The course will provide students with a theoretical and practical grounding in the treatment of partial differential equations: position of the problem, boundary conditions, classification, analytical solutions, numerical methods.
By the end of the course, students should be able to:
- analyse a problem formulated in terms of PDEs - partial differential equations - (analytical properties and classification),
- find general or specific analytical solutions, possibly in simplified cases
- develop and validate a numerical code for approximating the solutions of PDEs.
The theoretical course is divided into two main parts. The first deals with the theoretical aspects (position of a PDE problem, classification, analytical methods); the second presents two large families of basic methods for their numerical 'solution' (approximation), with solved examples. The practical work will cover both the analytical aspects ('hand solving') and the numerical aspects ('computer solving').
Part 1: Theoretical aspects of partial differential equations (PDEs) - C. Dubussy
Notions of PDEs and well-posed problems (Dirichlet and Neumann conditions).
Examples of easy solutions of PDEs of order 1.
Characterisation of PDEs of order 2.
The homogeneous wave equation, first without initial values, then with (d'Alembert).
The homogeneous diffusion equation, the maximum principle, the case of the line, the half-line, etc.
Inhomogeneous equations, method of characteristics, Duhamel's principle.
The case of segments: the method of separation of variables.
Laplace equation. Study in dimension 2 on a rectangle, a circle, etc.
Part 2: Introduction to numerical methods for solving PDEs (A. Füzfa)
Finite difference method: construction of approximation formulae; truncation and rounding errors by example.
Edge conditions (Dirichlet, Neumann, Sommerfeld-radiative) and implementation
Application of finite differences: line method, finite difference schemes, illustrations on first- and second-order PDEs, convergence and validation of codes (analytical solutions, conserved quantities, etc.), Courant-Friedrich stability conditions
Spectral methods; principles; function approximations by orthogonal decomposition in a Hilbert space; eigenfunctions of the Laplacian in different geometries and coordinates; application to the wave or heat equation
The assessment covers 3 learning activities: (1) theoretical concepts; (2) analytical methods and (3) numerical methods.
Analytical methods will be the subject of an in-session exercise examination.
Numerical methods will be the subject of work on a detailed subject given at the end of the course and to be presented individually in the second part of the oral examination. This work will consist of the implementation and validation of applied mathematical methods seen in the course for solving PDEs.
The theoretical aspects will form the first part of the in-session oral examination, the second part being dedicated to the individual defence of the work on numerical methods.
The two-part oral examination will be given by the two course teachers.
The distribution of the final mark between these three learning activities is balanced (1/3 theory; 1/3 exercises on analytical methods and 1/3 on work on implementing numerical methods) provided that all parts are passed. In the event of failure in one or more of the parts, these will have to be retaken in the following session. The postponement of a learning activity from one academic year to the next will be left to the discretion of the holders.
| Training | Study programme | Block | Credits | Mandatory |
|---|---|---|---|---|
| Bachelor in Mathematics | Standard | 0 | 5 | |
| Bachelor in Mathematics | Standard | 3 | 5 |