Mathematical Methods in Physics: Differential Equations 1

A.Y. 2019/2020
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.
Single course

This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.

Course syllabus and organization

Single session

Responsible
Lesson period
First semester
Course syllabus
Distributions, Fourier transformations, kernels, heat equation with and without source, heat kernel and Schroedinger kernel, separation of variables,
initial value problems and boundary value problems, one-sided and two-sided
Green's functions, classification of quasilinear partial differential equations,
Cauchy problem and theorem of Cauchy-Kowalevsky, time-ordered exponential,
minimal surfaces, method of Lagrange-Charpit, Korteweg-De Vries equation, solitons, Liouville and sine-Gordon equation, Baecklund transformations.
Prerequisites for admission
Ability to solve linear inhomogeneous systems of ordinary differential equations.
Teaching methods
Blackboard lectures
Teaching Resources
My lecture notes. Additional material will be distributed/indicated during the lectures.
Assessment methods and Criteria
Oral examination
FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 6
Lessons: 42 hours
Professor: Klemm Silke
Shifts:
-
Professor: Klemm Silke
Professor(s)