# Fractional View Analysis of Acoustic Wave Equations, Using Fractional-Order Differential Equations
## 3 The Procedure of VHPTM
\begin{split}& v_{0}(\xi,\eta)=3\alpha\sec h^{2}(\beta\xi),\\ & p^{1}v_{1}(\xi,\eta)=p^{1}v_{0}(\xi,\eta)-p^{1}\xi^{-1}\left[ \frac{1}{s^{\beta}}\xi\left\{s^{\beta}\frac{\partial v_{0}}{\partial\eta}+ \frac{\partial v_{0}}{\partial\xi}+v_{0}\frac{\partial v_{0}}{\partial\xi}- \frac{\partial\partial^{2}v_{0}}{\partial\eta\partial\xi^{2}}\right\}\right], \\ & v_{1}(\xi,\eta)=3\alpha\sec h^{2}(\beta\xi)+3\alpha\beta\{1+6 \alpha\beta+\cosh(2\beta\xi)\}\sec h^{4}(\beta\xi)\tanh(\beta\xi)\frac{\eta^{ \beta}}{\Gamma(\beta+1)},\\ & p^{2}v_{2}(\xi,\eta)=p^{2}v_{1}(\xi,\eta)-p^{2}\xi^{-1}\left[ \frac{1}{s^{\beta}}\xi\left\{s^{\beta}\frac{\partial v_{1}}{\partial\eta}+ \frac{\partial v_{1}}{\partial\xi}+v_{0}\frac{\partial v_{1}}{\partial\xi}+v_ {1}\frac{\partial v_{0}}{\partial\xi}-\frac{\partial\partial^{2}v_{1}}{ \partial\eta\partial\xi^{2}}\right\}\right],\\ & v_{2}(\xi,\eta)=3\alpha\sec h^{2}(\beta\xi)+3\alpha\beta\{1+6 \alpha\beta+\cosh(2\beta\xi)\}\sec h^{4}(\beta\xi)\tanh(\beta\xi)\frac{\eta^{ \beta}}{\Gamma(\beta+1)}\\ &-\frac{3}{32}\alpha\beta^{2}\{-8-96\alpha-576\alpha^{2}+3(-3-16 \alpha+144\alpha^{2})\cosh(2\beta\xi)+48\alpha\cosh(4\beta\xi)+\cosh(6\beta \xi)\}\\ &\sec h^{8}(\beta\xi)\frac{\eta^{2\beta}}{\Gamma(2\beta+1)},\\ &.\\ &.\\ &.\\ &.\\ &.\end{split} \]  
The analytical expression is therefore obtained in the following way  
\[ \begin{split}& v(\xi,\eta)=3\alpha\sec h^{2}(\beta\xi)+3\alpha \beta\{1+6\alpha\beta+\cosh(2\beta\xi)\}\sec h^{4}(\beta\xi)\tanh(\beta\xi) \frac{\eta^{\beta}}{\Gamma(\beta+1)}\\ &-\frac{3}{32}\alpha\beta^{2}\{-8-96\alpha-576\alpha^{2}+3(-3-16 \alpha+144\alpha^{2})\cosh(2\beta\xi)+48\alpha\cosh(4\beta\xi)+\cosh(6\beta\xi )\}\\ &\sec h^{8}(\beta\xi)\frac{\eta^{2\beta}}{\Gamma(2\beta+1)}+\frac {1}{32}\alpha\beta^{3}\{-85-1416\alpha-8496\beta^{2}-2937\beta^{3}+4(-31-432 \alpha-1584\alpha^{2}+3456\alpha^{3})\\ &\cosh(2\beta\xi)-4(11+54\alpha-540\alpha^{2})\cosh(4\beta\xi)-4 \cosh(6\beta\xi)+96\alpha\cosh(6\beta\xi)+\cosh(8\beta\xi)+\sec h^{8}(\beta \xi)\\ &\tanh(\beta\xi)\}\frac{\eta^{3\beta}}{\Gamma(3\beta+1)}+\cdots \end{split}  \tag{24}\]  
The exact solution at \(\beta=1\)  
\[v(\xi,\eta)=3\alpha\sec h^{2}(\beta(\xi-(1+\alpha)\eta)). \tag{25}\]  
In Figure 3, we compared the analytical solution of VHPTM with the exact solution of example 4.2. The comparison has shown the close contact between VHPTM solution and exact solution of the problems. Figure 4, represents VHPTM solution at different fractional-orders \(\beta=1,0.8,0.6\) and \(0.4\) The convergence analysis of fractional-order problems are convergent towards the integer-order problem of example 4.2, as observed. ### Example  
We consider time fractional-order linear RLW equation  
\[\frac{\partial^{\beta}v}{\partial\eta^{\beta}}+\frac{\partial v}{\partial\xi}-2 \frac{\partial\partial^{2}v}{\partial\eta\partial\xi^{2}}=0,\qquad 0<\xi\leq 1, \qquad 0<\beta\leq 1,\quad\eta>0, \tag{26}\]  
initial condition is  
\[v(\xi,0)=e^{-\xi}, \tag{27}\]  
Figure 4: Variational homotopy perturbation transform method solution of example 4.2 at different fractional order \(\beta=1,\,0.8,\,0.6,\,0.4\). Figure 3: Variational homotopy perturbation transform method solution of example 4.2 for \(\beta=1\). By using Equation (12), the fractional PDE given in Equation (26) can be written as  
\[\sum_{j=0}^{\infty}p^{j}v_{j+1}(\xi,\eta)=\sum_{j=0}^{\infty}p^{j}v_{j}(\xi,\eta) +p^{j}\ell^{-1}\left[\lambda(s)\xi\right.\left\{s^{\beta}\frac{\partial^{\beta}v _{j}}{\partial\eta^{\beta}}(\xi,\eta)+\frac{\partial v_{j}}{\partial\xi}-2 \frac{\partial\partial^{2}v_{j}}{\partial\eta\partial\xi^{2}}\right\}\right], \tag{28}\]  
where \(\lambda(s)\) is the Lagrange multiplier  
\[\lambda(s)=\frac{-1}{s^{\beta}}.\]  
Applying VHPTM using He's polynomials,  
\[