Advanced PDEs (L.7) (866G1)
Advanced Partial Differential Equations (L.7)
Module 866G1
Module details for 2022/23.
15 credits
FHEQ Level 7 (Masters)
Library
Indicative reading and resource list:
Evans, Lawrence C. Partial differential equations, 2nd ed. : Providence, R.I. : American Mathematical Society, 2010. ISBN: 978-0-8218-4974-3.
Salsa, Sandro, Partial differential Equations in Action: From Modelling to Theory (Universities), Springer Verlag, 2010. ISBN-13: 978-8847007512.
Module Outline
The students will be introduced to modern theory of linear and nonlinear Partial Differential Equations. Starting from the theory of Sobolev spaces and relevant concepts in linear operator theory, which provides the functional analytic framework, they will treat the linear second-order elliptic, parabolic, and hyperbolic equations (Lax-Milgram theorem, existence of weak solutions, regularity, maximum principles), e.g., the potential, diffusion, and wave equations that arise in inhomogeneous media. The emphasis will be on the solvability of equations with different initial/boundary conditions, as well as the general qualitative properties of their solutions. They then turn to the study of nonlinear PDE, focusing on calculus of variation.
Module learning outcomes
Understand the theory of Sobolev spaces and central properties of Sobolev functions (mollification, extension theorem, trace theorem, Poincare inequality, Gagliardo-Nirenberg-Sobolev inequality, Morrey’s inequality, Rellich-Kondrachov theorem).
Apply existence and uniqueness theorems for weak solutions of elliptic equations (Gaarding inequality, Galerkin’s method), evaluate (interior and boundary) regularity and use the maximum principle.
Apply theory of parabolic and hyperbolic equations of second order; use eigenfunction expansions, derive and use energy inequalities.
Understand and apply semigroup theory (Hille-Yosida theory) and the theory of calculus of variations, in particular explaining the concept of a minimiser and discussing the regularity of minimisers.
| Type | Timing | Weighting |
|---|---|---|
| Coursework | 20.00% | |
| Coursework components. Weighted as shown below. | ||
| Portfolio | T2 Week 11 | 40.00% |
| Problem Set | T2 Week 4 | 15.00% |
| Problem Set | T2 Week 7 | 15.00% |
| Problem Set | EASTER Week 1 | 15.00% |
| Problem Set | PS2 Week 1 | 15.00% |
| Computer Based Exam | Semester 2 Assessment | 80.00% |
Timing
Submission deadlines may vary for different types of assignment/groups of students.
Weighting
Coursework components (if listed) total 100% of the overall coursework weighting value.
| Term | Method | Duration | Week pattern |
|---|---|---|---|
| Spring Semester | Lecture | 1 hour | 11111111111 |
| Spring Semester | Lecture | 2 hours | 11111111111 |
How to read the week pattern
The numbers indicate the weeks of the term and how many events take place each week.
Prof Michael Melgaard
Convenor, Assess convenor
/profiles/36956
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