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Partial Differential Equations
Last Updated: 2026-06-03 00:07:54
Abstract
Introduction to first and second-order PDE: transport, wave, Laplace, and heat equations. PDE methods: superposition, reflection methods, representation formulae, Duhamel, separation of variables, etc. Introduction to existence and regularity theories. Some example results for nonlinear PDE.
Objective
At the end of the course, students should have a good understanding of: - First-order PDE: transport equation. Method of characteristics: theory and computations. - Standard theory of heat, Laplace, and wave equations. In particular, standard methods to represent/compute solutions. - Separation of variables, Laplace's eigenfunctions, and applications. - Challenges of existence theory and how to solve them.
Content
1. First-order equations. The Method of characteristics: linear and nonlinear, transport equation, Hamilton-Jacobi equation, scalar conservation laws. 2. Wave equation. D'Alembert formula, solutions by spherical means. 3. Laplace equation. Fundamental solution, harmonic functions and main properties, maximum principle. Poisson equation. Green functions. Regularity of solutions to the Poisson Equation. 4. Heat equation, fundamental solution, existence of solutions to the Cauchy problem and representation formulas, main properties, uniqueness and maximum principle, regularity. Methods with broad applicability (not specific to a single PDE): Duhamel principle. Separation of variables: "theory and practice". Eigenfunctions and eigenvalues of Laplace operator. Fourier transform. Representation formulae. Introduction to the existence and regularity theory: energy and viscosity approaches. Focus on estimates versus qualitative regularity. Discrete approximations of PDE and existence via a priori estimates. Schauder estimates. Barriers and regularity up to the boundary. More advanced topics: Functional analytic aspects of PDE; Calculus of Variations; existence and regularity of Dirichlet eigenfunctions: ABP estimate, Courant Lemma, Weyl law, etc.
Resources
Literature
L. Evans, Partial Differential Equations, AMS 2010 (2nd edition) Q. Han, F. Lin, Elliptic Partial Differential Equations: Second Edition, Courant Lecture Notes. W. Strauss, Partial Differential Equations: An Introduction. 2nd ed. Hoboken, NJ: John Wiley & Sons, 2007. F. John, Partial Differential Equations, Springer, 1995. H. Brezis Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. Y. Pinchover & J. Rubinstein: An Introduction to Partial Differential Equations, Cambridge University Press, 2005.
General Information
- Language
- English
- Levels
- BSC , MSC
- Frequency
- Yearly recurring
Examination
- Type
- session examination
- Mode
- oral 30 minutes
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture | Partial Differential Equations | No time listed | 4 h weekly |
| exercise | Partial Differential Equations | No time listed | 2 h weekly |
Offered In
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Electives (For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 14 of the required 26 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.)
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