MAGIC056: Introduction to the theory of pdes for applied mathematics

Course details

Semester

Spring 2010
Monday, January 11th to Friday, March 19th

Hours

Live lecture hours
20
Recorded lecture hours
0
Total advised study hours
0

Timetable

Thursdays
12:05 - 12:55
Fridays
10:05 - 10:55

Description

Description: Introduction to the theory of pdes for applied mathematics
Outline Syllabus: Systems of first order pdes and single pdes of higher order, examples from continuum mechanics
Symbol of a pde and of systems; characteristics; existence, uniqueness and continuous dependence on the data; well- and ill-posedness.
(Brief exposition of necessary functional analysis, e.g. operator theory, distributions, Sobolev spaces, see below *).
Weak and strong solutions.
Maximum principles for elliptic and parabolic pde's, existence of solutions.
Linear elliptic pde's, coercivity/energy estimates; Lax-Milgram lemma, Garding's inequality, existence and uniqueness of weak solutions.
Evolutionary pde's - abstract parabolic initial value problems, energy methods, uniqueness and existence.
Nonlinear elliptic pde's, monotone operators, existence of a weak solution.
Systems of hyperbolic equations; Symmetrisable systems; well-posedness.
Introduction to semi-group methods.
Prerequisites
Undergraduate courses on real analysis and partial differential equations (methods courses) will be assumed without explicity mention. * Functional analysis is more problematic (as applied mathematics students may not have taken such options) but time constraints prevent assuming no prior knowledge. Probably the best way forward is to present some necessary functional analysis briefly during the lectures and to provide ädditional" notes online and together with careful page references to books covering the material in the hope that students who have little or no functional analysis will wish to learn more in ßelf-study" as a means to coming to a deeper understanding of the "theory" of pde's

Prerequisites

(eg other courses which this course fits in with)
Other Magic courses will/may provide useful pre-requisites:
MAGIC003: Introduction to Linear Analysis
MAGIC018: Linear Differential Operators in Mathematical Physics

Syllabus

No syllabus information is available yet.

Lecturer

  • DH

    Dr David Harris

    University
    University of Manchester

Bibliography

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Assessment

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Lectures

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