PARTIAL DIFFERENTIAL EQUATIONS
Academic year and teacher
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- Versione italiana
- Academic year
- 2016/2017
- Teacher
- ANDREA CORLI
- Credits
- 6
- Didactic period
- Secondo Semestre
- SSD
- MAT/05
Training objectives
- The aim of the course is to give a simple analytic introduction to partial differential equations (PDEs), with particular reference to first-order equations.
The student shall know the physical phenomena modelized by PDEs, the
classification of PDEs, the theory of integration along the characteristics, weak and strong solutions, the theory of systems of conservation laws in one space dimension.
He/she will be able to interpret the physical phenomena modelized by a PDE, to integrate a PDE along the characteristics, solve a Riemann problem for a simple system of conservation laws. Prerequisites
- Calculus for one and several real variables. Ordinary differential equations. Linear algebra. Elementary knowledge of Matlab or analogous software.
Course programme
- Generalities about PDEs. The theory of characteristics. The theorem of local existence for noncharacteristic problems
The scalar conservation law. Weak solutions, shock and rarefaction waves, entropy conditions. The Riemann problem.
Systems of conservation laws. Weak solutions; strict hyperbolicity. Traveling waves. Genuinely nonlinear and linearly degenerate eigenvalues. Simple waves. Rarefaction and shock waves, contact discontinuities. The Riemann problem. The Riemann problem for the p-system. Didactic methods
- The course is organized through classroom lectures and tutorials. The exercises, at various levels, proposed week to week, are corrected individually by the teacher and discussed the following week with the students. Students are strongly advised to engage in this activity, both to have direct control of their level of learning, and not to have a merely theoretical knowledge of the course.
Learning assessment procedures
- The exam consists in a discussion on the topics of the course and shall be evaluated as follows:
-) up to 20 points for the theoretical part;
-) up to 10 points for the practical part: the teacher shall evaluate the exercises done by the students during the course and, in case the exercises were missing or with wrong solutions, the teacher shall assign some simple exercises to be solved during the exam.
The student must show of having understood both the technical aspects of the course (differential calculus for several real variables, linear algebra) and the physical-geometric interpretations of the results.
Exams can take place during the usual periods (January-February, June-July, September) as well as by fixing an appointment with the teacher. Reference texts
- L.C. Evans: Partial differential equations, second edition. American Mathematical Society (2010).
More detailed text:
A. Bressan: Hyperbolic systems of conservation laws. Oxford (2000).
J. Smoller: Shock waves and reaction-diffusion equations. Springer (1994).
