On the Study of Nonlinear Murray Equation in Non-Newtonian Fluids: Fractional Solitary Wave Structures, Chaos, and Sensitivity Demonstration

This article introduces fractional solitary wave structures to the nonlinear Murray equation by applying advanced techniques, namely the generalized Arnous method and the modified generalized Riccati equation mapping method (MGREMM). This equation is known as a generalization of the nonlinear reaction–diffusion equation, which describes the diffusion of chemical reactions in a medium. For such equations, solitary wave solutions are of the utmost importance in numerical and analytical theories. Despite the widespread use of numerical methods, a deeper understanding of the dynamics requires the enhancement of analytical methods used to derive analytical solutions. In order to achieve the desired solutions, the governing equation is transformed into an ordinary differential equation by utilizing an appropriate wave transformation with the β-derivative. The solitary wave solutions of various types, such as mixed, singular, dark, bright, bright–dark, and combined solitons, are extracted. Additionally, we plot a variety of graphs with various parameters to examine the solution’s behavior at various parameter values. Moreover, another important aspect of this study is to discuss the chaotic and sensitivity analysis of the studied model by the assistance of the Galilean transformation. The 2D phase portraits, Poincaré maps, time series analysis, and sensitivity analysis graphs based on the initial conditions have been sketched. This study achieves a significant milestone by applying the aforementioned methods with the assistance of β-fractional derivatives to the proposed equation for the first time and contributes significantly to the existing literature. In many fields, the equation in concern can introduce improved algorithms that model the evolution of certain types of biological and physical systems, particularly in contexts like vascular network growth and fluid dynamics. By demonstrating the effectiveness of the applied approaches, the outcomes of this research could improve understanding of the nonlinear dynamic properties shown by the given system.