ECTS credits ECTS credits: 4.5
ECTS Hours Rules/Memories Student's work ECTS: 74.2 Hours of tutorials: 2.25 Expository Class: 18 Interactive Classroom: 18 Total: 112.45
Use languages Spanish, Galician
Type: Ordinary Degree Subject 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
This is an introduction to the study and practical resolution of partial differential equations that model real life physical processes such as vibrations, heat transfer and potential distribution. As a necessary tool for this study, some basic concepts and results on Fourier series and their convergence is studied. We also study the basic aspects of the relation between Fourier series and functional analysis in Hilbert spaces, with special attention to L2.
ITEM 1. Fourier series of functions in L1(-L, L), with L> 0. Dirichlet kernels. Pointwise convergence criteria: Dini's test and some of its consequences and Carleson's Theorem. Fejér Theorem, absolutely continuous functions and a first result of uniform convergence of Fourier series. Presentation of Hilbert spaces and their fundamental properties, with particular attention to L2(-L, L). Completeness of the trigonometric system and convergence of Fourier series in L2. The space H1(-L, L) and other results of uniform convergence of Fourier series. (5 expositive sessions.)
ITEM 2. The wave equation in one spatial dimension. General solution as superposition of waves. D'Alembert formulas for solving initial value problems. Compatibility conditions and d'Alembert formulas to solve problems involving initial homogeneous boundary conditions (Dirichlet, Neumann and mixed type). Uniqueness of solution by the method of energy. Continuous dependence of the solution with respect to the initial data. Series expression of d'Alembert solutions by Fourier series. Resolution of some inhomogeneous problems. (4 expositive sessions.)
ITEM 3. Heat equation in one spatial dimension. Resolution by the Fourier method of separation of variables of different equations (second order, linear and constant coefficients) with different types of homogeneous boundary conditions (Dirichlet, Neumann and periodic). Maximum principle and consequences: uniqueness and continuous dependence of the solution initial problems with boundary conditions of Dirichlet type. Resolution of some inhomogeneous problems. (3 expositive sessions.)
ITEM 4. Laplace equation in two dimensions. Fundamental theory of harmonic functions. Hadamard's example of an ill posed problem. Resolution by separation of variables with different types of boundary conditions in rectangular domains of the plane. Maximum principle and consequences: uniqueness and continuous dependence. (2 expositive sessions.)
Basic bibliography:
CAO LABORA, D., FERREIRO SUBRIDO, M. e LÓPEZ POUSO, R. (2023). Series de Fourier: introducción ás ecuacións en derivadas parciais. Esenciais, USC Editora.
LÓPEZ POUSO, R. (2019). Series de Fourier y ecuaciones en derivadas parciales. Manuais Universitarios, USC Editora.
MYINT-U, T. e DEBNATH, L. (2007). Linear partial differential equations for scientists and engineers, Cuarta Edición. Boston. Birkhäuser. Available on-line through SpringerLink.
Complementary bibliography:
EVANS, L. (2002). Partial differential equations. Providence, American Mathematical Society.
HABERMAN, R. (2003). Ecuaciones en derivadas parciales con series de Fourier y problemas de contorno, Tercera Edición. Madrid. Pearson Educación S. A.
KOLMOGOROV, A. N. e FOMÍN, S. V. (1978). Elementos de la Teoría de Funciones y del Análisis Funcional, Ed. Mir.
STROMBERG, K. R. (1981). Introduction to classical real analysis. Belmont, CA, Wadsworth Inc.
WEINBERGER, H. F. (1979). Ecuaciones diferenciales en derivadas parciales: con métodos de variable compleja y de transformaciones integrales. Barcelona, Reverté.
After successful study of this course, students should understand and be able to express rigorously every relevant concept, and should also be able to apply the adequeate techniques in the resolution of problems. In particular, they should be able to apply the results on the space L2, to distinguish the different types of convergence of functional series, to compute the sums of some numerical series, to study the existence and uniqueness of solution partial differential equations of 2nd order classical physics and, from a practical point of view, compute the solution of problems which involve vibrations of a string, heat transfer in a bar or potential distribution in a plate.
There are three types of sessions: expositive, seminars, and laboratories.
Expositive classes consist mostly on the teacher's explanations about theoretical aspects of the subject, but there is also room for solving exercises and work examples out.
Seminars are devoted specially to exercises and specially difficult parts of the theory which want slower pace in their explanation.
Finally, laboratories are devoted to the use of the program MAPLE as a tool for computation and graphical representations for faster resolution of exercises and better comprehension of the theory.
The Final Mark (FM) is computed using the Final Exam Mark (FEM, up to 10 points) and the Continuous Evaluation Mark (CEM, up to 10 points) according the following formula:
FM=Max{FEM,0,65*CEM+0,35*FEM}
Total working hours in classroom 45. Total working hours outside classroom 67.5
Before taking this course, students should have some knowledge about differentiation of functions of several real variables, sequences and series of functions, and Lebesgue integration.
Rodrigo Lopez Pouso
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813166
- rodrigo.lopez [at] usc.es
- Category
- Professor: University Professor
Lucia Lopez Somoza
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- lucia.lopez.somoza [at] usc.es
- Category
- Professor: LOU (Organic Law for Universities) PhD Assistant Professor
| Monday | |||
|---|---|---|---|
| 09:00-10:00 | Grupo /CLIL_05 | Spanish | Computer room 3 |
| Tuesday | |||
| 09:00-10:00 | Grupo /CLE_02 | Spanish | Classroom 06 |
| 10:00-11:00 | Grupo /CLE_01 | Spanish | Classroom 03 |
| Wednesday | |||
| 10:00-11:00 | Grupo /CLIS_04 | Spanish | Classroom 02 |
| 11:00-12:00 | Grupo /CLIS_03 | Spanish | Classroom 07 |
| 13:00-14:00 | Grupo /CLIL_02 | Spanish | Computer room 2 |
| Thursday | |||
| 09:00-10:00 | Grupo /CLIL_06 | Spanish | Computer room 3 |
| 10:00-11:00 | Grupo /CLIL_04 | Spanish | Computer room 3 |
| 12:00-13:00 | Grupo /CLIL_03 | Spanish | Computer room 4 |
| 13:00-14:00 | Grupo /CLIL_01 | Spanish | Computer room 4 |
| Friday | |||
| 10:00-11:00 | Grupo /CLIS_02 | Spanish | Classroom 03 |
| 11:00-12:00 | Grupo /CLIS_01 | Spanish | Classroom 02 |
| 05.31.2024 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |
| 07.08.2024 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |