M8101 An introduction to partial differential equations

Faculty of Science
Spring 2020

The course is not taught in Spring 2020

Extent and Intensity
2/2/1. 3 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Teacher(s)
Phan Thanh Nam, Ph.D. (lecturer)
Guaranteed by
Phan Thanh Nam, Ph.D.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Prerequisites
There is no strict pre-requisites. In general, it would be an advantage if students know some basic concepts in Functional Analysis, Measure Theory and Probability Theory. However, all necessary backgrounds will be explained quickly in the course.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The goal of this course is to provide an overview of partial differential equations and their applications. In the first part, we will focus on some standard concepts such as weak solution, weak compactness and regularity. In the second part, we will introduce some basic background for stochastic differential equations, for example Brownian motion, Ito formula and Black-Scholes model. The main feature of the course is that the mathematical theories will be built up naturally via concrete examples from physics and finance.
Syllabus
  • 1. Notions of weak solutions and Sobolev spaces 2. Weak topology and compactness 3. Variational method: examples in Hydrogen atom equation, Thomas-Fermi equation and Gross-Pitaevskii equation 4. Regularity of weak solutions 5. Brownian motion 6. Stochastic integrals and Ito's formula 7. Stochastic differential equations: example of Black–Scholes equation References: - Elliott H. Lieb and Michael Loss, Analysis, Graduate Studies in Mathematics, Vol. 14, published by American Mathematical Society in 2001. ISBN-13: 978-0-8218-2783-3 -Lawrence C. Evans, An Introduction to Stochastic Differential Equations, published by American Mathematical Society in 2013. ISBN-13: 978-1-4704-1054-4
Teaching methods
Lectures, tutorial/exercise discussion and homework
Assessment methods
There will be two mandatory homework assignments and one final written examination, each of them contributes 1/3 to the total grade.
Language of instruction
English
Further comments (probably available only in Czech)
The course can also be completed outside the examination period.
The course is taught annually.
The course is taught: every week.
The course is also listed under the following terms Spring 2017, spring 2018, Spring 2019.
  • Enrolment Statistics (Spring 2020, recent)
  • Permalink: https://is.muni.cz/course/sci/spring2020/M8101