Study-Unit Description

Study-Unit Description


CODE MAT5616

 
TITLE Partial Differential Equations - A Functional Analysis Approach

 
UM LEVEL 05 - Postgraduate Modular Diploma or Degree Course

 
MQF LEVEL 7

 
ECTS CREDITS 15

 
DEPARTMENT Mathematics

 
DESCRIPTION Distributions; their differentiation and multiplication; compact supports
Convolution and the Fourier transform; approximations of the identity; tempered distributions
Sobolev spaces and generalized derivatives; the Sobolev inequality
Fundamental solutions of a differential equation; Green's functions; Hilbert space methods
Elliptic equations; Schauder's theorem; spectral decomposition of the Laplacian

Study-unit Aims:

To further a student's knowledge of partial differential equations, and Sobolev spaces;
To prepare the student for research in the well-posedness of non-linear partial differential equations.

Learning Outcomes:

1. Knowledge & Understanding:

By the end of the study-unit the student will be able to:
- Describe and compare the connections between differential equations and functional analysis;
- Distinguish between classical and weak-type solutions;
- Recognise the issues of existence and uniqueness of solutions of partial differential equations.

2. Skills:

By the end of the study-unit the student will be able to:
- Prove the existence and uniqueness of solutions of various simple second-order partial differential equations;
- Work confidently with distributions, their generalized derivatives and their Fourier transform.

Main Text/s and any supplementary readings:

W. Rudin: "Functional Analysis", 2nd edition McGraw-Hill.
F. Hirsch and G. Lancombe: "Elements of Functional Analysis", Springer.
A. Giniatoulline: "An Introduction to Spectral Theory", RT Edwards.
M. Miklaveic: "Applied Functional Analysis and Partial Differential Equations", World Scientific.
J. Cornway: "A Course in Functional Analysis".

 
ADDITIONAL NOTES Pre-requisite Qualifications: B.Sc. with Mathematics as a main area

Follows from: MAT3211 and MAT3712

 
STUDY-UNIT TYPE Lecture and Independent Study

 
METHOD OF ASSESSMENT
Assessment Component/s Assessment Due Sept. Asst Session Weighting
Examination (3 Hours) SEM2 Yes 100%

 
LECTURER/S Joseph Muscat

 

 
The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints.
Units not attracting a sufficient number of registrations may be withdrawn without notice.
It should be noted that all the information in the description above applies to study-units available during the academic year 2024/5. It may be subject to change in subsequent years.

https://www.um.edu.mt/course/studyunit