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| Module Availability |
Autumn |
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| Assessment Pattern |
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Unit(s) of Assessment
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Weighting Towards Module
Mark (%)
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Coursework
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40
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Final Test
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60
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Qualifying Condition(s)
An overall aggregate mark of 50% for the module is required to pass the module. |
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| Module Overview |
| This module introduces students to basic concepts and methods of the functional analysis with applications to partial differential equations. |
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| Prerequisites/Co-requisites |
MAT1016 Linear Algebra
MAT2004 Real Analsis II
MAT3010 Function Spaces |
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| Module Aims |
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| Learning Outcomes |
At the end of the module, a student should:
• have an understanding of basic properties of Hilbert and Banach spaces and linear functional s on them;
• understand a concept of a distributional solution of a differential equation and to be able to give a weak formulation for the Dirichlet and Neumann problems for the simplest differential operators;
• be able to prove the existence and uniqueness of a solution for some classical partial differential equations using the methods of functional analysis.
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| Module Content |
The contents of the lectures will include:
• Hilbert and Banach spaces, linear functionals, dual spaces, reflexivity.
• Compact sets. Compactness criteria for C and Lp.
• Weak and strong convergences. Weak compactness of a unit ball in reflexive spaces.
• Linear operators: adjoint, symmetric and compact operators. Fredholm operators.
• Convex functional s and minimization problems
• Introduction to distributions and Sobolev spaces.
• Weak formulation of Dirichlet and Neumann problems for the Laplacian.
• Variational formulation of these problems.
Introduction to non-linear partial differential equations. |
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| Methods of Teaching/Learning |
| Teaching is by lectures and tutorials. Learning takes place through lectures, exercises (tutorials), and background reading. |
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| Selected Texts/Journals |
N. Young, An Introduction to Hilbert Space, Cambridge University Press (1988)
L. Evans, Partial Differential Equations, AMS Press (2002)
E. Zeidler, Applied Functional Analysis: Applications of Mathematical Sciences (Applied Mathematical Sciences, 108), Springer Verlag (1995)
V. Hutson, J. Pym, M. Cloud, Applications of Functional Analysis and Operator Theory, Second Edition. Mathematics in Science and Engineering, 200. Elsevier B. V., Amsterdam, 2005.
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| Last Updated |
| 27 July 2009 |
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