Fourier Series and Introduction to Partial Differential Equations

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 L^2.

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.

In this section and the next ones we will take into account the list of competences according to the memory of the degree in Mathematics, which can be found at the web page of the Faculty of Mathematics of USC.

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.

During expositive sessions the theoretical aspects of the subject will be shown, illustrating them with examples in order to make it more comprehensible. Moreover, some time will be dedicated to solve exercises and propose questions for the students to discuss. In this sense, we will work the basic competences CB1 to CB5, the general competences CG1 to CG5 and the specific competences CE1 to CE6.

Regarding lessons in small groups, the maximum participation of students will be promoted. During these lessons we will analyze in detail some problems and aspects of the subject which are not considered in expositive lessons as they are the most difficult ones to understand. In this sessions, we will work the same competences as in expositive lessons as well as the specific competences CE7 and CE8.

Finally, laboratory lessons will take place in computer classrooms and 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. In these sessions we will work the specific competences CE7 to CE9 and the transversal competences CT1 and CT4.

Theoretical and interactive teaching will be in person and will be complemented with the virtual course of the subject, where the students will find bibliographical tools, exercises, explanatory videos, etc.

Tutorials will be in person or via email or MS Teams.

Avaliation will take into account the continuous assessment as well as the final exam.

The continuous assessment will consist on three tasks made during lessons. They will consist on the resolution of problems, writing of proofs or theoretical results, tasks on the virtual course, etc. The dates of these tasks will be previously announced in order to ensure the maximum participation and advantage, as they are not only assessment tools but also training exercises to reinforce the competences worked in previous sessions. Each student will get a mark between 0 and 10 points for each of the tasks. The lack of attendance to these sessions will only be recoverable in cases properly justified. The continuous assessment tasks will be similar in all the groups.

The continuous assessment mark will be the average of the marks of all the tasks.

In the final exam, the theoretical part of the task will be, at least, 3 points over 10. The final exam will be the same for all the groups.

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 to the following formula:

FM=Max{FEM,0,65*CEM+0,35*FEM}

Remark: It is possible to pass the subject without doing the final exam (previous formula with F=0) and in such case it will be understood that the subject has been passed in the first opportunity. It will be considered as NOT PRESENTED who is not in conditions of passing the subject without doing the final exam and does not attend to such exam.

For the second opportunity, the final qualification will be computed using the same formula.

WORKING HOURS IN CLASSROOM
Expositive lessons (14 h)
Seminary lessons (14 h)
Laboratory lessons (14 h)
Tutorials in very small groups (2 h)
Total working hours in classroom: 44

PERSONAL WORK OF THE STUDENT
Autonomous individual or in group study (42h)
Writing of exercises, conclusions or other works (15h)
Works with computer (7.5 h)
Recommended lectures (5 h)
Total working hours outside classroom: 68.5

TOTAL: 112.5 hours

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.
Students should work with regularity and accuracy, attend to lessons and participate actively, making questions, both during lessons as in tutorials.