Generalized Projective Riccati Equation Method for Exact Solutions in Fractional Gas Dynamics and Nonlinear Wave Propagation

In this work, the exact solutions of nonlinear wave equations are studied in the framework of fractional gas dynamics. Two powerful methods, such as the generalized projective Riccati equation method (GPREM) and modified extended simple equation method (MESM), are applied to solve the fractional gas dynamics equations, which are used to model complex wave phenomena, exactly and analytically. A central role in this approach is played by the Riccati equation, which we can use to obtain the soliton solutions, periodic waves, and other nonlinear wave patterns. We show that both GPREM and MESM give a systematic way to find exact and soliton solutions that are essential to understanding the dynamics of fractional systems and facilitate the introduction of a novel order of the gas dynamics equations, which poses a unique set of challenges. The physical interpretation of these solutions is discussed in terms of wave propagation, stability, and interaction of nonlinear waves. The findings demonstrate the applicability of the Riccati-based techniques to the solution of fractional nonlinear evolution equations and will help in the development of theoretical models in fractional dynamics. This paper sets the groundwork for additional research into the use of fractional calculus in the study of fluid dynamics and wave propagation and could prove useful in other branches of physics and engineering.