A differential equation is a relation between an unknown function (to be determined) and its derivatives. While for ordinary differential equations the unknown function depends on a single independent variable, it depends on several variables for partial differential equations.

A huge variety of processes in science and technology is described by partial differential equations, which therefore belong to the most important objects of investigation in Applied Mathematics.

The number of phenomena occurring in the context of partial differential equations, and the number of methods and techniques to investigate them, is by far too complex to be the content of a one semester course. The lecture course can therefore only be of an introductory type. Topics to be treated are e.g. the classical wave-, Poisson-, and heat equation, maximum principles, separation of variables, classification of quasilinear second-order equations and first-order systems, normal forms, a fixed-point approach for second-order hyperbolic equations. Strong emphasis will be put on many examples from physics and engineering.

The lecture course addresses students in their fifth semester (third year) or higher, with substantial knowledge in analysis and linear algebra. It is suitable for students of mathematics, and for students of other subjects who have strong mathematical interests.

The lectures will be accompanied by exercise lessons. Attendance of these exercises is strongly recommended to all participants.

As already mentioned, this lecture course can cover only a small portion of the overall topic of partial differential equations. Deeper knowledge can be aquired in further subsequent courses.

# References

Textbooks:

Evans, L. C.: Partial Differential Equations; Graduate Studies in Mathematics 19, American Mathematical Society.

John, F.: Partial Differential Equations; Springer.

Copson, E. T.: Partial Differential Equations; Cambridge University Press.

Courant, R., Hilbert, D.: Methods of Mathematical Physics, Vol. I + II; Wiley Classics.

Hellwig, G.: Partial Differential Equations; Teubner.

For those who understand German (or want to learn it):

Hellwig, G.: Partielle Differentialgleichungen; Teubner.

Leis, R.: Vorlesungen über partielle Differentialgleichungen zweiter Ordnung; Bibliographisches Institut, Mannheim.