Degree | Type | Year | Semester |
---|---|---|---|
2500149 Mathematics | OB | 3 | 1 |
Linear Algebra
Calculus in Several variables.
The Theory of Differential Equations is distinguished both by the richness of ideas and methods as well as by its applicability. Thus the subject Differential Equations and Modeling I has a theoretical aspect (that will be used in theory and problem lessons) as well as a very applied aspect (which will be introduced in the theory sessions and will be developed in problems and practical sessions). Practical lessons will be carried out in the computer lab. On the one hand we will emphasize the presentation of the theory and the demonstration of the results and on the other hand the students will learn how to model real situations that allow them to predict the studied behaviors.
We think that this subject is good to show to the students that certain theoretical results that they already know about other subjects (topological properties of normed spaces and Jordan canonical forms, for example) can be applied to develop the theory of differential equations.
1. Differential Equations of the first order.
1.1 Introduction to differentia equations. Separable equatiuons. Exact equations.
1.2 Applications to modelling.
2. The linear equation.
2.1 Uniqueness and existence theorems. Algebraic properties of the space of solutions. Liouville's theoreme.
2.2 The autonomous case: Exponencial of a matrix.
2.3 The linear equation of order n.
3. Uniqueness and existence theorems
3.1 The Cauchy's problem. Picard and Peano theorems.
3.2 Prolongation of solutions. Wintner's lemma
3.3 Continuous and differentiable dependence on initial data and parameters.
4. Qualitative theory of autonomous systems.
4.1 Dynamical systems.Critical points and periodic orbits. Stability. Conjugation of dynamical systems.
4.2 Tubular flow theorem. Hartman's theorem.
4.3 Qualitative study of the autonomous linear equation.
Fundamental in the learning process of the subject is the work by the student, who can count on the guidance of the teacher at each moment.
There will be three types of guided activities:
Theory Classes: The teacher introduces the basic concepts of the subject matter showing examples, demostrating properties and fundamental results. The student must complement the teacher's explanations with personal study.
Classes of Problems: We work on the understanding and application of the concepts and tools introduced to theory, with the realization of theoretical and/or practical exercises. It is well known that the only way to learn mathematics is by solving lots and lots of problems. For this reason we think that the student must dedicate a minimum of 5 hours a week to solving problems in this subject. The student will have a list of problems for each theme, which he must think about, try to solve and which will be worked on in the problem classes. A delivery of problems is requested for each theme to ensure that this work is done continuously.
Computer practices: in each session a different type of differential equation will be dealt with to model a real situation and predict future behaviors depending on circumstancial parameters.
The exercises that appear in the lists of Problems or Computer Practices and that have not finished in the corresponding session will have to solve them the student like part of his autonomous work.
The notes on the Theory, the lists of Problems and Computer Practices will be posted on the subject's Moodle Aules website; a summary of the Theory and Problem classes is also posted weekly.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|
Type: Directed | |||
Practical modelization problems | 12 | 0.48 | |
Theory classes | 30 | 1.2 | |
Type: Supervised | |||
Problem classes | 30 | 1.2 | |
Type: Autonomous | |||
Study of the theory and resolution of problems | 114 | 4.56 |
50% of the course will be evaluated in a continuous way.
Continuous evaluation:
Evaluation that can be repeated to get a new oportunity:
To pass the course it is necessary to have obtained an average note of at least 35% for the partial exams (resp. new opportunity)
More concrete, we apply the following averaging to obtain the final note for the course:
In each case, when going for a second opportunity, obtaining the score ER/10 for this exam, then the final and definite score NF2 will be:
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
Exam of practical sessions | 10% | 3 | 0.12 | 5, 4, 3, 6 |
Handout of practical work | 20% | 12 | 0.48 | 5, 4, 3, 6, 7 |
Handout of problemes | 10% | 12 | 0.48 | 2, 1, 4, 3, 6, 7 |
Partial exams | 60% | 8 | 0.32 | 2, 1, 4, 3, 6 |
Repeated final exam | 50% | 4 | 0.16 | 2, 1, 4, 3, 6 |
P. Blanchard, and R.L. Devaney. Differential Equations. G.R. Hall, 2002. Traduït al castellà: "Ecuaciones Diferenciales". International Thomson Editores, México, 1999.
E. Boyce, y R.C. Di Prima. Ecuaciones Diferenciales y Problemas con Valores en la Frontera. Ed. Limusa, México, 1967.
R.L. Borrelli and C.S. Coleman. Differential equations: a modeling perspective. Prentice-Hall, 1987.
M. Braun. Ecuaciones diferenciales y sus aplicaciones. Grupo Editorial Iberoamérica. México, 2000.
R. Cubarsí. Equacions diferencials i la transformada de Laplace. Iniciativa Digital Politècnica, 2012. (http://hdl.handle.net/2099.3/36610)
C. Fernandez y J.M. Vegas. Ecuaciones diferenciales. Pirámide, Madrid, 1996.
G. Fulford, P. Forrester, A. Jones. Modelling with differential and difference equations. Cambridge University Press, New York, 1997.
M. Guzmán. Ecuaciones diferenciales ordinarias. Ed. Alhambra, Madrid, 1978.
M. W. Hirsch , S. Smale, R. Devaney. Differential Equations, Dynamical Systems: An Introduction to Chaos. Elsevier, 2003.
V. Jimenez. Ecuaciones diferenciales. Serie: enseñanza. Universidad de Murcia, 2000.
M.C. Leseduarte, M. D. Llongueras, A. Magaña, R. Quintanilla de Latorre. Equacions Diferencials: Problemes resolts. Iniciativa Digital Politècnica, 2012. (http://hdl.handle.net/2099.3/36607)
F. Mañosas. Apunts d'Equacions diferencials. Campus virtual.
R. Martínez. Models amb Equacions Diferencials. Materials de la UAB, Servei de Publicacions de la UAB, no. 149. Bellaterra, 2004.
R.K. Nagle, E.B. Saff and A.D. Snyder. Fundamentos de Ecuaciones diferenciales. Addison Wesley, 1992.
C.Perelló. Càlcul infinitesimal amb mètodes numèrics i aplicacions. Enciclopèdia Catalana, 1994.
G.F. Simmons. Ecuaciones diferenciales con aplicaciones y notas históricas. Mc Graw-Hill, 1977.
H. Ricardo. Ecuaciones diferenciales: una introducción moderna. Editorial Reverté, Barcelona, 2008.
D.G. Zill. Ecuaciones diferenciales con aplicaciones de modelado. International Thomson Editores, México, 2001.
In the computer sessions the students makes use of the programs SAGE and Excel; to resolve the problems proposed in these sessions students often receive information in the language of the computer algebra package of Mathematica.
There will also made use of the program P4 to show the behavior in the neighbourhood of critical points for polinomial systems in two dimensions.