Mathematical Methods in Physics: Differential Equations 1

This course represents an introduction to partial differential equations. Particular emphasis is given to the linear case (e.g. heat equation, Helmholtz and Laplace equations), where a solution can be constructed using kernels. A part of the class is dedicated to nonlinear partial differential equations such as Korteweg-De Vries or sine-Gordon, and some tools to solve them, like Baecklund transformations, are introduced.

At the end of the course the students are expected to have the following skills:
1. construction of the kernel for the most important partial differential equations like the heat equation or the Helmholtz and Laplace equations;
2. knows the method of separation of variables;
3. knows some important special functions like Euler's Gamma function or the Bessel functions;
4. ability to classify quasilinear partial differential equations, knows the Cauchy problem and the Cauchy-Kowalewsky theorem;
5. knows some techniques to solve nonlinear differential equations, like e.g. the method of characteristics or the Baecklund transformations.