Dynamics of fluid mechanics with the modified 1/G'- expansion method of sharma-tasso-olver-burgers equation

This thesis comprehensively examines the Sharma-Tasso-Olver-Burgers equation, a family of nonlinear partial differential equations. The Burgers equation, which holds a fundamental place in fluid mechanics, has been used for many years to model the dynamic behavior of fluids and waves. Derived from its combination with the Sharma-Tasso-Olver equation, the Sharma-Tasso-Olver-Burgers equation allows for more complex and realistic representations of nonlinear wave motions. Therefore, this equation has become an important research topic not only from a theoretical perspective but also in applied fields such as engineering, plasma physics, fiber optics, and fluid mechanics. The focus of this study is the systematic investigation of traveling wave solutions to the Sharma-Tasso-Olver-Burgers equation. In this context, two methods widely used in recent years are employed in the solution process: the classical 1/G'-expansion method and its improved version. While the classical expansion method provides simpler and less computationally demanding solutions, it represents the solution space in a limited way. In contrast, the improved expansion method generates more parametric, flexible, and physically meaningful solutions by expanding the derivative terms. Thus, a unified model encompassing not only the Sharma-Tasso-Olver-Burgers equation but also its special cases was created. The analytical solutions obtained as a result of the applications were examined graphically and evaluated in two main classes: singular and hyperbolic solutions. By transforming between these forms at different parameter values, the solutions reveal the rich nature of nonlinear dynamics. In this respect, the study provides a methodologically robust framework for obtaining solutions for nonlinear systems and contributes to a deeper understanding of the physical aspects of wave interactions. Hyperbolic-type solutions, in particular, play a critical role in modeling fluid behavior, optical wave packets, and plasma oscillations. Furthermore, the obtained solutions were visualized using computer-aided software and presented in three-dimensional, contour, and two-dimensional graphics. These visualizations make the dynamic structure of the equation and the physical effects of wave solutions more understandable and concrete. In this context, the study not only provides a theoretical analysis but also reveals important findings in the field of applied mathematics. In conclusion, this study makes the mathematical structure and physical interpretations of the Sharma-Tasso-Olver-Burgers equation more understandable through both the obtained solutions and their graphical representations; thus, it provides practical and visually rich contributions to the literature on nonlinear differential equations.