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Theory of Partial Differential Equations 34 (MATH M6000)

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Administrative Information

  1. Unit number and title: MATH M6000 Theory of Partial Differential Equations 34
  2. Level: M/7
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 2005-06
  6. Unit Organiser: Michiel van den Berg
  7. Lecturer: Michiel van den Berg
  8. Teaching block: 1
  9. Prerequisites: Measure Theory and Integration (MATH 34000)

Unit aims

To introduce basic tools of pure mathematics that are used in studying partial differential equations.

General Description of the Unit

Partial Differential Equations (PDE's) play a central role both in pure and applied mathematics. They arise in mathematical models in mechanics, physics, natural sciences, and finance. The course will introduce the main types of partial differential equations and develop the theory of solvability and properties of solutions. The rigorous approach to PDE's will show the relevance of abstract methods of Analysis in studying problems arising in applied mathematics.

Teaching Methods

A standard lecture course of 33 lectures plus 3 exercise classes

Learning Objectives

At the end of the course the student should know and understand the definitions and theorems (and their proofs) and should be able to use the ideas of the course in unseen situations.

Assessment Methods

The assessment mark for Theory of Partial Differential Equations 34 is calculated from a 2½-hour written examination in January consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT allowed in this examination.

Award of Credit Points

Credit points are gained by:

  • by passing the unit, i.e. gaining a mark at least 50 on the examination;

Transferable Skills

Assimilation of abstract ideas and reasoning in an abstract context. Setting out a sustained argument in a form comprehensible to others.

Texts

L. C. Evans, Partial Differential Equations. AMS 2004.

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equatons of second order. Springer-Verlag 1977

Syllabus

  1. Classification of second-order PDEs
  2. Transport Equation
    • Initial Value problem.
    • Nonhomogenous problem.
  3. Laplace Equation.
    • Fundamental solution.
    • Mean Value Theorems.
    • Properties of Harmonic Functions
    • Green's functions.
    • Energy methods.
  4. Heat Equation
    • Fundamental Solution.
    • Mean value Formula
    • Properties of solutions
    • Energy methods
  5. Wave Equation
    • Solutions by spherical means.
    • Nonhomogenous problem.
    • Energy methods.
  6. Sobolev Spaces
    • Holder spaces
    • Sobolev spaces
    • Approximation
    • Extension
    • Trace
    • Sobolev inequality
    • Compactness