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)
* Acquire fundamental knowledge about solving parabolic and hyperbolic partial differential equations.
* Understand the concept of generalized solution for differential equations within their functional context.
* Master the fundamental principles of the variational formulation of partial differential equations.
1. Classical Solutions of Second-Order Partial Differential Equations (4 lecture hours). Parabolic, Hyperbolic, and Elliptic Equations. Fourier Transform.
2. Theoretical Foundations (2 lecture hours). Distributions and calculus with distributions. Sobolev Spaces.
3. Weak Solutions and Applications (3 lecture hours). Variational formulation of boundary problems for elliptic equations. Evolution problems: heat equation and wave equation.
Basic Bibliography
* Cabada, A. (2011). Problemas resueltos de ecuaciones en derivadas parciales. Universidad de Santiago de Compostela.
* Brezis, H. (2010). Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer.
* González Burgos, M. (2015). Apuntes de ecuaciones en derivadas parciales. Universidad de Sevilla.
* Peral, I. (2000). Primer curso de ecuaciones en derivadas parciales. Universidad Autónoma de Madrid.
Complementary Bibliography
* Brezis, H. (1983). Analyse fonctionnelle. Théorie et applications. Masson.
* Grossinho, M. and Tersian, S. A. (2001). An Introduction to Minimax Theorems and Their Applications to Differential Equations. Kluwer Academic Publishers.
* Haberman, R. (1998). Ecuaciones en derivadas parciales. Prentice Hall.
* John, F. (1982). Partial Differential Equations (4th ed.). Springer-Verlag.
* Jost, J. (2002). Partial Differential Equations. Springer.
* Kesavan, S. (1989). Topics in Functional Analysis and Applications. New Age International.
* Kesavan, S. (1989). Nonlinear Functional Analysis: A First Course. Hindustan Book Agency.
* Mijailov, V. P. (1978). Ecuaciones diferenciales en derivadas parciales. Mir.
* Raviart, P. A. and Thomas, J. M. (1983). Introduction à l'analyse numérique des équations aux dérivées partielles. Masson.
* Stavroulakis, I. P. and Tersian, S. A. (2004). Partial Differential Equations: An Introduction with Mathematica and MAPLE. World Scientific.
* Strichartz, R. S. (1994). A Guide to Distribution Theory and Fourier Transforms. CRC Press.
* Struwe, M. (2008). Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (4th ed.). Springer.
This course contributes to the development of the competencies established in the USC Master's Degree in Mathematics:
General Competencies
* CG1: Acquisition of high-level mathematical tools for various applications.
* CG2: Knowledge of the current mathematical landscape: research lines, methodologies, resources, and problems.
* CG3: Ability to analyze, formulate, and solve problems in new or unfamiliar environments.
* CG4: Skill for decision-making, organization, and planning of complex issues.
Specific Competencies
* CE1: Capacity for study and research in developing mathematical theories.
* CE2: Application of mathematical tools in various scientific and technological fields.
* CE3: Development of skills for oral and written mathematical communication, with formal correctness and communicative effectiveness.
Transversal Competencies
* CT1: Use of general and specific bibliographic resources in Mathematics.
* CT2: Optimal management of time and available resources.
* CT3: Ability to work in cooperative and multidisciplinary environments.
Specific Learning Outcomes
Upon completion of the course, students will be able to:
* Understand and express with rigor the concepts and techniques of the program.
* Explicitly solve linear second-order Partial Differential Equations.
* Apply differential equations to problems in physics and other sciences.
* Use the concept of generalized derivative and weak solution.
* Handle basic properties of integral transforms and Functional Analysis.
* Apply basic concepts from critical point theory.
Special emphasis will be placed on:
* Rigorous and clear expression
* Logical reasoning and error identification
* Capacity for abstraction and creativity
* Teamwork
* Analysis in problem-solving
* Critical attitude toward different solutions
The course follows the methodological guidelines established in the USC Master's Degree in Mathematics Program.
Training Activities
* Lectures (9 hours): Presentation and development of essential theoretical content.
* Seminars (6 hours): Presentation of examples and solving theoretical and applied problems.
* Laboratories (6 hours): Individual or group work to solve proposed problems, with subsequent discussion and debate.
Teaching will be in-person and complemented with a virtual course that will include:
* Bibliographic materials
* Problem sheets
* Explanatory videos
* Tests for continuous assessment
Tutoring will be conducted in person or via email/Teams platform.
The general criterion established in the USC Master's Degree in Mathematics Program will be applied.
The final grade (FG) will be calculated using the formula: FG = max{CA, FE}
Where:
* CA: Continuous assessment grade
* FE: Final exam grade
This system will be applicable in both evaluation opportunities.
Continuous Assessment
It will be based on the results obtained in written tests or assignments on practical or theoretical aspects, which may be individual or group work. Students will complete two evaluable tasks during the course that will determine the continuous assessment grade (CA).
Additional Considerations
* Students will be considered NOT PRESENTED if they are unable to pass the course without taking the final exam and do not take that exam.
* The second opportunity will use the same evaluation system but with a specific test for this call.
In-person work (21 hours)
* Lectures: 9 hours
* Seminars: 6 hours
* Laboratories: 6 hours
Student's autonomous work (54 hours)
* Individual or group study: 45 hours
* Preparation of exercises and assignments: 9 hours
Total: 75 hours
For optimal use of this course, students should:
1. Master prior knowledge from undergraduate Mathematics courses related to differential equations and functional analysis.
2. Have completed the Master's courses "Functional Analysis" and "Real and Complex Analysis."
3. Maintain regular and rigorous work, preferably with daily dedication.
4. Actively participate in classes, both theoretical and practical.
5. Ask questions to resolve doubts about the subject.
6. Use tutoring sessions to consolidate learning.
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.
Rodrigo Lopez Pouso
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813166
- rodrigo.lopez [at] usc.es
- Category
- Professor: University Professor
Fernando Adrian Fernandez Tojo
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- fernandoadrian.fernandez [at] usc.es
- Category
- Professor: University Lecturer
| Tuesday | |||
|---|---|---|---|
| 11:00-12:00 | Grupo /CLE_01 | Spanish | Classroom 10 |
| 05.18.2026 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |
| 06.26.2026 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |