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". |
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ADDITIONAL NOTES | Pre-requisite Qualifications: B.Sc. with Mathematics as a main area Follows from: MAT3211 and MAT3712 |
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STUDY-UNIT TYPE | Lecture and Independent Study | ||||||||
METHOD OF ASSESSMENT |
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LECTURER/S | Joseph Muscat |
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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. |