Fractional Differential Equations

Winter Term 2009/10

Fractional Differential Equations

Dr. Martin Welk


Winter Term 2009/10

Seminar (2h)

Notice for bachelor/master students of mathematics: This is a »Hauptseminar« in the sense of these study programs.

Important DatesDescriptionAdministrative detailsRequirementsOverview of Topics

No meeting on January 18!

First meeting: Thursday, July 30, 2009, 13:00h (1:00 p.m.), Bld. E1.1, room 3.06

Sign-up deadline: Tuesday, July 28, 2009. Details see below.

Regular meetings during winter term 2009/10:
Mondays, 16:15h (4:15 p.m.), Bld. E1.1, room 306, starting November 2, 2009

Please direct all organisatorial requests to Martin Welk.

Contents: The idea to generalise differentiation to non-integer derivative order goes back to the very beginnings of calculus, as already Leibniz investigated possible definitions of e.g. half-order derivatives. Nowadays there exist various approaches to their definition, and though they have not become part of freshman courses in calculus, fractional derivatives (and more generally, pseudo-differential operators) have turned into an established branch of analysis. Fractional differential equations formulated with their help have found surprising applications in the modelling of physical systems but also in image processing.

In this seminar, we will introduce different approaches to the definition of fractional derivatives. We will study their properties, part of which resembles standard derivatives while others turn out to be fundamentally different. We will consider important fractional differential equations and understand how the differences between standard and fractional differentiation influence their behaviour, and how the concept of an initial value problem transfers to them. Based on selected examples, we will see the application of fractional differential equations in physics, engineering, and image processing.

Prerequisites: The seminar is designed for graduate students (i.e. advanced Bachelor, Master, and post-Vordiplom) of mathematics, or graduate students of computer science or visual computing with solid mathematical background.

Language: Understanding scientific text in English is indispensable, though for part of the topics some material in German is available. Talks should be preferably in English (if the talk is in German, slides should be in English). Write-up can be in English or German.

Sign-up: If you are interested to participate, please sign up by e-mailing not later than July 28, 2009. Please include your name, student ID (Matrikelnummer), date of birth, study programme, and valid e-mail address.
Since the number of talks is limited, we ask for your understanding that participants are considered strictly in the order of incoming e-mails – no exceptions.

Sessions: First meeting takes place Thursday, July 30, 2009, 13:00h (1:00 p.m.), Bld. E1.1, room 3.06.
Regular sessions will be weekly during the winter term. Exact schedule to be determined at first meeting.

Regular attendance: It is expected that you attend all seminar meetings, except for provable important reasons (we check).

Talk: Talk duration is 30 min, plus 15 min for discussion. Please do not deviate significantly from this time schedule.
You may give a presentation using a data projector (»beamer«), overhead projector or blackboard, or mix these media appropriately. English is preferred as language of presentation; if you should decide to talk in German anyway, you must use slides in English.

Write-up: The write-up has to be handed in till the end of the lecture period (i.e. February 5, 2010. It should summarise your talk, so some 5+ pages per speaker will be adequate in most cases.
Electronic submission is preferred. File format for electronic submissions is PDF – text processor files (like .doc) are not acceptable. We recommend using LaTeX.
Adhere to the standards of scientific referencing: Quotations and copied material (such as images) must be clearly marked as such, and a bibliography is required.

Mandatory consultation: Talk preparation has to be presented to seminar supervisor not later than in the week before the talk is given. It's your responsibility to approach us timely and make your appointment!

Please note that the presentation dates given here are preliminary and subject to change.

No. Topic Presented by            Date            
1 Fractional derivatives: definition, comparison, properties S. Galliani November 2
2 Important examples of fractional ordinary differential equations S. John November 9
3 Causality and initial value problems S. Alshabben November 23
4 Application to viscoelastic materials S. Schwarzbach November 23
7 Fractional diffusion processes D. Mahmoud December 7
10 Fractional calculus and finance mathematics M. Zeshan Afzal December 14
5 Fractional derivatives in image processing: α scale spaces E. Pirmoradian January 4
8 Fractional higher order scale spaces in image processing P. Ochs January 11
6 Modelling of coils using fractional derivatives V. Kremer January 25

Martin Welk / July 24, 2009–February 25, 2010

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