Generation of traveling wave solutions of the nonlinear evolution equation with the improved sub equation method

Differential equations are fundamental tools for modeling and analyzing problems arising in various disciplines, including physics, engineering, and biology. Nonlinear evolution equations, in particular, are among those whose analytical solutions are difficult to obtain due to their complex structure. Therefore, investigating analytical solutions of these equations using various methods is of great importance for both theoretical and applied sciences. This thesis examines methods for solving differential equations and consists of five chapters in total. The first chapter presents a general framework for the fundamental concepts of nonlinear evolution equations and their solution approaches. Fundamental definitions are also provided through examples. The second chapter discusses the concept of solitons, which constitute an important class of nonlinear wave solutions. A literature-based summary of the theoretical foundations, physical meanings, and application areas of solitons is presented. The third chapter focuses on the history of Sharma Tasso Olver-Burgers type differential equations and their mathematical significance. The fourth chapter provides theoretical explanations for the sub equation method and the improved sub equation method, and the role and applicability of these methods in the solution process are evaluated in detail. The fifth chapter applies these methods to the nonlinear Sharma Tasso Olver-Burgers equations, and traveling wave solutions are obtained. These solutions are analyzed using Mathematica software and visualized using 3D, 2D, and contour plots. In the conclusion, the findings are interpreted, and the advantages and limitations of the applied methods are discussed. Furthermore, the potential of the methods in solving nonlinear evolution equations is evaluated in general terms.