The rapid development of computers and computing power has led to advances in simulation and computational physics. Various numerical methods have been developed or are still being developed, including well-established methods such as the Finite Volume Method (FVM) or the Finite Element Method (FEM). Modern computational methods such as neural networks (NN) or new "exotic" formulations of the FEM are emerging and being incorporated into computational physics and commercial software. The aim of this thesis is to implement solvers for several partial differential equations (PDEs) derived from fluid dynamics and heat transfer, using well-established numerical methods, and to compare their results with benchmark analytical solutions. Furthermore, selected problems will also be addressed using emerging computational techniques that employ neural networks and machine learning.