ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 102 Hours of tutorials: 6 Expository Class: 18 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician
Type: Ordinary subject Master’s Degree RD 1393/2007 - 822/2021
Departments: Applied Mathematics
Areas: Applied Mathematics
Center Faculty of Mathematics
Call: First Semester
Teaching: With teaching
Enrolment: Enrollable | 1st year (Yes)
I. NUMERICAL METHODS FOR INITIAL VALUE PROBLEMS (IVP) ASSOCIATED TO ORDINARY DIFFERENTIAL EQUATIONS (ODEs):
1. To know the most common methods for solving IVP associated to EDOs.
2. To become familiar with the concepts of convergence and order, related to accuracy, and with the one of numerical stability, related to error blowup.
3. To inspect the phenomena mentioned in the previous item, and also the effect of rounding errors on convergence, by implementing on the computer some of the methods studied.
II. DYNAMICAL SYSTEMS:
1. To handle with ease some analytical methods for the integration of ODEs.
2. To understand and to be able to analyze the low-dimensional dynamical systems.
3. To understand the basic concepts on bifurcations and to know how to apply them to specific problems.
4. To use dynamical systems to model and analyze problems of industrial interest.
I. NUMERICAL METHODS FOR INITIAL VALUE PROBLEMS (IVP) ASSOCIATED TO ORDINARY DIFFERENTIAL EQUATIONS (ODEs):
1. Concept of initial value problem (IVP) for ODEs. Idea of numerical solution of an IVP.
2. MATLAB® commands for solving IVPs.
3. Definition of convergence and order of convergence. Discretization error and rounding error; effect of rounding errors on convergence.
4. Description of Euler methods: explicit (forward) and implicit (backward).
5. High order methods:
5.a. One-step non-linear methods: Runge-Kutta (RK) methods.
5.b. Linear multistep methods (LMM):
5.b.i. Concept of LMM. Starting procedure. Order theorem.
5.b.ii. LMM based on numerical integration:
• Adams-Bashforth methods.
• Adams-Moulton methods.
• Nyström methods.
• Milne-Simpson methods.
5.b.iii. LMM based on numerical differentiation: BDF methods.
II. DYNAMICAL SYSTEMS:
1. Linear dynamical systems.
1.a. Linear vector fields.
1.b. Calculus of the exponential of a matrix. Jordan canonical form.
1.c. Fundamental theorem of existence and uniqueness of solution for linear systems.
1.d. Invariant subspaces: stable, unstable and central spaces.
2. Basic theorems related to the general theory of differential equations.
2.a. The fundamental theorem of existence and uniqueness of solution. Dependence on the parameters and on the initial conditions.
2.b. The problem of extension of solutions. Maximal solutions
2.c. Flux associated to a differential field. Singular and regular points. Orbits. alpha-limit and omega-limit sets.
3. Local theory.
3.a. Lyapunov stability. Lyapunov functions.
3.b. Concepts of equivalence and topological conjugacy. Structural stability.
3.c. The invariant manifold theorem.
3.d. The Hartman-Grobman theorem.
3.e. Gradient and Hamiltonian systems.
4. Global theory.
4.a. The concept of limit cycle.
4.b. Electric circuits. Liénard systems. The Van der Pol equation.
4.c. The Poincaré map.
I. NUMERICAL METHODS FOR INITIAL VALUE PROBLEMS (IVP) ASSOCIATED TO ORDINARY DIFFERENTIAL EQUATIONS (ODES):
BASIC BIBLIOGRAPHY:
1. ASCHER, URI M.; PETZOLD, LINDA R. (1998) Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia, PA.
2. HAIRER, ERNST; NØRSETT, SYVERT PAUL; WANNER, GERHARD (1987) Solving Ordinary Differential Equations I. Nonstiff Problems. Springer, Berlin.
3. ISAACSON, EUGENE; KELLER, HERBERT BISHOP (1994, unabridged, corrected republication) Analysis of Numerical Methods. Dover Publications, New York, NY. [Original edition: Wiley, 1966].
4. ISERLES, ARIEH (2008, second edition) A first course in the numerical analaysis of differential equations. Cambridge Texts in Applied Mathematics. Cambridge University Press. Cambridge. [First edition: 1997.]
5. LAMBERT, JOHN DENHOLM (1991) Numerical Methods for Ordinary Differential Systems. Wiley, Chichester.
6. STOER, JOSEF; BULIRSCH, ROLAND (1993, second edition) Introduction to Numerical Analysis. Springer, New York, NY. [First edition: 1980].
COMPLEMENTARY BIBLIOGRAPHY:
1. BUTCHER, JOHN CHARLES (2008, second edition) Numerical Methods for Ordinary Differential Equations. Wiley, Chichester. [First edition: 2003.]
2. CROUZEIX, MICHEL; MIGNOT, ALAIN L. (1989, second edition) Analyse Numérique des Équations Différentielles. Masson, Paris. [First edition: 1984.]
3. DEKKER, KEES; VERWER, JAN G. (1984) Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. Elsevier Science Publishers B. V., Amsterdam.
4. HAIRER, ERNST; WANNER, GERHARD (1991) Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer, Berlin.
5. HENRICI, PETER (1962) Discrete Variable Methods in Ordinary Differential Equations. Wiley. New York, NY.
6. KINCAID, DAVID RONALD; CHENEY, ELLIOT WARD (1991) Numerical Analysis. Brooks/Cole, Pacific Grove, CA.
7. LAMBERT, JOHN DENHOLM (1973) Computational Methods in Ordinary Differential Equations. Wiley, London.
8. QUARTERONI, ALFIO; SACCO, RICCARDO; SALERI, FAUSTO (2000) Numerical Mathematics. Springer, New York, NY.
II. DYNAMICAL SYSTEMS:
BASIC BIBLIOGRAPHY:
1. PERKO, LAWRENCE (2000, third edition). Differential Equations and Dynamical Systems. Texts in Applied Mathematics 7. Springer.
2. HIRSCH, MORRIS W.; SMALE, STEPHEN (1974). Differential Equations, Dynamical Systems and Linear Algebra. Pure and Applied Mathematics. Academic Press.
COMPLEMENTARY BIBLIOGRAPHY:
1. GUCKENHEIMER, JOHN; HOLMES, PHILIP (1983). Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag New York.
2. HALE, JACK K.; KOÇAK, HÜSEYIN (1991). Dynamics and Bifurcations. Springer-Verlag New York.
3. HAIRER, ERNST; NØRSETT, SYVERT PAUL; WANNER, GERHARD (1987) Solving Ordinary Differential Equations I. Nonstiff Problems. Springer, Berlin.
Basic and general:
CG1 - Have knowledge that provide a basis or opportunity for originality in developing and / or applying ideas in a research context, knowing how to translate industrial needs in terms of R + D + i in the field of Industrial mathematics.
CG4 - Knowing how to communicate the conclusions to specialist and non-specialist audiences clearly and unambiguously.
CG5 - Possess learning skills that enable them to continue studying in a way that will be largely self-directed or autonomous, and able to successfully undertake doctoral studies.
Specific:
CE3 - Determine if a model of a process is mathematically well posed and well formulated from the physical standpoint.
Specialty "modeling":
CM1 – To be able to extract, using different analytical techniques, both qualitative and quantitative information from the models.
The previous competences, as well as those described on page 8 of the memory of the degree on the link
https://www.usc.gal/export9/sites/webinstitucional/gl/servizos/sxopra/m…,
are developed in class and evaluated according to the system described in the "Assessment system" section.
1. Planning for the contents of each class.
2. Explanation on blackboard (lecture) or equivalent by using videoconferencing.
3. Programming some methods on the computer.
Skills CG1, CG4 and CG5, as well as CE3 and CM1, are assessed by means of the following process:
To pass the course it is compulsory to hand in the exercises and programming practices commissioned by the teachers within the timeframes set to it. The final grade will result from a written examination in which:
• Each of the two parts of this subject, namely Numerical Methods for ODEs on the one hand and Dynamical Systems on the other hand, weighs the 50% of the total mark.
• The part of the exam devoted to Numerical Methods for ODEs reserves 30% of its value for questions related to the programming practices.
To attend or not to attend classes will not have influence in the final mark.
Personal working hours, including class time: about 150 hours (25 hours per ECTS).
Teachers are willing to teach in English. As of today, this can be done only in case every student accept the change.
The order in which the two parts of this subject are explained, namely Numerical Methods for ODEs on the one hand and Dynamical Systems on the other hand, will be communicated to the students at the beginning of each academic course.
CONTINGENCY PLAN for the adaptation of this guide to the document "Bases para o desenvolvemento dunha docencia presencial segura no curso 2020-2021", approved by the "Consello de Goberno" of the USC in ordinary session held on June 19, 2020:
The virtual course of the USC platform will remain active. It will be the responsibility of the students to check that they have access to the virtual course, and, if not, let the teacher know.
If necessary, the teacher will be in charge of providing the students with notes or other appropriate bibliography so that they can prepare the subject and carry out the tasks assigned to them.
Classes will continue through videoconference whenever possible.
In the event that it is possible to access the university buildings, but there is a limited capacity that prevents the presence in class of all the students, shifts will be established for the classes. These classes will be used to answer questions and explain the most difficult parts of the subject. In addition to programming practices for the EDO part, students will be asked to submit exercises from the SD part. Together, the sum of both tasks (which is the "continuous assessment") will have a weight of 30% in the final grade. The written final exam will be maintained, which will also be taken in turns if necessary, and will have a weight of 70% in the final grade.
In the event of closure of facilities, the same teaching model will be followed as described in the previous paragraph, but classes will be followed by students from their homes. Regarding the evaluation model, students will receive assignments that must be delivered within the term established by the teacher. These tasks should be carried out in groups that the teacher will be in charge of creating. The final grade will result from the evaluation of these tasks. The teacher reserves the right, however, to carry out telematic evaluation sessions on the tasks assigned.
Different combinations of the situations described above may occur. In such cases, the teacher, in agreement with the M2i Academic Commission, will let the students know at all times what the teaching system and the evaluation system are, taking into account that there will always be a written final exam if it is possible to do it, and that the continuous evaluation, in cases of sanitary emergency, will have a weight of not less than 30% in the final grade.
In cases of fraudulent performance of exercises or tests, the provisions of the “Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións” of the USC will apply.
Óscar López Pouso
Coordinador/a- Department
- Applied Mathematics
- Area
- Applied Mathematics
- Phone
- 881813228
- oscar.lopez [at] usc.es
- Category
- Professor: University Lecturer
| Wednesday | |||
|---|---|---|---|
| 09:00-10:00 | Grupo /CLE_01 | Spanish | Computer room 5 |
| Thursday | |||
| 11:00-13:00 | Grupo /CLE_01 | Spanish | Computer room 5 |
| 10.22.2019 10:00-14:00 | Grupo /CLE_01 | Computer room 5 |
| 11.06.2019 16:00-20:00 | Grupo /CLE_01 | Computer room 5 |
| 06.15.2020 10:00-14:00 | Grupo /CLE_01 | Computer room 5 |