- The student is able to work with partial differential equations, as well as to distinguish between various classes of PDEs. For such classes, the student is able to discuss solution methods and theoretical considerations.
- The student is familiar with a few classical partial differential equations such as the Laplace, heat and wave equation.
- The student is acquainted with classical, weak and distributional solutions, and is able to perform calculations involving those.
- The student is acquainted with maximum principles and the energy method.
- The student is able to solve simple first order partial differential equations explicitly using the method of characteristics.
Partial differential equations describe a wide range of phenomena (heat, sound, fluid dynamics etc.) and thus play an important role in many applications. In this course we will introduce and study the basic types of partial differential equations. Solution methods, representation formulas for solutions and properties of solutions for classical linear equations of second order (Laplace, heat and wave equation) will be discussed. We are mainly concerned with existence, uniqueness and regularity of solutions. This involves the use of fundamental solutions, the maximum principle for elliptic equations etc. Furthermore, nonlinear partial differential equations of first order will be considered via the method of characteristics.