# Fractional View Analysis of Acoustic Wave Equations, Using Fractional-Order Differential Equations  
[PERSON]  
1 Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan;  
[EMAIL_ADDRESS] (I.A. ); [EMAIL_ADDRESS] (H.K. ); [EMAIL_ADDRESS] (R.S. );  
[EMAIL_ADDRESS] (M.A.) Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey;  
[EMAIL_ADDRESS]  
Institute of Space Sciences, 077125 Magurele, Romania  
Center of Excellence in Theoretical and Computational Science (TaCS-CoE) & Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uhit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand  
Department of Medical Research, China Medical University Hospital, China Medical University,  
Taichung 40402, Taiwan  
November 5, 2021 November  
In this paper, Laplace Variational Homotopy Perturbation Method (LVHPM) is implemented to solve the following linear and non-linear fractional-order regularized long wave equations. \[\frac{\partial^{\beta}v}{\partial\eta^{\beta}}+\frac{1}{2}\frac{\partial v^{2}}{ \partial_{\xi}^{2}}-\frac{\partial}{\partial\eta}\left(\frac{\partial^{2}v}{ \partial\xi^{2}}\right)=0,\qquad 0<\xi\leq 1,\qquad 0<\beta\leq 1,\quad\eta>0, \tag{1}\]  
with initial condition  
\[v(\xi,0)=\xi,\]  
\[\frac{\partial^{\beta}v}{\partial\eta^{\beta}}+\frac{\partial v}{\partial\xi} +v\frac{\partial v}{\partial\eta}-\frac{\partial}{\partial\eta}\left(\frac{ \partial^{2}v}{\partial\xi^{2}}\right)=0,\qquad 0<\xi\leq 1,\qquad 0<\beta\leq 1, \quad\eta>0, \tag{2}\]  
with initial condition  
\[v(\xi,0)=3\alpha\sec h^{2}(\beta\xi),\quad\alpha>0,\quad\beta=\frac{1}{2} \sqrt{\frac{\alpha}{1+\alpha}}.\]  
\[\frac{\partial^{\beta}v}{\partial\eta^{\beta}}+\frac{\partial v}{\partial\xi} -2\frac{\partial}{\partial\eta}\left(\frac{\partial^{2}v}{\partial\xi^{2}} \right)=0,\qquad 0<\xi\leq 1,\qquad 0<\beta\leq 1,\quad\eta>0, \tag{3}\]  
with initial condition  
\[v(\xi,0)=e^{-\xi},\]  
and  
\[\frac{\partial^{\beta}v(\xi,\eta)}{\partial\eta^{\beta}}+\frac{\partial^{4}v( \xi,\eta)}{\partial\xi^{4}}=0,\qquad 0<\xi\leq 1,\qquad 0<\beta\leq 1,\quad\eta>0, \tag{4}\]  
with initial condition  
\[v(\xi,0)=\sin\xi. \tag{5}\]  
Equation (1) is known as the fractional-order non-linear regularized long wave equation (RLWE); Equation (2) is known as the fractional-order non-linear general regularized long wave equation (GRLWE) and Equations (3) and (4) are known as fractional-order linear regularized long wave equations (RLWEs) [11]. The Benjamin Bona Mahony equation (BBME) also identified the regularized long wave (RLW) equation. This equation is the updated version of Korteweg-de Vries equation (KdV) for the modeling of tiny amplitude lengthy surface gravitational waves spreading unidirectionally in two dimensions. RLW equations have several implementations in certain areas of science, such as ion-acoustic waves in plasma, longitudinal dispersive waves in elastic rods, magneto-hydrodynamic waves in plasma, rotating tube flow and stress waves in compressed gas bubble mixes, etc. The RLW equations are described as useful models in applied physics and engineering for many significant physical structures. They also design many liquid flow nature issues where diffusion is significant, either in viscous or shock situations. It can be used to model any dissipation-related non-linear wave diffusion problem. Chemical reaction, heat conduction, mass diffusion, viscosity, thermal radiation or other sources may result from this dissipation, depending on problem modeling [12]. The RLW problem is a family of non-linear growth models that provides excellent designs for predicting natural phenomena. The algorithm was initially introduced to define the undular bore behavior [13]. It was also obtained from the research of acoustic plasma waves of water and ion. An analytical solution for the RLW equation was identified under restricted initial and boundary conditions in [14].