# Fractional View Analysis of Acoustic Wave Equations, Using Fractional-Order Differential Equations
## 3 The Procedure of VHPTM
\begin{split}&\sum_{j=0}^{\infty}p^{j}v_{j}(\xi,\eta)=\sum_{j=0}^{ \infty}p^{j}v_{j}(\xi,\eta)-p\ell^{-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_{2}}{\partial\eta}+\cdots\right)\right. \right.\\ &\left.\left.+\frac{\partial}{\partial\xi}\left\{v_{0}+pv_{1}+p^{ 2}v_{2}+\cdots\right\}-2\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{29}\]  
Comparing the coefficients of \(p\)  
\[ \begin{split}& v_{0}(\xi,\eta)=e^{-\xi},\\ & p^{1}v_{1}(\xi,\eta)=p^{1}v_{0}(\xi,\eta)-p^{1}\ell^{-1}\left[ \frac{1}{s^{\beta}}\xi\left\{s^{\beta}\frac{\partial v_{0}}{\partial\eta}(\xi,\eta)+\frac{\partial v_{0}}{\partial\xi}-2\frac{\partial\partial^{2}v_{0}}{ \partial\eta\partial\xi^{2}}\right\}\right],\\ & v_{1}(\xi,\eta)=e^{-\xi}+e^{-\xi}\frac{\eta^{\beta}}{\Gamma( \beta+1)},\\ & p^{2}v_{2}(\xi,\eta)=p^{2}v_{1}(\xi,\eta)-p^{2}\ell^{-1}\left[ \frac{1}{s^{\beta}}\xi\left\{s^{\beta}\frac{\partial v_{1}}{\partial\eta}( \xi,\eta)+\frac{\partial v_{1}}{\partial\xi}-2\frac{\partial\partial^{2}v_{1} }{\partial\eta\partial\xi^{2}}\right\}\right],\\ & v_{2}(\xi,\eta)=e^{-\xi}+e^{-\xi}\frac{\eta^{\beta}}{\Gamma( \beta+1)}+e^{-\xi}\frac{\eta^{2\beta}}{\Gamma(2\beta+1)},\\ & p^{3}v_{3}(\xi,\eta)=p^{3}v_{2}(\xi,\eta)-p^{3}\ell^{-1}\left[ \frac{-1}{s^{\beta}}\xi\left\{s^{\beta}\frac{\partial v_{2}}{\partial\eta}( \xi,\eta)+\frac{\partial v_{2}}{\partial\xi}-2\frac{\partial\partial^{2}v_{2} }{\partial\eta\partial\xi^{2}}\right\}\right],\\ & v_{3}(\xi,\eta)=e^{-\xi}+e^{-\xi}\frac{\eta^{\beta}}{\Gamma( \beta+1)}+e^{-\xi}\frac{\eta^{2\beta}}{\Gamma(2\beta+1)}+e^{-\xi}\frac{\eta^{ 3\beta}}{\Gamma(3\beta+1)}.\\ &.\\ &.\\ &.\end{split} \]  
The analytical expression is therefore obtained in the following way  
\[v(\xi,\eta)=e^{-\xi}+e^{-\xi}\frac{\eta^{\beta}}{\Gamma(\beta+1)}+e^{-\xi} \frac{\eta^{2\beta}}{\Gamma(2\beta+1)}+e^{-\xi}\frac{\eta^{3\beta}}{\Gamma(3 \beta+1)}+\cdots. \tag{30}\]  
If \(\beta=1\) the series form is  
\[v(\xi,\eta)=e^{-\xi}\left(1+\eta+\frac{\eta^{2}}{2!}+\frac{\eta^{3}}{3!}+\cdots\right) \tag{31}\]  
The exact solution at \(\beta=1\)  
\[v(\xi,\eta)=e^{\eta-\xi}. \tag{32}\]  
In Table 2, the analytical solutions of VHPTM and HPSTM are compared in terms of absolute error. The accuracy has been measured for both the methods. By comparison it has shown that the proposed method VHPTM has a higher degree of accuracy than HPSTM. ### Example  
We consider time fractional-order linear RLW equation  
\[\frac{\partial^{\beta}v(\xi.\eta)}{\partial\eta^{\beta}}+\frac{\partial^{4}v( \xi.\eta)}{\partial\xi^{4}}=0,\qquad 0<\xi\leq 1,\qquad 0<\beta\leq 1,\quad\eta>0, \tag{33}\]  
with initial condition  
\[v(\xi,0)=\sin\xi, \tag{34}\]  
By using Equation (12), the fractional PDE given in Equation (33) can be written as  
\[\sum_{j=0}^{\infty}p^{j}v_{j+1}(\xi,\eta)=\sum_{j=0}^{\infty}p^{j}v_{j}(\xi, \eta)+\sum_{j=0}^{\infty}p^{j}\xi^{-1}\left[\lambda(s)\mathcal{L}\left\{s^{ \beta}\frac{\partial^{\beta}v_{j}(\xi.\eta)}{\partial\eta^{\beta}}(\xi,\eta)+ \frac{\partial^{4}v_{j}(\xi.\eta)}{\partial\xi^{4}}\right\}\right], \tag{35}\]  
where \(\lambda(s)\) is the Lagrange multiplier  
\[\lambda(s)=\frac{-1}{s^{\beta}},\]  
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}}\mathcal{L} \left\{s^{\beta}\left(\frac{\partial v_{0}}{\partial\eta}+p\frac{\partial v_ {1}}{\partial\eta}+p^{2}\frac{\partial v_{2}}{\partial\eta}+\cdots\right) \right.\\ &\left.+\frac{\partial^{4}}{\partial\xi^{4}}\left\{v_{0}+pv_{1}+p ^{2}v_{2}+\cdots\right\}\right\}\right]\end{split}