PARTIAL DIFFERENTIAL EQUATIONS
Academic year and teacher
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- Versione italiana
- Academic year
- 2020/2021
- Teacher
- ANDREA CORLI
- Credits
- 6
- Didactic period
- Primo Semestre
- SSD
- MAT/05
Training objectives
- The aim of the course is to give a simple analytic introduction to some aspects of partial differential equations (PDEs), which are motivated by applications. In particular the focus is on
(A): first-order PDEs and
(B): second-order parabolic PDE.
According to the students' interests, the topics of the lectures shall focus either on (A) or on (B). Motivations and applications for (A) come from fluid dynamics and traffic flows; for (B), from biomathematics.
The student shall know the physical phenomena modeled by PDEs, the
classification of PDEs. (A) the theory of integration along the characteristics, weak and strong solutions, the theory of systems of conservation laws in one space dimension. (B) Analytic theory of second-order parabolic equations; stability and (Turing) instability; traveling waves.
He/she will be able to interpret the physical phenomena modelized by a PDE. (A): to integrate a PDE along the characteristics; solve a Riemann problem for a simple system of conservation laws. (B): solve and discuss the stability of a parabolic equation or of a system of parabolic equations; to study the existence and the property of a traveling wave. 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 (about 4 hours).
(A) The theory of characteristics. The theorem of local existence for noncharacteristic problems (about 12 hours). The scalar conservation law. Weak solutions, shock and rarefaction waves, entropy conditions. The Riemann problem (about 12 hours). 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. Several applications to vehicular traffic flows (about 14 hours).
(B): Reaction-diffusion equations: well-posedness and global existence of solutions (about 8 hours). Asymptotic behavior and Turing instability (about 10 hours). Applications to some models of chemical reactions. Traveling waves; the system of chemotaxis (about 20 hours). 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 geometric or physical or biological 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).
(B): C. Mascia, E. Montefusco and A. Terracina: Biomat 1.0. La Dotta (2018).
More detailed texts:
J.D. Murray: Mathematical Biology: I. An Introduction. Third Edition. Springer (2001).
