## 4 Physical Numerical Applications
\]  
In particular, the CFPS solution of Equations (20) and (21) at \(\alpha=1\) is \(u(x,t)=1+\cosh(2x)e^{-4t}\), which is exactly in accordance with the results obtained by homotopy perturbation method [42], variational iteration method [43], and Adomian decomposition method [44]. In the light of showing the agreement between the exact solutions and CRPS solutions at \(\alpha=1\) of Equations (20) and (21), the absolute and relative errors are listed in Table 11 (Appendix A) for \(n=9\) and at some selected grid points \(\left(x_{i},t_{i}\right)\) in the domain \([0,1]\times[0,1]\) with step-size \(0.1\) for time and space directions. From the table, it can be noted that the CRPS approximate solutions are in good agreement with the exact solutions over the domain of interest. Figure 11 (Appendix B) shows the three-dimensional surface plots of the exact and CRPS solutions at \(\alpha=1\) and \(n=6\) for \(t\in[0,1]\) and \(x\in[-\pi,\pi]\). Figure 12 depicts the surface plots behavior of CRPS solutions of Example 1 at different levels of fractional order \(\alpha\) for each \(t\in[0,1]\) and \(x\in[-\pi,\pi]\) such that \(\alpha=0.5\) and \(\alpha=0.75\). Further, the curves of CRPS solutions for Example 1 at different values of \(\alpha\) for each \(x\in[-\pi,\pi]\) with \(t=0.1\) are shown in Figure 13, where blue, red, green, yellow, and gray indicate the solution curve when \(\alpha=1\), \(0.9\), \(0.8\), \(0.7\), and \(\alpha=0.6\), respectively. From these graphs, it can be indicated that the behavior of the approximate solutions for Equations (20) and (21) smoothly correspond to each other for different values of the fractional order \(\alpha\). **Example 2**. : _Consider the following linear fractional Schrodinger equation with zero trapping potential [45; 46]:_  
\[iT_{t}^{\alpha}u(x,t)+u_{xx}(x,t)+2|v(x,t)|^{2}v(x,t)=0,\ t\geq 0,\ 0<\alpha\leq 1, \tag{30}\]  
_subject to the initial condition_  
\[u(x,0)=e^{ix},x\in\mathbb{R}. \tag{31}\]  
This model arises in the modeling of free particle state, where the particle moves without any potential energy. In particular, the exact solution of this model at \(\alpha=1\) is \(u(x,t)=e^{i(x+t)}\). Evidently, the corresponding coupled system of Equations (30) and (31) is given by  
\[ \begin{array}{ll}T_{t}^{\alpha}v(x,t)+w_{xx}(x,t)+2\big{(}v^{2}(x,t)w(x,t)+ w^{3}(x,t)\big{)}=0,\\ T_{t}^{\alpha}w(x,t)-v_{xx}(x,t)-2\big{(}v^{3}(x,t)+w^{2}(x,t)v(x,t)\big{)}=0, \end{array}  \tag{32}\]  
subject to the following initial conditions  
\[v(x,0)=\cos(x)\ \text{and}\ w(x,0)=\sin(x). \tag{33}\]  
According to CRPS algorithm, the \(n\)th-truncated series solutions are  
\[ \begin{array}{ll}v_{n}(x,t)=\cos(x)+\sum\limits_{k=1}^{n}a_{k}(x)\frac{t^{ \alpha}}{a^{k}k!},\\ w_{n}(x,t)=\sin(x)+\sum\limits_{k=1}^{n}b_{k}(x)\frac{t^{\alpha}}{a^{k}k!},\end{array}  \tag{34}\]  
Hence, the first few CRPS solutions for Equations (30) and (31) are given as follows:  
\[ \begin{array}{ll}u_{1}(x,t)=&\left(\cos(x)-\sin(x)\frac{t^{\alpha}}{\alpha} \right)+i\left(\sin(x)+\cos(x)\frac{t^{\alpha}}{\alpha}\right),\\ u_{2}(x,t)=&\left(\cos(x)-\frac{\sin(x)t^{\alpha}}{\alpha}-\frac{\cos(x)t^{ 2\alpha}}{2a^{2}}\right)+i\left(\sin(x)+\frac{\cos(x)t^{\alpha}}{\alpha}- \frac{\sin(x)t^{2\alpha}}{2a^{2}}\right),\\ u_{3}(x,t)=&\left(\cos(x)-\frac{\sin(x)t^{\alpha}}{\alpha}-\frac{\cos(x)t^{ 2\alpha}}{2a^{2}}+\frac{\sin(x)t^{3\alpha}}{3t^{3}}\right)\\ &\qquad+i\left(\sin(x)+\frac{\cos(x)t^{\alpha}}{a}-\frac{\sin(x)t^{2\alpha}}{ 2a^{2}}-\frac{\cos(x)t^{3\alpha}}{3t^{3}}\right),\\ u_{4}(x,t)=&\left(\cos(x)-\frac{\sin(x)t^{\alpha}}{\alpha}-\frac{\cos(x)t^{ 2\alpha}}{2a^{2}}+\frac{\sin(x)t^{3\alpha}}{3t^{3}}+\frac{\cos(x)t^{\alpha}}{ 4t^{\alpha}}\right)\\ &\qquad+i\left(\sin(x)+\frac{\cos(x)t^{\alpha}}{a}-\frac{\sin(x)t^{2\alpha}}{ 2a^{2}}-\frac{\cos(x)t^{3\alpha}}{3t^{3}}+\frac{\sin(x)t^{4\alpha}}{4!a^{4}} \right),\end{array}  \tag{38}\]  
By continuing this procedure, the multiple CFPS solution \(u(x,t)\) in term of infinite series can be obtained by  
\[