# Fractional View Analysis of Acoustic Wave Equations, Using Fractional-Order Differential Equations
## 3 The Procedure of VHPTM  
To demonstrate, the fundamental concept of the present method [21; 22], we are considering  
\[D_{\eta}^{\beta}v(\xi,\eta)+\bar{R}v(\xi,\eta)+\bar{N}v(\xi,\eta)=f(\xi,\eta), \tag{6}\]  
with initial condition  
\[v(\xi,0)=g(\xi),\]  
where \(f(\xi,\eta)\) is an inhomogeneous term, \(\bar{R}\) and \(\bar{N}\) are particular linear and non-linear differential operators and \(D_{\eta}^{\beta}v(\xi,\eta)\) is the Caputo fractional derivative of \(v(\xi,\eta)\). By taking Laplace transform of Equation (6) on both sides, we get  
\[\mathcal{L}_{\eta}\{v(\xi,\eta)\}-\sum_{k=0}^{m-1}s^{\beta-1-k}\frac{\partial^ {k}v(\xi,\eta)}{\partial^{k}\eta}|_{t=0}=-\mathcal{L}\left\{\bar{R}v(\xi,\eta )+\bar{N}v(\xi,\eta)-f(\xi,\eta)\right\},\]  
where \(v(s)=\mathcal{L}_{\eta}(v(\xi,\eta))=\int_{0}^{\infty}e^{-s\eta}v(\eta)d\eta\). We can build a functional correction according to the variation iteration method  
\[\mathcal{L}_{\eta}\{v_{j+1}(\xi,\eta)\}=\mathcal{L}\{v_{j}(\xi,\eta)\}+ \lambda(s)\left[s^{\beta}\mathcal{L}\left\{v_{\eta}(\xi,\eta)-\sum_{k=0}^{m-1 }s^{\beta-1-k}\frac{\partial^{k}v(\zeta,\eta)}{\partial^{k}\eta}|_{t=0}+ \mathcal{L}\left\{\bar{R}v(\xi,\eta)+\bar{N}v(\xi,\eta)-f(\xi,\eta)\right\} \right\}\right], \tag{7}\]  
where \(\lambda(s)\) is the Lagrange multiplier. Here we put \(\lambda(s)=\frac{-1}{s^{\beta}}\)[22]. Applying inverse Laplace of Equation (7)  
\[v_{j+1}(\xi,\eta)=v_{j}(\xi,\eta)-\mathcal{L}^{-1}\left[\frac{1}{s^{\beta}} \mathcal{L}\left\{s^{\beta}\frac{\partial v}{\partial\eta}+\bar{R}v_{j}(\xi, \eta)+\bar{N}v_{j}(\xi,\eta)-f(\xi,\eta)\right\}\right]. \tag{8}\]The basic idea in the procedure of homotopy perturbation method is that the solution can be written as a series in powers of \(p\):  
\[v(\xi,\eta)=\sum_{j=0}^{\infty}p^{j}v_{j}(\xi,\eta)=v_{0}+pv_{1}+p^{2}v_{2}+p^{3}v _{3}+\cdots, \tag{9}\]  
where the non-linear expression can be expressed as  
\[\bar{N}v(\xi,\eta)=\sum_{j=0}^{\infty}p^{j}\bar{H}_{j}(v). \tag{10}\]  
\(\bar{H}_{j}\) is He's polynomials,  
\[\bar{H}_{j}(v_{0}+v_{1}+\cdots+u_{j})=\frac{1}{j!}\frac{\partial^{j}}{ \partial p^{j}}\left[\bar{N}\left(\sum_{i=0}^{\infty}p^{i}v_{i}\right)\right]. \tag{11}\]  
The technique of fractional VHPTM of Equation (8) with He's polynomials. \[\sum_{j=0}^{\infty}p^{i}v_{j}(\xi,\eta)=\sum_{j=0}^{\infty}p^{i}v_{j}(\xi,\eta )+\mathcal{L}^{-1}\left[\lambda(s)\mathcal{E}\left\{\sum_{j=0}^{\infty}p^{j} \frac{\partial^{\delta}v_{j}}{\partial\eta^{\delta}}(\xi,s)+\sum_{j=0}^{ \infty}p^{j}\bar{R}v_{j}(\xi,\eta)+\sum_{j=0}^{\infty}p^{j}\bar{H}_{j}(v)-f( \xi,\eta)\right\}\right]. \tag{12}\]  
By comparing the coefficients of like power of \(p\) on both sides of Equation (12), we get the VHPTM solution of the given problem. **Theorem 2**. : _Let \(\xi\) and \(\mathcal{Y}\) be two Banach spaces and \(T:\xi\to\mathcal{Y}\) be a contractive nonlinear operator, such that for all \(v;v^{*}\in;\xi,||T(v)-T(v^{*})||\leq K||v-v^{*}||,\qquad 0<K<1\,y\)[31]. Then, in view of Banach contraction theorem, \(T\) has a unique fixed point \(v\), such that \(Tv=v\): Let us write the generated series (12), by the Laplace decomposition method as_  
\[\xi_{m}=T(\xi_{m-1}),\quad\xi_{m-1}=\sum_{m=1}^{m-1}v_{j},\quad m=0,1,2,\cdots\]  
_and supposed that \(\xi_{0}=v_{0}\in\mathcal{S}_{p}(v)\), where \(\mathcal{S}_{p}(v)=\{v^{*}\in\xi:||v-v^{*}||<p\}\) then, we have_  
\[v(\xi,0)=g(\xi),\]0 \[v(\xi,0)=g(\xi),\]1  
Proof. : \((B_{1})\) In view of mathematical induction for \(m=1\), we have  
\[v(\xi,0)=g(\xi),\]2  
Let the result be true for \(m-1\), then  
\[v(\xi,0)=g(\xi),\]3  
We have  
\[v(\xi,0)=g(\xi),\]4  
Hence, using \((B_{1})\), we have  
\[v(\xi,0)=g(\xi),\]5  
which implies that \(\xi_{m}\in\mathcal{S}_{p}(v)\). The analytical expression is therefore obtained in the following way  
\[v(\xi,0)=g(\xi),\]6  
If \(\beta=1\) the series form is  
\[v(\xi,0)=g(\xi),\]7  
The exact solution at \(\beta=1\)  
\[v(\xi,0)=g(\xi),\]8  
Figure 1, shows the plot of exact and VHPTM solutions of example 4.1, at integer-order \(\beta=1\).