =r-1+\frac{4}{\sqrt{\pi}}(r-1)\xi^{\frac{1}{2}}+(4(r-1)\xi+2(r-1))\] \[\quad+(\frac{32}{3\sqrt{\pi}}(r-1)\xi^{\frac{3}{2}}+\frac{16}{ \sqrt{\pi}}(r-1)\xi^{\frac{1}{2}})\] \[\quad+(16(r-1)\frac{\xi^{2}}{2}+8(r-1)(2\xi+1))+\cdots,\]  
_and_  
\[\overline{\phi}(\xi,r)=\overline{\psi}(\xi,r) =1-r+\frac{4}{\sqrt{\pi}}(1-r)\xi^{\frac{1}{2}}+(4(1-r)\xi+2(1-r))\] \[\quad+(\frac{32}{3\sqrt{\pi}}(1-r)\xi^{\frac{3}{2}}+\frac{16}{ \sqrt{\pi}}(1-r)\xi^{\frac{1}{2}})\] \[\quad+(16(1-r)\frac{\xi^{2}}{2}+8(1-r)(2\xi+1))+\cdots.\]  
**Example 3**. : _Consider the system_  
\[ \begin{cases}\frac{c}{\mathbb{P}}D_{0}^{1/2}\phi(\xi)=(\frac{2}{\sqrt{\pi}} \sqrt{\xi}-1)\oplus\phi^{\prime}(\xi),\\ \frac{c}{\mathbb{P}}D_{0}^{1/2}\psi(\xi)=\xi^{2}\oplus\phi^{2}(\xi)\oplus\psi^ {\prime}(\xi),\end{cases} \]  
_where \(\phi(\xi)\geq 0\), with initial conditions_  
\[ \begin{cases}\phi(0)=0,\\ \psi(0)=c=(0,1,1.5),\end{cases} \]  
_where \(c\) is a triangular fuzzy number. In this system, there is a nonlinear expression \(\phi^{2}(\xi)\), so Adomian polynomials are used. Like the previous example, we have lower and upper systems as follows:_\[ \begin{cases}\underline{\phi}_{n+1}(\xi,r)=^{RL}I_{0}^{\frac{1}{2}}\underline{ \phi}_{n}^{\prime}(\xi,r),\\ \underline{\psi}_{n+1}(\xi,r)=-^{RL}I_{0}^{\frac{1}{2}}A_{n}(\xi,r)+\underline{ \psi}_{n}^{\prime}(\xi,r),\end{cases} \]  
_with (IC)_  
\[ \begin{cases}\underline{\phi}_{0}(\xi,r)=^{RL}I_{0}^{\frac{1}{2}}(\frac{2}{ \sqrt{\pi}}\sqrt{\xi}-1),\\ \underline{\psi}_{0}(\xi,r)=r+^{RL}I_{0}^{\frac{1}{2}}(\xi^{2}),\end{cases} \]  
_and_  
\[ \begin{cases}\overline{\phi}_{n+1}(\xi,r)=^{RL}I_{0}^{\frac{1}{2}}\overline{ \phi}_{n}^{\prime}(\xi,r),\\ \overline{\psi}_{n+1}(\xi,r)=-^{RL}I_{0}^{\frac{1}{2}}B_{n}(\xi,r)+\overline{ \psi}_{n}^{\prime}(\xi,r),\end{cases} \]  
_with (IC)_  
\[ \begin{cases}\overline{\phi}_{0}(\xi,r)=^{RL}I_{0}^{\frac{1}{2}}(\frac{2}{ \sqrt{\pi}}\sqrt{\xi}-1),\\ \overline{\psi}_{0}(\xi,r)=1.5-0.5r+^{RL}I_{0}^{\frac{1}{2}}(\xi^{2}),\end{cases} \]  
_where \(A_{n}(\xi,r)\) and \(B_{n}(\xi,r)\) are the Adomian polynomials for functions \(\overline{\phi}_{n}^{2}(\xi,r)\) and \(\underline{\phi}_{n}^{2}(\xi,r)\), respectively. Next, putting \(n=0,1\cdots\), solving both systems, and finally, summing all functions, we have_  
\[\underline{\phi}(\xi,r)=\overline{\phi}(\xi,r)=(\xi-\frac{2}{\sqrt{\pi}}\xi^{ \frac{1}{2}})+(\frac{2}{\sqrt{\pi}}\xi^{\frac{1}{2}}-1)+1+0+0+\cdots=\xi,\]  
_and_  
\[\underline{\psi}(\xi,r) =r+(\frac{16}{15\sqrt{\pi}}\xi^{\frac{5}{2}})+(-\frac{16}{15 \sqrt{\pi}}\xi^{\frac{5}{2}}+\frac{5}{2}\xi^{2}-\frac{16}{3\pi\sqrt{\pi}}\xi^ {\frac{3}{2}})\] \[\quad+(-\frac{5}{2}\xi^{2}+(7+\frac{8}{\pi})\frac{4}{3\sqrt{\pi} }\xi^{\frac{3}{2}}-(\frac{4}{\pi}+2)\xi)+((-7-\frac{4}{\pi})\frac{4}{3\sqrt{ \pi}}\xi^{\frac{3}{2}}\] \[\quad+11\xi+\frac{8}{\pi}\xi-(3+\frac{4}{\pi})\frac{2}{\sqrt{\pi }}\xi^{\frac{1}{2}})+\cdots,\]  
_and_  
\[\overline{\psi}(\xi,r) =(1.5-0.5r)+(\frac{16}{15\sqrt{\pi}}\xi^{\frac{5}{2}})+(-\frac{16 }{15\sqrt{\pi}}\xi^{\frac{5}{2}}+\frac{5}{2}\xi^{2}-\frac{16}{3\pi\sqrt{\pi}} \xi^{\frac{3}{2}})\] \[\quad+(-\frac{5}{2}\xi^{2}+(7+\frac{8}{\pi})\frac{4}{3\sqrt{\pi} }\xi^{\frac{3}{2}}-(\frac{4}{\pi}+2)\xi)+((-7-\frac{4}{\pi})\frac{4}{3\sqrt{ \pi}}\xi^{\frac{3}{2}}\] \[\quad+11\xi+\frac{8}{\pi}\xi-(3+\frac{4}{\pi})\frac{2}{\sqrt{\pi }}\xi^{\frac{3}{2}})+\cdots.\]  
_Note that while summing, each term is simplified by the previous one, so after some steps, all terms will be removed and we only have the constant phrase \(r\) and \(1.5-0.5r\) for \(\underline{\phi}\) and \(\overline{\psi}\), respectively.