Mathematical methods in physics: differential equations 1

A.Y. 2021/2022
6
Max ECTS
42
Overall hours
SSD
FIS/02
Language
Italian
Learning objectives
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.
Expected learning outcomes
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.
Course syllabus and organization

Single session

Responsible
Lesson period
First semester
Course syllabus
Distributions, Fourier transform, classification of linear partial differential equations, Cauchy problem, Cauchy-Kowalevsky theorem, method of Lagrange-Charpit, KdV equation, Bäcklund transformations.
Prerequisites for admission
Analysis, (systems of) linear ordinary differential equations with constant coefficients.
Teaching methods
Blackboard lectures
Teaching Resources
My lecture notes as well as any textbook on partial differential equations.
Assessment methods and Criteria
Oral examination
FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 6
Lessons: 42 hours
Professor: Klemm Silke