Animations on the internet illustrate, for example, the wave equation through a vibrating membrane, heat conduction through a decreasing temperature distribution, and the potential equation through a potential imposed by the boundary conditions. What methods are behind these animations, and how are they created? The reader should be able to provide a comprehensive answer to this. Detailed formal proofs are omitted. The concepts are illustrated and motivated through examples and graphics to convey their fundamental ideas. The reader is expected to gain background knowledge and problem-solving skills so that they do not have to settle for mere manipulation of formulas. Students should be empowered to tackle problems arising from their bachelor's or master's thesis or from practical applications.