Exact solutions of the (1+1)-dissipative Westervelt equation using an optimal system of Lie sub-algebras and modified simple equation method

Abstract

The Westervelt model is a non-linear partial differential equation that models sound propagation and its effects in non-linear media. In this paper, we obtain exact invariant solutions of the (1+1)-dimensional dissipative Westervelt equation using Lie symmetry analysis with modified simple equation method. By utilizing an optimal system of Lie sub-algebras, the model is reduced to an ordinary differential equation. The modified simple equation method leverages that the studied model admits the travelling wave solution, i.e., $u(x,t)=g\left(\phi \right)$, where, $\phi =x-\alpha t$, to obtain solitary wave solutions. The constructed solutions have applications in high-intensity focused ultrasound (e.g., cancer detection) as the different parameters are varied.