# MATS 230 Partial differerential equations, fall 2019, 9 cr

tekijä: Mikko Johannes Parviainen Viimeisin muutos torstai 28. marraskuuta 2019, 16.47

### Announcements

Matlab and numerics is not asked in the exam; however you get exercise points and computer skills are crucial in the job market.

Observe that the second computer exercise is at the time of the Tuesday-lecture.

The last two exercise sessions are in the computer classrooms, and the idea is that you work with the problems there and return a written report. You have about a week to complete the report after the exercise session. Participance gives you 2 points.

Exercise points are denoted to TIM system

First lecture: MaD380, Tue 10.9, at 10.15-12.

### Sisältö

Osittaisdifferentiaaliyhtälö (ODY) on yhtälö, jossa tyypillisesti esiintyy tuntemattomana kahden tai useamman muuttujan funktio, ja sen osittaisderivaatat. Osittaisdifferentiaaliyhtälöillä on lukemattomia sovelluksia geometriassa, stokastiikassa, rahoitusmatematiikassa, fysiikassa ja kuvankäsittelyssä.

Kurssi sisältää johdatuksen osittaisdifferentiaaliyhtälöihin. Kurssilla tutustutaan ratkaisujen esityslauseisiin lineaarisille yhtälöille, sekä käsitellään esimerkkeinä kuljetusyhtälö, Laplace-yhtälö, lämpöyhtälö ja aaltoyhtälö. Tyypillisiä osittaisdifferentiaaliyhtälöiden teoriassa käsiteltäviä kysymyksiä ovat ratkaisun olemassaolo, yksikäsitteisyys, stabiilius ja säännöllisyys. Lopuksi tutustutaan osittaisdifferentiaaliyhtälöiden numeerisen ratkaisemisen alkeisiin käyttäen Matlab-ohjelmistoa.

Kurssi soveltuu ensimmäiseksi osittaisdifferentiaaliyhtälöiden kurssiksi. Muita Matematiikan ja tilastotieteen laitoksella luennoitavia osittaisdifferentiaaliyhtälöiden kursseja ovat MATS340 Osittaisdifferentiaaliyhtälöt 2 ja MATS424 Viskositeettiteoria.

### 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

Lectures Tue 10.15-12 and  Thu 14.15-16 at MaD380 starting Tue 10.9. Exercises Thu 12.15-14 starting Thu 19.9 at MaD380, except the last two exercises which are at a computer classroom.

Lectures and exercises:  Mikko Parviainen, MaD306.

### Lectures

See the lecture note in the Material folder.

1. week: Introduction of PDEs, derivation of transport equation, classifications of PDEs, Examples: minimal surface equation
2. week: Hamilton-Jacobi equation and its connections to optimal control, first order linear equations with constant and non constant coefficients, transport equation, derivation of Laplace equation
3. week: Examples, fundamental solution to the Laplace equation, solution to Poisson equation through the convolution, mollifiers
4. week: Mean value property, max principle, uniqueness results, smoothness of harmonic functions, derivative estimates for harmonic functions, Liouville's theorem
5. Harnack's inequalities, Green functions, Green function on the half space, Green function on the ball B(0,1)
6. Green function on the ball B(0,r), eigenvalue problem, Rayleigh's principle, intro to heat equation, derivation of the fundamental solution to the heat equation
7. Solutions to the homogenous and inhomogeneous Cauchy problems, max principles in a bounded domain
8. Uniqueness in a bounded domain, max principle in an unbounded domain with the growth condition, uniqueness in an unbounded domain with the growth condition, uniqueness through the energy method. Backward in time uniqueness and integral regularity was skipped! Smoothness of solutions, parabolic Harnack. Intro to wave equation, d'Alembert's formula (n=1).
9. Reflection method, solution for wave equation in 3D (Kirchhoff's formula) and 2D, inhomogeneous problem, conservation of energy
10. Uniqueness by energy method for the wave equation, Fourier sums (not going into L2 theory), separation of variables, intro to Fourier transform .
11. Fourier transform continued: convolution, Bessel potential, d'Alembert formula by Fourier transform, fundamental solution to the heat equation by Fourier transform, basic Matlab usage
12. Programming strategy with matlab, numerics (discretization of Laplacian in 1D and 2D), solving Laplace equation and wave equation.

### 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

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 exams on  11.12 or 19.12.

### Exercises

Exercises appear in the Material folder during the course.

### Material

The course follows a lecture note (see Material folder) mainly based on the beginning of 'Evans: Partial Differential Equations'.