# MATS 230 Partial differerential equations, fall 2021, 9 cr

### Announcements

Exercise points are denoted to TIM system

**Update**: Tuesday's lecture has been canceled and Monday's lecture (MaD381, 14:15-16:00) has been repurposed to an instruction session (ohjaukset). The lecture videos will be updated during Wednesdays as usual.

First lecture: 6.9.2021, 14.15–16.00, MaD381

### Contents

A partial differential equation (PDE) is an equation involving an unknown function of two or more variables, and its partial derivatives. Partial differential equations have numerous applications to geometry, stochastics, mathematical finance, physics and image processing.

This course gives an introduction to PDEs. We derive representation formulas for linear equations, and deal with in particular the transport, Laplace, heat and wave equations. Typical questions that arise in the theory of PDEs are existence, uniqueness, stability, and regularity. At the end of the course we look at some examples of solving PDEs numerically using Matlab software.

This is first of the three PDE couses (others being MATS340 Osittaisdifferentiaaliyhtälöt 2 and MATS424 Viscosity theory) lectured at the math department.

### Teaching

Instruction session (ohjaukset) Monday 14.15–16.00 at MaD381.

Exercises Tuesday 8.30-10.00 at MaD380, starting at 14.9 (except the last two exercises which are at a computer classroom).

Lectures and exercises: Jarkko Siltakoski, MaD250, jarkko.siltakoski@jyu.fi.

Lectures

See the lecture note in the Material folder. It will be updated as the course progresses. Videos that cover the material of a week will be updated Wednesdays.

- week: Introduction of PDEs, derivation of transport equation, classifications of PDEs, Examples: minimal surface equation.
- week: Linear first order equations, method of characteristics, transport equation. Introduction to Laplace's equation.
- week: Fundamental solution, Poisson equation, mollification.
- week: Mean value property, maximum principle, uniqueness to Laplace's equation.
- week: Harnack's inequality, Green's function.

### Completion mode

Course exam and exercises, or alternatively final exam.

In the exercises, the exercises are denoted on the list, and one of the students works out the exercise on the board. You get points from exercises to the course exam according to the following:

35% exercises -> 1 point

....

85% exercises -> 6 points

If you cannot get to an exercise session, then you can return scanned exercises using this webpage or by sending them to the email address jarkko.siltakoski@jyu.fi.

Two last exercises are computer exercises using Matlab, which are to be returned and each exercise is graded on 0-1 scale. Participation gives you 2 points.

### Prerequisites

Bachelor level courses in mathematics are recommended, at the minimum in particular Vector calculus 2 or equivalent.

### Exam days

Course exam on 8.12.2021

### Exercises

Exercises appear in the Material folder during the course.

### Material

The course follows a lecture note that is updated as the lectures progress (see Material folder) and is mainly based on the beginning of 'Evans: Partial Differential Equations'.

Additional reading

- E. DiBenedetto: Partial differential equations
- W. Strauss: Partial differential equations. An introduction.
- J. Kinnunen: Partial differential equations