Solutions of nonlinear equations with the help of some analytical methods and artificial neural networks

This thesis examines exact solutions of some nonlinear partial differential equations that play an important role in mathematics and physics, as well as stochastic solutions of nonlinear ordinary differential equation systems addressing certain biological problems. Firstly, general information about differential equations, which form the foundation of mathematics, is provided, and some major methods previously introduced in the literature to obtain their exact and numerical solutions are briefly mentioned. General information about solitons, which emerge when exact solutions are obtained, is given. Subsequently, general information is provided about the artificial intelligence method we used while investigating the optimization of numerical solutions. Detailed information is given about the functioning of the analytical and stochastic solution methods used. Solutions of the specified equations using the given methods were investigated, and solutions that had not previously been introduced to the literature were obtained.