Minicourse on Partial Differential Equations: Analytical and Numerical Tools
May 1 – June 30, 2025
ICTP-SAIFR, São Paulo, Brazil
ICTP-SAIFR/IFT-UNESP
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In this minicourse, we aim to introduce the participants to the expansive field of Partial Differential Equations (PDEs). These equations serve as powerful tools for modeling a remarkable spectrum of natural phenomena—ranging from quantum effects at nanometer spatial scales and femtosecond timescales, to fluid dynamics at scales familiar from everyday life, and even up to the evolution of galaxies and cosmic structures across billions of years.
Here, perhaps at the expense of some depth, we propose a comprehensive survey of the different types of PDEs. We will not only study their analytical properties, but also explore their numerical approximations, which will, in turn, help us visualize these properties and behaviors more clearly. To this end, we will delve into fundamental analytical tools that are essential not only for demonstrating the existence, uniqueness, and stability of solutions for a given type of problem, but also for establishing the convergence of the numerical approximations we will be producing.
Ultimately, the aim is to provide a balanced understanding of both the theory and practical techniques for solving PDEs. By combining analytical and numerical approaches, we seek to equip the participants with the tools necessary for addressing real-world problems and advancing their study of these powerful equations.
- Oscar Reula (National University of Córdoba, Argentina)
Teaching Assistants:
- Joaquín Pelle (AEI-Max-Planck Institute at Golm, Germany)
- Pablo Montes (National University of Córdoba, Argentina)
Click HERE for online application to the school
Deadline: April 1, 2025
Main topics
| Theory | Numerical | Labs | |
|---|---|---|---|
| Topology and Linear Algebra (Classes 1 and 2) | Functional Analysis and Fourier Theory (Classes 3 and 4) | Lab 1A: Getting familiar with Julia and its environment Lab 1B: The logistic map | |
| 1. Vectors, covectors, tensors, symmetries, complexification 2. Quotient spaces 3. Norms, induced norms 4. Inner product 5. Linear maps, invariant subspaces, eigenvalues-eigenvectors, exponentials, adjoint and unitary operators 6. Topological Spaces 7. Examples, Continuity, Compactness, Sequences, convergence 8. Stability of fixed points | 1. Basic Elements of Functional Analysis 2. Completing a normed space 3. Hilbert spaces 4. Riesz Representation Theorem 5. Sobolev spaces of positive integer indices and the Poincaré-Hardy theorem 6. Fourier Series and Sobolev Spaces 7. Basic properties of Fourier Series 8. Sobolev spaces of real indices, two important theorems | ||
| Ordinary Differential Equations – Analytical Studies (Classes 5 and 6) | Ordinary Differential Equations – Numerical Studies (Classes 7 and 8) | Lab 2A: Solving ordinary differential equations with numerical methods Lab 2B: Computing the stability region of some numerical integration schemes | |
| 1. Definition, examples, uniqueness, existence 2. Reduction to first-order systems 3. Geometric interpretation 4. First integrals 5. Fundamental theorem, dependence on parameters, variational equation 6. Linear systems, general solution 7. Stability | 1. Defining the problem 2. Various approximation methods 3. Existence proof using Euler’s method 4. Stability regions | ||
| Evolutionary Partial Differential Equations – Analytical Studies (Classes 9 and 10) | Evolution Partial Differential Equations – Numerical Studies (Periodic Case) (Classes 11 and 12) | Lab 3A: Solving a single wave equation in a periodic domain Lab 3B: Solving a simple hyperbolic system | |
| 1. Examples: advection and Burgers’ equation 2. The Cauchy problem 3. Symmetric-hyperbolic systems: Wave equations, Maxwell’s equations, Einstein’s equations, etc. 4. Propagation cones 5. Existence and uniqueness, maximum propagation speed | 1. Method of lines 2. Discretizing space, finite differences 3. Discretizing time 4. Stability of evolution operators and the CFL condition | ||
| Evolutionary Partial Differential Equations (Classes 13 and 14) | Evolution Partial Differential Equations – Numerical Studies (Boundary Conditions) (Classes 15 and 16) | Lab 4: Solving the second-order wave equation with boundaries and discontinuous interfaces | |
| 1. The initial-boundary-value problem 2. Energy estimates with boundaries | 1. Operators satisfying summation by parts 2. Applying boundary conditions using penalty methods | ||
| Non-linear Theory (Classes 17 and 18) | Other Evolutionary Equations (Classes 19 and 20) | Lab 5: Solving the heat equation | |
| 1. An example (Burgers equation) 2. The general theory | 1. Parabolic equations (Heat equation) 2. Mixed systems (Navier-Stokes) 3. Schrödinger equation | ||
| Weak Solutions, Shocks (Classes 21 and 22) | Approximating Weak Solutions (Classes 23 and 24) | Lab 6: Solving Burgers equation | |
| 1. Examples 2. Juncture conditions 3. Propagation Speeds 4. Lack of uniqueness 5. Entropy conditions | Lax-Friedrich and Weno algorithms for approximating weak solutions | ||
| Stationary Partial Differential Equations – Analytical Studies (Classes 25 and 26) | Stationary Partial Differential Equations – Numerical Studies (Classes 27 and 28) | Lab 7: Solving the Laplacian with Dirichlet boundary conditions | |
| 1. The boundary problem 2. Ellipticity 3. Example: the Laplacian 4. Weak existence and uniqueness 5. Generalizations | 1. Finite element theory 2. Solving problems in their weak formulation using Gridap | ||
| Further Topics on Hyperbolic Systems (Classes 29 and 30) | Further Topics on Stationary Systems (Classes 31 and 32) | ||
| 1. Strongly Hyperbolic Systems 2. Constraints | 1. Lax convergence theorem 2. Finite element approximation theory 3. Variations, non-elliptic systems (min-max) |
Registration
Program
TBA
The schedule might be changed.
Additional Information
Attention! Some participants in ICTP-SAIFR activities have received email from fake travel agencies asking for credit card information. All communication with participants will be made by ICTP-SAIFR staff using an e-mail “@ictp-saifr.org”. We will not send any mailings about accommodation that require a credit card number or any sort of deposit.
BOARDING PASS: All participants, whose travel has been provided or will be reimbursed by ICTP-SAIFR, should bring the boarding pass upon registration. The return boarding pass (PDF, if online check-in, scan or picture, if physical) should be sent to secretary@ictp-saifr.org by e-mail.
COVID-19: Brazilians and foreigners no longer have to present proof of vaccination before entering the country.
Visa information: Nationals from several countries in Latin America and Europe are exempt from tourist visa. Nationals from Australia, Canada and USA are exempt from tourist visa until April 10, 2025. Please check here which nationals need a tourist visa to enter Brazil.
Accommodation: Participants, whose accommodation will be provided by the institute, will stay at The Universe Flat. Hotel recommendations are available here.
Poster presentation: Participants who are presenting a poster MUST BRING A PRINTED BANNER . The banner size should be at most 1 m (width) x 1,5 m (length). We do not accept A4 or A3 paper. Click here to see what a banner looks like: http://designplast.ind.br/produtos/detalhe/impressao-digital/banner/119/9