ECTS credits ECTS credits: 3
ECTS Hours Rules/Memories Student's work ECTS: 51 Hours of tutorials: 3 Expository Class: 9 Interactive Classroom: 12 Total: 75
Use languages Spanish, Galician
Type: Ordinary subject Master’s Degree 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 | 1st year (Yes)
Know some basic aspects of solving partial differential equations of parabolic and hyperbolic type.
Become familiar with the concept of generalized solution for a differential equation and place it within its functional context.
Know the fundamental principles of the variational formulation of a partial differential equation.
1.- (4 hours of presentation approximately) Classical solutions of Partial Differential Equations of second order: Parabolic, Hyperbolic and Elliptic. Fourier transform.
2.- (2 hours of exhibition approximately) Distributions. Calculation with distributions. Sobolev spaces.
3.- (3 hours of presentation approximately) Concept of weak solution: Variational formulation of boundary problems for partial differential equations of elliptic type. Evolution problems: the heat equation and the wave equation.
A. CABADA, Problemas Resueltos de Ecuaciones en Derivadas Parciales, http://webspersoais.usc.es/export9/sites/persoais/persoais/alberto.caba…
H. BREZIS, Analyse Fonctionnelle. Théorie et applications, Masson, 1996.
H. BREZIS, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2011.
M. GONZÁLEZ BURGOS, Apuntes de Ecuaciones en Derivadas Parciales. http://departamento.us.es/edan/php/asig/LICMAT/LMAEDP/ApuntesAEDP1213.p…
M. GROSSINHO; S. A. TERSIAN, An Introduction to Minimax Theorems and their Applications to Differential Equations. Nonconvex Optimization and its Applications. Kluwer Academic Publishers, 2001.
R. HABERMAN, Ecuaciones en Derivadas Parciales (3ª Ed.). Prentice Hall, 2003.
F. JOHN, Partial Differential Equations (4ª Ed.), Springer-Verlag, 1982
J. JOST, Partial Differential Equations (2ª Ed.), Springer, 2007
S. KESAVAN, Topics in Functional Analysis and Applications, John Wiley & Sons, 1989.
S. KESAVAN, Nonlinear Functional Analysis. A First Course, Hindustan Book Agency, 2004.
V. P. MIJAILOV, Ecuaciones Diferenciales en Derivadas Parciales, Mir. 1978.
I. PERAL, Primer curso de Ecuaciones en Derivadas Parciales. Addison-Wesley, 1995.
P. A. RAVIART; J. M. THOMAS, Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles, Masson, 1988.
I. P. STAVROULAKIS; S. A. TERSIAN, Partial Differential Equations. An introduction with Mathematica and MAPLE (2ª Ed.). World Scientific Publishing Co., 2004.
R. S. STRICHARTZ, A Guide to Distribution Theory and Fourier Transforms (2ª Ed.). World Scientific Publishing Co., 2003.
M. STRUWE, Variational methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (4ª Ed.). Springer, 2008.
In this subject, all the competences included in the Report of the Master's Degree in Mathematics of the USC will be worked on.
General competences:
(GC1) Acquisition of high-level mathematical tools for various applications, covering the expectations of graduates in mathematics and other basic sciences.
(GC2) Know the broad panorama of current mathematics, both in its lines of research, and in methodologies, resources and problems addressed in various areas.
(GC3) Train for the analysis, formulation and resolution of problems in new or unfamiliar environments, within broader contexts.
(GC4) Prepare for decision-making based on abstract considerations, to organize and plan and to solve complex issues.
Specific competences:
(EC1) Train for study and research in developing mathematical theories.
(CE2) Apply the tools of mathematics in various fields of science, technology and social sciences.
(SG3) Develop the necessary skills for the transmission of mathematics, oral and written, both in terms of formal correction and in terms of communicative effectiveness, emphasizing the use of appropriate ICT.
Transversal competences:
(CT1) Use bibliography and search tools for general and specific bibliographic resources of Mathematics, including Internet access.
(CT2) Optimally manage working time and organize available resources, establishing priorities, alternative paths and identifying logical errors in decision-making.
(CT3) Enhance the capacity for work in cooperative and multidisciplinary environments.
In addition, the training activities to be developed aim for students to acquire the following competences and learning outcomes related to Partial Differential Equations:
Understand, learn, and know how to express rigorously the concepts and techniques that are developed in the program.
Be able to explicitly solve linear second-order Partial Differential Equations.
Know relevant examples of differential equations in physics and other sciences.
Translate in terms of Partial Differential Equations some problems of the applied sciences (physics, chemistry, biology, medicine, etc.).
Master the concept of generalized derivative and weak solution.
Know basic properties of integral transforms and Functional Analysis.
Know basic concepts of critical point theory.
They will work in a special way: The rigorous and clear expression, both oral and written, logical reasoning and identification of errors in procedures, the capacity for abstraction, creativity, teamwork, the development of analytical skills in problem-solving and critical attitude towards different solutions.
The general methodological indications established in the Report of the Master's Degree in Mathematics of the University of Santiago de Compostela (USC) will be followed.
Teaching is scheduled in expository and interactive classes.
Expository Teaching (9 hours): The expository classes will be dedicated to the presentation and development of the essential contents of the subject.
Seminar and laboratory teaching (12 hours): The interactive seminar classes (6 hours) will be dedicated to the presentation of examples and problem solving (both theoretical and in the field of applications). In the interactive laboratory classes (6 hours) individual or group work will be organized and problems will be proposed to be solved by the students. In these classes, the discussion and debate with the students, as well as the resolution and presentation on their part of the proposed tasks, is essential for the knowledge to be practiced and strengthened and some of the aforementioned competences to be worked on.
The expository and interactive teaching will be face-to-face and will be complemented by the virtual course of the subject, in which students will find bibliographic materials, problem bulletins, explanatory videos, etc. Through the virtual course, students will also be able to take tests and tests for continuous evaluation, as described in the corresponding section.
The tutorials will be face-to-face or through email or the Teams platform.
The general evaluation criteria established in the Report of the Master's Degree in Mathematics of the USC will be followed.
For the calculation of the final grade (CF) the grade of the continuous assessment (AC) and the grade of the final exam (EF) will be taken into account, and the formula CF = AC/2 (1-AC/20)*EF will be applied.
This calculation will be applicable on both occasions.
The continuous evaluation will be based on the results obtained in the written controls or works commissioned by the teacher on practical or theoretical aspects of the subject, which may be individual or in groups. More specifically, students must deliver two works throughout the course, whose assessment will be the grade of the continuous evaluation AC. It will make it possible to check the degree of achievement of the specific competences mentioned above.
The final examination will consist of an individual presentation of a proposed topic well in advance. The knowledge obtained by the students in relation to the concepts and results of the subject will be measured, both from the theoretical and practical point of view, also valuing the clarity and logical rigor shown in the presentation of the same. The achievement of the basic, general and specific competences referred to in the Report of the Master in Mathematics of the USC and which were previously indicated will be evaluated.
It will be understood as NOT PRESENTED who at the end of the teaching period is not in a position to pass the subject without taking the final test and does not present himself to said test.
In the second opportunity, the same evaluation system will be used but with the test corresponding to the second opportunity, which will be an exam of the same type as that of the first.
CLASSROOM WORK
Expositive classes (9 h)
Seminar classes (6 h)
Laboratory classes (6 h)
Total hours classroom: 21
PERSONAL STUDENT WORK
Individual or group self-study (45 h)
Writing exercises, conclusions and other work (9 h)
Total hours of student work: 54
TOTAL: 75 hours
The students must handle with ease the subjects studied in the subjects of the degree in mathematics related to differential equations and functional analysis. He must also know the master’s subjects "Functional Analysis" and "Real and Complex Analysis".
Starting from this situation, they should work regularly (daily) and rigorously. It is essential to participate actively in the learning process of the subject, regularly attend classes, both theoretical and practical, in a participatory manner, especially in interactive classes, asking the pertinent questions to clarify any doubts that may arise in relationship with this subject.
Students must handle with ease the topics studied in the subjects of the degree in mathematics related to differential equations and functional analysis. You must also master the subjects of the Master "Functional Analysis" and "Real and Complex Analysis".
Starting from this situation, you must work regularly (daily) and rigorously. It is essential to participate actively in the process of learning the subject, regularly attend classes, both theoretical and practical, in a participatory way, especially in interactive classes, asking the relevant questions that allow you to clarify any doubts that may arise in relation to the subject.
Alberto Cabada Fernandez
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813206
- alberto.cabada [at] usc.gal
- Category
- Professor: University Professor
Fernando Adrian Fernandez Tojo
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- fernandoadrian.fernandez [at] usc.es
- Category
- Professor: University Lecturer
| Tuesday | |||
|---|---|---|---|
| 11:00-12:00 | Grupo /CLIL_01 | Galician, Spanish | Classroom 10 |
| 05.19.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |
| 06.27.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |