Partial Differential Equations: Modelling, Analysis and Numerical Approximation
Code: 45561 ECTS Credits: 6| Degree | Type | Year |
|---|---|---|
| Modelización para la Ciencia y la Ingeniería / Modelling for Science and Engineering | OP | 1 |
Contact
- Name:
- Francisco Javier Mora Gine
- Email:
- xavier.mora@uab.cat
Teachers
- Susana Serna Salichs
Teaching groups languages
You can view this information at the end of this document.
Prerequisites
Students should have basic knowledge of calculus, algebra and ordinary differential equations, as well as basic notions of programming.
Objectives and Contextualisation
Many phenomena that unfold in space and/or time can be modelled by means of partial differential equations. The purpose of this course is to provide the main concepts about such models as well as numerical methods for computing their solution.
Learning Outcomes
- CA18 (Competence) Computationally implement numerical analysis techniques to roughly solve partial differential equations.
- CA19 (Competence) Integrate partial differential equations into other modelling tools in the context of multidisciplinary projects.
- CA19 (Competence) Integrate partial differential equations into other modelling tools in the context of multidisciplinary projects.
- CA20 (Competence) Incorporate, using partial differential equations, sustainability and/or environmental efficiency criteria in mathematical modelling projects.
- CA20 (Competence) Incorporate, using partial differential equations, sustainability and/or environmental efficiency criteria in mathematical modelling projects.
- KA15 (Knowledge) Identify the mathematical analysis methods of partial differential equations.
- KA16 (Knowledge) Recognise the role and usefulness of partial differential equations in the construction of mathematical models.
- SA18 (Skill) Apply models based on partial differential equations to solve specific problems.
- SA19 (Skill) Interpret the meaning and phenomenology associated with the parameters present in partial differential equations, in order to describe specific processes.
- SA20 (Skill) Interpret the results obtained from applying a formalised model with partial differential equations.
Content
PART I: PDE MODELS AND THEIR MAIN PROPERTIES
I.0. Introduction: Examples, different types of equations.
I.1. The heat equation. The solution formula for the pure initial value problem; the Gauss kernel. Solution by means of the Fourier method in the case of a bounded interval with Dirichlet or Neumann boundary conditions. Dissipative character of the heat equation. The parabolic maximum principle.
I.2. The wave equation. The solution formula for the pure initial value problem. Solution by means of the Fourier method in the case of a bounded interval with Dirichlet or Neumann boundary conditions. Conservative character of the wave equation.
I.3. Laplace's equation with Dirichlet or Neumann boundary conditions. Variational principle. The elliptic maximum principle. The Poisson kernel. Solution by means of the Fourier method in the case of a rectangle, a circle or a sphere.
I.4. Turing's "chemical basis of morphogenesis".
I.5. Travelling-wave solutions of non-linear heat equations.
I.6. The traffic equation and scalar conservation laws. Shocks. Weak solutions. Rankine-Hugoniot and entropy conditions.
I.7. The Navier-Stokes equations.
PART II: NUMERICAL METHODS
II.1. Finite difference methods for scalar parabolic equations: Euler explicit, Euler implicit and Crank-Nicholson methods: Von Neumann stability test. Parabolic stability Courant-Friedrichs-Lewy condition. Examples.
II.2. Numerical methods for elliptic equations.
II.3. Numerical methods for scalar conservation laws: Finite difference methods in conservation form. Shock-capturing schemes. Monotone schemes: Lax-Friedrichs and upwind schemes. Convergence and stability conditions. Entropy-satisfying schemes. Examples.
Activities and Methodology
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Classes of theory and exercises | 30 | 1.2 | CA18, CA19, CA20, KA15, KA16, SA18, SA19, SA20, CA18 |
| Type: Supervised | |||
| Internship classes | 8 | 0.32 | CA18, CA19, CA20, KA15, KA16, SA18, SA19, SA20, CA18 |
| Type: Autonomous | |||
| Studies and practical work by the student. | 96 | 3.84 | CA18, CA19, CA20, KA15, KA16, SA18, SA19, SA20, CA18 |
The aim of the classes of theory, problems and practices is to give to the students the most basic knowledge about partial differential eauqtions and their applications.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Assessment
Continous Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| First partial exam | 30% | 4 | 0.16 | CA18, CA19, CA20, KA15, KA16, SA18, SA19, SA20 |
| Second partial exam | 30% | 4 | 0.16 | CA18, CA19, CA20, KA15, KA16, SA18, SA19, SA20 |
| Solution of a problem with a computer | 40% | 8 | 0.32 | CA18, CA19, CA20, KA15, KA16, SA18, SA19, SA20 |
The assessment will consist of two partial exams and the delivery of the resolution of a problem by means of the computer.
Bibliography
Bibliography
L.C. Evans, Partial differential equations, Graduate Studies in Mathematics 19 (2nd ed.), Providence, R.I., American Mathematical Society, (2010).
B. Gustafson, H-O. Kreiss and J. Oliger, Time Dependent Problems and Difference Methods, Wiley-Intersciences, (1996).
F. John, Partial Differential equations, vol. 1, Applied Math Sciences, Springer, (1978).
P.D. Lax, Hyperbolic systems of Conservation Laws and The Mathematical Theory of Shock Waves SIAM, 1973.
R.J. LeVeque, Finite Volume Methods for Hyperbolic problems, Cambridge University Press, 2002.
Y.Pinchover, J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge 2005.
S. Salsa, Partial differential equations in action : from modelling to theory Springer, 2008.
G. Strang, Introduction to Applied Mathematics, Wellesley-Cambridge Press, (1986).
E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A practical Introduction, Springer-Verlag, 2009.
G.B. Whitham Linear and nonlinear Waves, Wiley-Intersciences, (1999).
Software
We leave full freedom to students to use the language that suits them best to do the numerical exercises of this course.
Groups and Languages
Please note that this information is provisional until 30 November 2025. You can check it through this link. To consult the language you will need to enter the CODE of the subject.
| Name | Group | Language | Semester | Turn |
|---|---|---|---|---|
| (TEm) Theory (master) | 1 | English | second semester | afternoon |