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## Partial Differential Equations

2019-2020

### Course Objective

After this course the student...

1. ...will know basic PDE's from classical physics and the properties of
their solutions.

2. ... will be able to derive solution formulas and methods, as well as
describe the qualitative properties of solutions in relation to the
models in which the PDE's arise.

3. ...will be able to apply Fourier series and transforms to solve
linear PDE's explicitly, and master techniques to mathematically
validate the resulting solution formulas.

4. ... will be able to apply some of the basic techniques and results
from real and functional analysis, such as the implicit function theorem
and certain properties of integral operators.

### Course Content

An overwhelming number of physical phenomena can be described by partial
differential equations (PDEs). This course discusses these equations and
methods for their solution. For first order equations we discuss the
method of characteristics, and the solution of such PDEs by methods from
ordinary differential equations. For second order equations, in
particular for the heat and wave equation, we discuss the method of
separation of variables. This ties in with the remarkable result of
Fourier that almost any periodic function can be represented as a sum of
sines and cosines, called its Fourier series. An analogous
representation for non-periodic functions is provided by the Fourier
transform, to be discussed briefly in part 2 of the course, together
with some theoretical background for Fourier series. We discuss some of
the background for generalised Fourier series: the role of eigenvalue
problems and some basic spectral theory. Potential methods and
fundamental solutions will be discussed for the standard examples: heat,
wave and Poisson equation. Harmonic functions will be discussed in
relation to mean value properties.

Topics:

- Classical examples
- First order equations and characteristics
- d'Alembert's solution for the wave equation
- Separation of variables for second order equations
- Fourier Series
- Fundamental solutions for heat and wave equation in one spatial
dimension
- The Dirac delta-function
- Fourier theory
- Laplace and Poisson equation through potential methods
- Eigenvalue problems and some spectral theory
- Harmonic functions
- Fundamental solutions in 2 and 3 spatial dimensions

The Riesz representation Theorem, and the spectral theorem for compact
symmetric real operators in inner product spaces will be introduced on
the side if time permits.

### Teaching Methods

Lectures and exercise classes (both 2 hours per week)

### Method of Assessment

Two written exams and two homework sets, with weights 4+4+1+1=10, make
cancelled in case the student takes the resit exam, which counts for
100%.

### Literature

Peter J. Olver, "Introduction to Partial Differential Equations",
Springer-Verlag, New York, 2014. ISBN 978-3-319-02099-0.

### Target Audience

Bachelor Mathematics Year 3

### Custom Course Registration

Registration via Canvas

### Recommended background knowledge

Calculus (in particular vector calculus), Mathematical analysis, and
Linear algebra

### General Information

Course Code X_400163 6 EC P4+5 300 English Faculty of Science dr. O. Fabert dr. O. Fabert dr. O. Fabert

### Practical Information

You need to register for this course yourself

Last-minute registration is available for this course.

Teaching Methods Seminar, Lecture
Target audiences

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