## 4 Physical Numerical Applications
\begin{array}{ll}u(x,t)=&\left(\cos(x)-\frac{\sin(x)t^{\alpha}}{\alpha}- \frac{\cos(x)t^{2\alpha}}{2!a^{2}}+\frac{\sin(x)t^{3\alpha}}{3!a^{3}}+\frac{ \cos(x)t^{4\alpha}}{4!a^{4}}+\ldots\right)\\ &\qquad+i\left(\sin(x)+\frac{\cos(x)t^{\alpha}}{a}-\frac{\sin(x)t^{2\alpha}}{ 2!a^{2}}-\frac{\cos(x)t^{3\alpha}}{3!a^{3}}+\frac{\sin(x)t^{4\alpha}}{4!a^{4} }+\ldots\right)\\ &=\left(\cos(x)+i\sin(x)\right)\left(\begin{array}{cc}\sum\limits_{n=0}^{ \infty}\frac{1}{n!}&\left(\frac{t^{\alpha}}{a}\right)^{n}\end{array}\right). \end{array} \]  
In particular, the CFPS solution of Equations (30) and (31) at \(\alpha=1\) is \(u(x,t)=e^{i(x+\frac{t^{\alpha}}{\alpha})}\), which is fully compatible with solutions found in other existing methods [42; 43; 44]. To show the geometric behaviors of the fourth CFPS approximations of Equations (30) and (31), the numerical comparison of exact and approximate solutions at different values of \(\alpha\) such that \(\alpha\in\{1,0.9,0.8,0.7\}\) is presented in Table 11 for each value of \(t_{i}\) in \([0,1]\) with step-size \(0.2\) when \(x=0.1\) and \(n=5\). From these numerical comparisons, it can be concluded that the results obtained by the proposed method are excellent in comparison with the exact solution. For further analysis, the surface plots of the exact \(v(x,t)\) and \(w(x,t)\) of Example 2 and fourth CRPS solutions \(v_{4}(x,t)\) and \(w_{4}(x,t)\) at different values of fractional order \(\alpha\) for each \(t\in[0,2]\) and \(x\in[-2\pi,2\pi]\) are illustrated in Figures 12 and 13, respectively. From these figures, it can be seen that the behavior of the fourth CRPS approximate solutions is in good agreement with each other at different values of \(\alpha\). According to CRPS algorithm, the \(n\)th-truncated series solutions are  
\[ \begin{array}{l}v_{n}(x,t)=\sin(x)+\sum\limits_{k=1}^{n}a_{k}(x)\frac{t^{k \alpha}}{a^{k}k!},\\ w_{n}(x,t)=\sum\limits_{k=1}^{n}b_{k}(x)\frac{t^{k\alpha}}{a^{k}k!},\end{array}  \tag{43}\]  
and the \(n\)th-residual functions are  
\[Res_{v}^{n}(x,t)=T_{t}^{\alpha}v_{n}(x,t)+\frac{1}{2}\big{(}w_{n}\big{)}_{xx}( x,t)-cos^{2}(x)w_{n}(x,t)-\big{(}v_{n}^{2}(x,t)w_{n}(x,t)+w_{n}^{3}(x,t) \big{)},\]  
\[Res_{w}^{n}(x,t)=T_{t}^{\alpha}v_{n}(x,t)-\frac{1}{2}(v_{n})_{xx}(x,t)+cos^{2} (x)v_{n}(x,t)+\big{(}w_{n}^{2}(x,t)v_{n}(x,t)+v_{n}^{3}(x,t)\big{)}.\]  
By applying the former iteration process of CRPS algorithm, the first few terms of Equation (43) are given as follows:  
\[ \begin{array}{l}a_{1}(x)=0,\ b_{1}(x)=-\frac{3}{2}\sin(x),\\ a_{2}(x)=-\frac{9}{4}\sin(x),\ b_{2}(x)=0,\\ a_{3}(x)=0,\ b_{3}(x)=\frac{27}{8}\sin(x),\end{array} \]  
Hence, the first few CRPS solutions for Equations (39) and (40) are given as follows:  
\[ \begin{array}{l}u_{1}(x,t)=\sin(x)-\frac{3i}{2}\sin(x)\frac{t^{\alpha}}{a},\\ u_{2}(x,t)=\left(\sin(x)-\frac{9}{8}\frac{\sin(x)t^{2\alpha}}{a^{2}}\right)- \frac{3i}{2}\sin(x)\frac{t^{\alpha}}{a},\\ u_{3}(x,t)=\left(\sin(x)-\frac{9}{8}\frac{\sin(x)t^{2\alpha}}{a^{2}}\right)+i \big{(}-\frac{3}{2}\frac{\sin(x)t^{\alpha}}{a}+\frac{9}{16}\frac{\sin(x)t^{3 \alpha}}{a^{3}}\big{)},\end{array}  \tag{44}\]  
in which the multiple CFPS solution \(u(x,t)\) in term of infinite series can be obtained by  
\[ \begin{array}{l}u(x,t)=\left(\sin(x)-\frac{9}{8}\frac{\sin(x)t^{2\alpha}}{ a^{2}}+\ldots\right)+i\big{(}-\frac{3}{2}\frac{\sin(x)t^{\alpha}}{a}+\frac{9}{16} \frac{\sin(x)t^{3\alpha}}{a^{3}}+\ldots\big{)}\\ =\sin(x)\bigg{(}\left(1-\frac{1}{2i}\left(\frac{3t^{\alpha}}{2a}\right)^{2}+ \ldots\right)-i\left(\left(\frac{3t^{\alpha}}{2a}-\frac{1}{3i}\left(\frac{3t^ {\alpha}}{2a}\right)^{3}+\ldots\right)\right)\\ =\sin(x)\bigg{(}\sum\limits_{n=0}^{\infty}\frac{1}{(2n)!}\ \left(\frac{3t^{ \alpha}}{2a}\right)^{2n}-i\sum\limits_{n=0}^{\infty}\frac{1}{(2n+1)!}\ \left(\frac{3t^{ \alpha}}{2a}\right)^{2n+1}\bigg{)}.\end{array} \]  
In particular, the CFPS solution of Equations (39) and (40) at \(\alpha=1\) is \(u(x,t)=\sin(x)e^{-3 it/2}\), which is fully compatible with solutions found in other existing methods [42; 43; 44].