ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 99 Hours of tutorials: 3 Expository Class: 24 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician, English
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Statistics, Mathematical Analysis and Optimisation
Areas: Mathematical Analysis
Center Faculty of Mathematics
Call: Second Semester
Teaching: With teaching
Enrolment: Enrollable
To introduce students to certain global aspects of the qualitative theory of ordinary differential equations, including those related to periodic orbits and, in the case of dynamical systems in the plane, the Poincare-Bendixon theory and fixed point index theory.
To familiarize students with the classical theory of partial differential equations. Getting to know those techniques necessary in order to solve equations of first and second order. To sort equations of second order. To know existence and uniqueness results for parabolic, hyperbolic and elliptic problems.
1.- Dynamical systems. Tube flux theorem. First integrals. Poincaré-Bendixon Theory. (9h)
2. -Fixed point index theory / degree theory. Applications. (10h)
3.- Linear and quasilinear first order partial differential equations. The method of first integrals. First order nonlinear partial differential equations. Method of characteristics. Second order partial differential equations. Canonical forms. (9h)
In-library material, with reference:
ARNOLD, V. Ecuaciones Diferenciales Ordinarias, Rubiños, 1995 (1202 78, 34 466)
CARTAN, H., Cálculo Diferencial, Omega, 1978 (1202 3)
COURANT, R.; HILBERT, D. Methods of Mathematical Physics, Vol. I e II. Wiley – Interscience, 1962. (00 9)
DOU, A Ecuaciones en Derivadas Parciales. Dossat, 1970. (35 139)
EVANS, L. C. Partial Differential Equations. AMS, 1998. (1202 347, 35 402)
FONSECA, I. Degree theory in analysis and applications. Oxford, 1995. (55 231)
GOCKENBACH, M. S., Partial differential equations. Analytical and numerical Methods, Siam, 2011. (35 512)
HYUN-KU, R. First-order partial differential equations. Dover Publications 2001 (35 442)
JOHN, F. Partial Differential Equations. Springer – Verlag, 1991. (35 101)
KEVORKIAN, J. Partial differential equations: analytical solution techniques. Chapman & Hall, 1990 (35 421)
LLOYD, N. G. Degree theory. Cambridge, 1978. (55 2)
MCOWEN, R. Partial differential equations: methods and applications. Upper Saddle River, 2003 (35 459)
OUTERELO DOMÍNGUEZ, E. Mapping degree theory. Real Sociedad Matemática Española, 2009. (47 264)
PERAL, I. Primer Curso de Ecuaciones en Derivadas Parciales. Addison – Wesley, 1995. (1202 261, 35 216)
PERKO L., Differential Equations and Dinamical Systems, Springer, 1996. (1202 287, 34 400)
PETROVSKY, I.G., Lectures on Partial Differential Equations. Interscience, 1964. (35 45)
SOTOMAYOR, J., Liçoes de Equaçoes Diferenciais Ordinarias, IMPA, 1979. (1202 83, 34 165)
STAVROULAKIS, I. P.; TERSIAN, S. A. Partial Differential Equations. An Introduction with Mathematica and MAPLE. World Scientific, 2003. (35 473)
STRAUSS, W. A. Partial Differential Equations, an Introduction. John Wiley, 1992. (35 318)
WEINBERGER, H. F. Ecuaciones Diferenciales en Derivadas Parciales. Reverté, 1992. (1202 13, 35 142)
On-line material:
• CABADA, A. Problemas Resueltos de Ecuaciones en Derivadas Parciales. http://webspersoais.usc.es/persoais/alberto.cabada/materialdocente.html
• Teschl, Gerald. Ordinary Differential Equationsand Dynamical Systems. URL: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.444.2949&rep=r…
The following books are accessible through Springer Link (here they explain how to access them: https://www.youtube.com/watch?v=t8hPlEwNFLg&feature=emb_logo )
• David G. Schaeffer, John W. Cain. Ordinary Differential Equations: Basics and Beyond. URL: https://link.springer.com/book/10.1007/978-1-4939-6389-8
• Walter G. Kelley, Allan C. Peterson. The Theory of Differential Equations. URL: https://link.springer.com/book/10.1007/978-1-4419-5783-2
• Shankar Sastry. Nonlinear Systems. URL: https://link.springer.com/book/10.1007/978-1-4757-3108-8
• Hartmut Logemann, Eugene P. Ryan. Ordinary Differential Equations. https://link.springer.com/book/10.1007/978-1-4471-6398-5
• Colin Christopher, Chengzhi Li. Limit Cycles of Differential Equations. URL: https://link.springer.com/book/10.1007/978-3-7643-8410-4
• Qingkai Kong. A Short Course in Ordinary Differential Equations. URL: https://link.springer.com/book/10.1007/978-3-319-11239-8
• David Betounes. Differential Equations: Theory and Applications. URL: https://link.springer.com/book/10.1007/978-1-4419-1163-6
• Jan Willem Polderman, Jan C. Willems. Introduction to Mathematical Systems Theory. URL: https://link.springer.com/book/10.1007/978-1-4757-2953-5
We will develop those general competences in the Degree in Mathematics from USC (see link
http://www.usc.es/export9/sites/webinstitucional/gl/servizos/sxopra/mem…)
In particular, the basic competencies CB2, CB3, CB4 and CB5; the transverse competencies CT1, CT2, CT3 and CT5; as well as the totality of the general and specific ones.
Regarding the specific knowledge of the subject, we will try to make the students understand and express rigorously the concepts and techniques in the program and apply the theoretical and practical knowledge acquired in the field. We will work on the analysis and abstraction abilities in the definition, statement and search of solutions of the considered problems, both in the theoretical and applied contexts.
We will follow the general methodological indications established in the Report of the Degree in Mathematics of the USC.
We will deliver lectures, interactive seminars and labs. We will provide in the lectures the essential contents of the discipline, whereas in the seminars and labs we will devote our time to the solution of problems. The tutored meetings will be devoted to the answering of questions and the discussion and debate with students.
The different competences will be acquired by daily exposure to the various topics of the course.
The adaptation of the methodology to the other scenarios considered in the document “Directrices para o desenvolvemento dunha docencia presencial segura, curso 2020-21” appears in the section Comments.
We will follow the general assessment criteria established in the Report of the Degree in Mathematics from USC.
In the final examination, written, we will measure the knowledge gained by students in relation to the concepts and results of the field, both theoretical and practical, assessing the clarity and logical rigor shown in the exposition.
Continuous assessment: it will consist of three examinations to be taking during lecture time. The exact date of the examination will be announced in advance. Each of the examinations will take place once each of the three main chapters of the course is finished: Dynamical Systems, Index and Degree and Partial Differential Equations.
Calculation of the final grade: The numerical grade of the opportunity will be computed as max{E,0’4C+0’6E} where E is the grade of the final exam of the opportunity (which will take place at the dates indicated by the Faculty) and C is the average of the continuous assessment.
Those students who do not participate at the final exam of a given opportunity will be scored as “not presented” in that opportunity.
In those cases of fraudulent behavior regarding assessments the precepts gathered in the “Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións” will be applied.
The adaptation of the assessment system to the other scenarios considered in the document “Directrices para o desenvolvemento dunha docencia presencial segura, curso 2020-21” appears in the section Comments.
WORK IN CLASSROOM
Lecture hours (28 hours)
Hours of interactive seminars (15 hours)
Interactive laboratory hours (12 hours)
Tutoring in small groups (3 hours)
Total classroom attendance hours: 58.
PERSONAL WORK: About 92 h depending on the person and her background.
Students must have a good knowledge of the topics discussed in the subjects of Mathematical Analysis, in special with those regarding the courses of “Ordinary Differential Equations” and “Fourier Series and Introduction to Partial Differential Equations”.
Regular (daily) and rigorous work is expected. It is basic to take part actively in the learning process of the subject. To attend regularly to lectures both theoretical and practical, to participate in them, to formulate questions as well.
Considerations and modifications depending on the scenario:
Scenario II: The lecturer will provide written material, of theoretical content or exercises, so the students can follow the course’s blended learning model. For the on-line sessions, the students will have videos available on-line at the course site. Tutoring will take place in Microsoft Teams. The assessment system will be the same as in Scenario I.
Scenario III: The lecturer will provide written material, of theoretical content or exercises, so the students can follow the course on-line. The students will have videos available on-line at the course site. Tutoring will take place in Microsoft Teams. Assessment will have the same structure as in the other scenarios, but it will be on-line.
Fernando Adrian Fernandez Tojo
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- fernandoadrian.fernandez [at] usc.es
- Category
- Professor: LOU (Organic Law for Universities) PhD Assistant Professor
| Monday | |||
|---|---|---|---|
| 18:00-19:00 | Grupo /CLE_01 | Spanish | Ramón María Aller Ulloa Main Hall |
| Tuesday | |||
| 18:00-19:00 | Grupo /CLE_01 | Spanish | Ramón María Aller Ulloa Main Hall |
| Wednesday | |||
| 18:00-19:00 | Grupo /CLIS_01 | Spanish | Ramón María Aller Ulloa Main Hall |
| Thursday | |||
| 18:00-19:00 | Grupo /CLIL_01 | Spanish | Ramón María Aller Ulloa Main Hall |
| 05.27.2021 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |
| 07.09.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 02 |