- The student is able to work with partial differential equations, as well as to distinguish between various classes of pde's. For such classes the student is able to discuss solution methods as well as theoretical considerations.
- The student is acquainted with partial differential equations and classical, weak and distributional solutions
- The student is able to derive properties of distributions, and to perform calculations involving distributions
- The student is able to classify partial differential equations according to classification schemes ((semi/quasi/non-)linear and elliptic/hyperbolic/parabolic)
- The student is able to solve simple first order partial differential equations explicitly using the methods of characteristics
- The student is acquainted with a few classical partial differential differential equations such as the Laplace, heat and wave equation
- The student is acquainted with maximum principles and the energy method
A partial differential equation (PDE) describes a relation between the partial derivatives of an unknown function and given data. Such equations appear in all areas of physicsan engineering. More recently the use of PDEs in models in biology, pharmacy, imageprocessing, finance etc. have increased strongly. Since the origin of these models is very diverse and the results should be applicationdriven, the analysis of PDEs has many facets. The classical approach focused on finding explicit solutions. Since numerical methods and fast computers became available, the modern approach is more oriented to the application of functional analytic methods in order to find existence and uniqueness results and to show that solutions depend continuouslyon the given data. Having existence, uniqueness and stability under perturbations, a numerical method may be implemented to find an approximation of the solution one is interested in. The present course will be an introduction to the field. The elementary classical results will be explained and we will touch some of the more modern aspects.|