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 (IVPs) ASSOCIATED WITH ORDINARY DIFFERENTIAL EQUATIONS (ODEs):
1. To know the most common methods for the numerical resolution of IVPs for ODEs.
2. To become familiar with the concepts of convergence and order, related to accuracy, and with numerical stability, related to error blowup.
3. To observe the phenomena mentioned in the previous point, as well as the effect of rounding errors on convergence, by implementing some of the studied methods on a computer.
II. DYNAMICAL SYSTEMS:
1. To proficiently handle some analytical methods of integrating ordinary differential equations.
2. To understand and be able to analyze low-dimensional dynamical systems.
3. To understand the basic concepts of bifurcations and 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 (IVPs) ASSOCIATED WITH ORDINARY DIFFERENTIAL EQUATIONS (ODEs):
1. Concept of initial value problem (IVP) for ODEs. Idea of a numerical solution to an IVP.
2. MATLAB® commands for solving IVPs.
3. Definition of convergence and order of convergence. Discretization error and rounding error; effect of rounding error on convergence.
4. Description of Euler methods: explicit (forward) and implicit (backward).
5. Higher-order methods:
5.a. Non-linear one-step methods: Runge-Kutta (RK) methods.
5.b. Linear multistep methods (LMMs):
5.b.i. Concept of LMM. Starting procedure. Order theorem.
5.b.ii. LMMs based on numerical integration:
• Adams-Bashforth methods.
• Adams-Moulton methods.
• Nyström methods.
• Milne-Simpson methods.
5.b.iii. LMMs based on numerical differentiation: BDF methods.
II. DYNAMICAL SYSTEMS:
1. Linear dynamical systems.
1.a. Linear vector fields.
1.b. Calculation of the matrix exponential. 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 solutions. Dependence on initial conditions and parameters.
2.b. The problem of extending solutions. Maximal solutions.
2.c. Flux associated with 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. Hartman-Grobman theorem.
3.e. Gradient systems and Hamiltonian systems.
4. Global theory.
4.a. The concept of limit cycle.
4.b. Electrical circuits. Liénard systems. The Van der Pol equation.
4.c. The Poincaré map.
I. NUMERICAL METHODS FOR INITIAL VALUE PROBLEMS (IVPs) ASSOCIATED WITH 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 - To have knowledge that provides a foundation or opportunity to be original in the development and/or application of ideas, often in a research context, knowing how to translate industrial needs into terms of R&D projects in the field of Industrial Mathematics.
CG4 - To be able to communicate conclusions, along with the knowledge and ultimate reasons that support them, to specialized and non-specialized audiences in a clear and unambiguous manner.
CG5 - To possess the learning skills that allow them to continue studying in a largely self-directed or autonomous way, and to successfully undertake doctoral studies.
Specific:
CE3 - To determine whether a model of a process is well-posed mathematically and well-formulated from a physical point of view.
Specialization in "modeling":
CM1 - To be able to extract, using different analytical techniques, both qualitative and quantitative information from the models.
The aforementioned competencies, as well as those described on page 8 of the degree report at the link
https://www.usc.gal/export9/sites/webinstitucional/gl/servizos/sxopra/m…,
are covered in class and evaluated according to the system described in the "Assessment System" section.
1. Planning the content of each class.
2. Explanation on the board (lecture) or equivalent using videoconferencing.
3. Programming some methods on the computer.
The competencies CG1, CG4, and CG5, as well as CE3 and CM1, are evaluated through the following process:
To pass the course, it is mandatory to submit the assigned exercises and programming practices by the deadlines set by the instructors. The final grade will be determined by a written exam in which:
• Each of the two parts of the course, namely Numerical Methods for ODEs and Dynamical Systems, has a weight of 50% in the final grade.
• The part of the exam dedicated to Numerical Methods for ODEs reserves 30% of its value for questions related to the programming practices.
Attendance or non-attendance to classes will not affect the grade.
Personal work hours, including class hours: approximately 150 hours (25 hours per ECTS).
The professors are willing to teach the classes in English. However, as of today, this can only be implemented if every student accepts the change.
The order in which the two parts of the course are taught, namely Numerical Methods for ODEs on one hand and Dynamical Systems on the other, will be announced at the beginning of each course.
The first assessment session is divided into two exams: one at the end of the first part of the course, covering that part, and another at the end of the classes, covering the second part. It is noted that this second exam evaluates only the second part of the course, and that there is no separate exam for the first part.
Partial grades obtained in the first opportunity are not saved for the second. In particular, if a student passes one of the two exams but fails the course, she or he will have to take the entire course exam in the second assessment opportunity.
In cases of fraudulent completion 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 be applied.
Óscar López Pouso
Coordinador/a- Department
- Applied Mathematics
- Area
- Applied Mathematics
- Phone
- 881813228
- oscar.lopez [at] usc.es
- Category
- Professor: University Lecturer
| Monday | |||
|---|---|---|---|
| 09:00-10:00 | Grupo /CLE_01 | Spanish | Computer room 5 |
| Thursday | |||
| 11:00-13:00 | Grupo /CLE_01 | Spanish | Computer room 5 |