# Fractional View Analysis of Acoustic Wave Equations, Using Fractional-Order Differential Equations
## 3 The Procedure of VHPTM
### Example  
We consider time fractional-order non-linear GRLW equation  
\[\frac{\partial^{\beta}v}{\partial\eta^{\beta}}+\frac{\partial v}{\partial\xi}+v \frac{\partial v}{\partial\xi}-\frac{\partial\partial^{2}v}{\partial\eta \partial\xi^{2}}=0,\qquad 0<\xi\leq 1,\qquad 0<\beta\leq 1,\quad\eta>0, \tag{20}\]  
with initial condition  
\[v(\xi,0)=3\alpha\sec h^{2}(\beta\xi),\quad\alpha>0,\quad\beta=\frac{1}{2}\sqrt {\frac{\alpha}{1+\alpha}}. \tag{21}\]  
By using Equation (12), the fractional PDE given in Equation (20) can be written as  
\[\sum_{j=0}^{\infty}p^{j}v_{j+1}(\xi,\eta)=\sum_{j=0}^{\infty}p^{j}v_{j}(\xi, \eta)+p^{j}\xi^{-1}\left[\lambda(s)\xi\left\{s^{\beta}\frac{\partial v_{j}(\xi, \eta)}{\partial\eta}+\frac{\partial v_{j}(\xi,\eta)}{\partial\xi}+v_{j}(\xi, \eta)\frac{\partial v_{j}(\xi,\eta)}{\partial\xi}-\frac{\partial\partial^{2}v _{j}(\xi,\eta)}{\partial\eta\partial\xi^{2}}\right\}\right], \tag{22}\]  
where \(\lambda(s)\) is the Lagrange multiplier  
\[\lambda(s)=\frac{-1}{s^{\beta}}.\] \begin{table}
\begin{tabular}{c c c c c c c} \hline  & **VHPTM** & **VHPTM** & **VHPTM** & **Exact** & **Absolute Error** & **Absolute Error** \\ \hline \(\xi\) & \(\beta=0.55\) & \(\beta=0.75\) & \(\beta=1\) & & \(VIM(\beta=1)\) & \(VHPTM(\beta=1)\) \\ \hline
0.1 & 0.0966295721 & 0.0993928867 & 0.0999000999 & 0.0999001000 & \(1\times 10^{-10}\) & \(1\times 10^{-10}\) \\
0.2 & 0.1932591443 & 0.1987857735 & 0.1998001998 & 0.1998002000 & \(2\times 10^{-10}\) & \(2\times 10^{-10}\) \\
0.3 & 0.2898887165 & 0.2981786602 & 0.2997002997 & 0.2997003000 & \(3\times 10^{-10}\) & \(3\times 10^{-10}\) \\
0.4 & 0.3865182887 & 0.3975715470 & 0.3996003996 & 0.3996004000 & \(4\times 10^{-10}\) & \(4\times 10^{-10}\) \\
0.5 & 0.4831478608 & 0.4969644337 & 0.4995004995 & 0.4995005000 & \(5\times 10^{-10}\) & \(5\times 10^{-10}\) \\
0.6 & 0.5797774330 & 0.5963573204 & 0.5994005994 & 0.5994006000 & \(6\times 10^{-10}\) & \(6\times 10^{-10}\) \\
0.7 & 0.6764070052 & 0.6957502072 & 0.6993006993 & 0.6993007000 & \(7\times 10^{-10}\) & \(7\times 10^{-10}\) \\
0.8 & 0.7730365774 & 0.7951430939 & 0.7992007992 & 0.7992008000 & \(8\times 10^{-10}\) & \(8\times 10^{-10}\) \\
0.9 & 0.8696661495 & 0.8945359807 & 0.8991008991 & 0.8991009000 & \(9\times 10^{-10}\) & \(9\times 10^{-10}\) \\
1 & 0.9662957217 & 0.9939288674 & 0.9990009990 & 0.9990010000 & \(10\times 10^{-09}\) & \(10\times 10^{-09}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of VHPTM and variational iteration method (VIM) [19] of example 1 at \(\eta=0.1\). Figure 2: Variational homotopy perturbation transform method solution of example 4.1 at different fractional order \(\beta=1\), 0.8, 0.6, 0.4. Applying VHPTM using He's polynomials,  
\[ \begin{split}&\sum_{j=0}^{\infty}p^{j}v_{j}(\xi,\eta)=\sum_{j=0}^{ \infty}p^{j}v_{j}(\xi,\eta)-p\xi^{-1}\left[\frac{1}{s^{\beta}}\xi\left\{s^{ \beta}\left(\frac{\partial v_{0}}{\partial\eta}+p\frac{\partial v_{1}}{ \partial\eta}+p^{2}\frac{\partial v_{1}}{\partial\eta}+\cdots\right)+\left\{ \frac{\partial v_{0}}{\partial\xi}+p^{1}\frac{\partial v_{1}}{\partial\xi}+p^{ 2}\frac{\partial v_{2}}{\partial\xi}+\cdots\right\}\right.\\ &+\left\{v_{0}\frac{\partial v_{0}}{\partial\xi}+p(v_{0}\frac{ \partial v_{1}}{\partial\xi}+v_{1}\frac{\partial v_{0}}{\partial\xi})+p^{2}(v_ {0}\frac{\partial v_{2}}{\partial\xi}+v_{1}\frac{\partial v_{1}}{\partial\xi} +v_{2}\frac{\partial v_{0}}{\partial\xi})+\cdots\right\}-\left\{\frac{ \partial\partial^{2}v_{0}}{\partial\eta\partial\xi^{2}}+p^{1}\frac{\partial \partial^{2}v_{1}}{\partial\eta\partial\xi^{2}}+p^{2}\frac{\partial\partial^{2 }v_{2}}{\partial\eta\partial\xi^{2}}+\cdots\right\}\right\}\right].\end{split}  \tag{23}\]  
Comparing the coefficients of \(p\)  
\[