## 4 Physical Numerical Applications
 \tag{27}\]Consequently, by operating \(T_{t}^{\alpha}\) on both sides of Equation (27), using the fact that \(T_{t}^{\alpha}t^{p}=0\) for \(p>\alpha\) at \(t=0\) and equating the resulting equations to zero at \(t=0\) yields  
\[T_{t}^{\alpha}Res_{v}^{2}(x,t)_{|t=0} =a_{2}(x)+\left(16\cosh(2x)-\frac{b_{2}^{\alpha}(x)t^{\alpha}}{ \alpha}\right)_{|t=0}=0,\] \[T_{t}^{\alpha}Res_{w}^{2}(x,t)_{|t=0} =b_{2}(x)+\left(\frac{a_{2}^{\alpha}(x)t^{\alpha}}{\alpha}\right) _{|t=0}=0.\]  
That is, \(a_{2}(x)=-16\cosh(2x)\) and \(b_{2}(x)=0\). Therefore, the second CPS approximate solutions of Equations (22) and (23) are  
\[v_{2}(x,t)=1+\cosh(2x)-8\cosh(2x)\frac{t^{2\alpha}}{\alpha^{2}}\text{ and }w_{2}(x,t)=-4\cosh(2x)\frac{t^{\alpha}}{\alpha}.\]  
For \(n=3\), substitute the third truncated series solutions \(v_{3}(x,t)=1+\cosh(2x)-8\cosh(2x)\frac{t^{2\alpha}}{\alpha^{2}}+a_{3}(x)\frac {t^{3\alpha}}{3t^{3\alpha}}\) and \(w_{3}(x,t)=-4\cosh(2x)\frac{t^{\alpha}}{\alpha}+b_{3}(x)\frac{t^{3\alpha}}{3t ^{3\alpha}}\) into the third truncated residual functions \(Res_{v}^{3}(x,t)\) and \(Res_{w}^{3}(x,t)\) such that  
\[Res_{v}^{3}(x,t)=\left(\frac{-8\cosh(2x)t^{\alpha}}{\alpha}+\frac{a_{3}(x)t^{ 2\alpha}}{2a^{2}}\right)+\left(\frac{16\cosh(2x)t^{\alpha}}{\alpha}-\frac{b_{ 3}^{\alpha}(x)t^{3\alpha}}{3t^{3\alpha}}\right), \tag{28}\]  
Consequently, by operating \(T_{t}^{\alpha}\) twice on both sides of Equation (28) and equating the resulting equations to zero at \(t=0\) such that  
\[T_{t}^{2\alpha}Res_{v}^{3}(x,t)_{|t=0} =a_{3}(x)-\left(\frac{b_{3}^{\alpha}(x)t^{\alpha}}{\alpha}\right) _{|t=0}=0,\] \[T_{t}^{2\alpha}Res_{w}^{3}(x,t)_{|t=0} =b_{3}(x)-\left(64\cosh(2x)+\frac{a_{3}^{\alpha}(x)t^{\alpha}}{ \alpha}\right)_{|t=0}=0.\]  
That is, \(a_{3}(x)=0\) and \(b_{3}(x)=64\cosh(2x)\). Therefore, the third FPS approximate solutions of Equations (22) and (23) are  
\[v_{3}(x,t)=1+\cosh(2x)-8\cosh(2x)\frac{t^{2\alpha}}{\alpha^{2}},\] \[w_{3}(x,t)=-4\cosh(2x)\frac{t^{\alpha}}{\alpha}+32\cosh(2x)\frac {t^{3\alpha}}{3t^{3\alpha}}.\]  
As the former, the fourth unknown coefficients \(a_{4}(x)\) and \(b_{4}(x)\) can be obtained using the same procedure of CRPS with the help of the fact that \(T_{t}^{3\alpha}Res_{v}^{4}(x,0)=T_{t}^{3\alpha}Res_{w}^{4}(x,0)=0\) such that \(a_{4}(x)=256\cosh(2x)\) and \(b_{4}(x)=0\). Thus, the fourth FPS approximate solutions of Equations (22) and (23) are  
\[v_{4}(x,t)=1+\cosh(2x)-8\cosh(2x)\frac{t^{2\alpha}}{\alpha^{2}}+32\cosh(2x) \frac{t^{4\alpha}}{3\alpha^{4}},\] \[w_{4}(x,t)=-4\cosh(2x)\frac{t^{\alpha}}{\alpha}+32\cosh(2x)\frac {t^{3\alpha}}{3\alpha^{3}}.\]  
Hence, the first few CRPS solutions for Equations (20) and (21) are given as follows:  
\[T_{t}^{\alpha}Res_{w}^{2}(x,t)_{|t=0} =b_{2}(x)+\left(\frac{a_{2}^{\alpha}(x)t^{\alpha}}{\alpha}\right) _{|t=0}=0.\]0 \[T_{t}^{\alpha}Res_{w}^{2}(x,t)_{|t=0} =b_{2}(x)+\left(\frac{a_{2}^{\alpha}(x)t^{\alpha}}{\alpha}\right) _{|t=0}=0.\]1 \[T_{t}^{\alpha}Res_{w}^{2}(x,t)_{|t=0} =b_{2}(x)+\left(\frac{a_{2}^{\alpha}(x)t^{\alpha}}{\alpha}\right) _{|t=0}=0.\]2 \[T_{t}^{\alpha}Res_{w}^{2}(x,t)_{|t=0} =b_{2}(x)+\left(\frac{a_{2}^{\alpha}(x)t^{\alpha}}{\alpha}\right) _{|t=0}=0.\]3by continuing this procedure, the multiple CFPS solution \(u(x,t)\) in term of infinite series can be obtained by  
\[ \begin{array}{ll}u(x,t)&=\left(1+\cosh(2x)-\frac{8\cosh(2x)}{a^{2}}t^{2a}+\frac {32\cosh(2x)}{3a^{4}}t^{4a}+\ldots\right)\\ &\qquad\qquad+i\left(\frac{-4\cosh(2x)}{a}t^{a}+\frac{32\cosh(2x)}{3a^{3}}t^{3 a}-\frac{12\cosh(2x)}{15a^{3}}t^{5a}+\ldots\right)\\ &=1+\cosh(2x)\bigg{(}\left(1-\frac{4^{2}t^{2a}}{2t^{2a}}+\frac{4^{4}t^{4a}}{4 ^{4}t^{4}}+\ldots\right)-i\left(\frac{4^{2}t^{2a}}{a}-\frac{4^{2}t^{2a}}{3a^{4} }+\frac{4^{5}t^{5a}}{5 ta^{5}}-\ldots\right)\bigg{)}\\ &=1+\cosh(2x)\bigg{(}\sum\limits_{n=0}^{\infty}\frac{(-1)^{n}}{(2n)!}\ \left(\frac{4a^{n}}{a}\right)^{2n}-i\left(\sum\limits_{n=0}^{\infty}\frac{(-1) ^{n}}{(2n+1)!}\ \left(\frac{4a^{n}}{a}\right)^{2n+1}\right)\bigg{)}.\end{array}