Introduction to Partial Differential Equations 

• 32 hrs lecture
• 32 hrs problem session

A partial differential equation 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.

• Introduction: some elementary models will be explained and different types of PDEs will be classified.
• First order equations: the method of characteristics, conservation laws and shock waves.
• Linear second order equations: the heat equation, the laplace equation and the wave equation are classical second order models.
• The wave equation for one space dimension: The Cauchy problem and d’Alembert’s solution.
• Seperation of variables. For special domains and special PDEs one may split the problem into a set of ODEs.
• Sturm-Liouville equations. Parameter dependent boundary value problems for ODEs.
• Elliptic equations. The maximum principle and uniqueness.
• Integral representations. For some special cases Green functions give an almost explicit solution.
• Equations in higher dimensions: the classification in parabolic, elliptic and hyperbolic equations. Some explicit solutions.
• Variational methods. Introducing the weak formulation.
• Some numerical methods: a first glance at finite differences and finite elements.

An Introduction to Partial Differential Equations
The course is based on the book with identical title by Pinchover and Rubinstein published by Cambridge University Press.