### General Information

Course Code | X_400163 |
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Credits | 6 EC |

Period | P4+5 |

Course Level | 300 |

Language of Tuition | English |

Faculty | Faculty of Science |

Course Coordinator | dr. O. Fabert |

Examiner | dr. O. Fabert |

Teaching Staff |
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 |
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Target audiences

This course is also available as:

### 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 partialdifferential 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, makeup your final grade. The grades for these exams and homework sets are

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.

Additional study material will be made available through Canvas.

### Target Audience

Bachelor Mathematics Year 3### Custom Course Registration

Registration via Canvas### Recommended background knowledge

Calculus (in particular vector calculus), Mathematical analysis, andLinear algebra