Course guide of Partial Differential Equations (27011B2)

It is convenient to have followed the courses of the basic matter Mathematics and the matters of Mathematical Analysis.

• Classical PDEs of Physics, methods for solving initial value problems, boundary value problems and mixed problems for elliptic, hyperbolic and parabolic equations.

1. The learning of the origin of the PDEs, including a historic perspective of the problems that motivated their study.
2. A deep understanding of the techniques of the proofs in this course, as well as their main applications.
3. To be able to distinguish the distinct types of PDEs (parabolic, hyperbolic and elliptic) as well as the different type of problems associated to them (initial value problems, boundary value problems, mixed problems).
4. To get familiar with the main applications of PDEs in different fields of Science, mostly in Physics and Engineering.
5. To get ready for later studies (i.e., Master studies) in Mathematical Analysis and other branches of mathematics, including research in mathematics.

Chapter I. Introduction and motivation

• The origin of the PDEs and their relations with problems in Physics.
• Diffusion and waves. Cauchy problems, boundary value problems, mixed problems.

Chapter II. Quasilinear equations of first order.

• The characteristic system.
• Cauchy problems, the method of characteristics.

Chapter III. The wave equation.

• The initial value problem and D’Alembert formula. Propagation of waves.
• The wave equation in higher dimensions.
• Fourier Series and mixed problems.

Chapeter IV. The heat equation

• The initial value problem.
• Qualitative properties: regularization, irreversibility.
• The maximum principle
• Fourier Series and mixed problems.

Chapter V. Laplace and Poisson equations.

• Laplace equation, harmonic functions.
• The Newtonian potential.
• Mean inequalities and maximum principle.
• Liouville Theorem.
• Fourier Series and the Dirichlet problem.
• Poisson equation.

• Cañada, A. Series de Fourier y Aplicaciones. Ediciones Pirámide, Madrid, 2002.
• Peral, I.: Primer curso de Ecuaciones en derivadas parciales. Addison-Wesley, 1995. La 2ª edición, de 2004, está disponible (libre) en http://matematicas.uam.es/~ireneo.peral/libro.pdf
• Evans, L. C.: Partial Differential Equations, AMS, 2002.

• Budak, B.M.; Samarski, A.A. y Tijonov, A.N.: Problemas de la Física Matemática. Volúmenes I y II. Mir, 1984.
• Gilbarg, D. y Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin-Heidelberg, 1983.
• Strauss W.A: Partial differential equations an introduction. New York: John Wiley and Sons, 2008.
• Tijonov, A.N. y Samarsky, A.A. Ecuaciones de la Física Matemática. Mir, 1980.

Notes of the PDE course of the MIT:

https://ocw.mit.edu/courses/mathematics/18-152-introduction-to-partial-differential-equations-fall-2011/

PDE solvers: http://math.uchicago.edu/~luis/index.html

• MD01. Lección magistral/expositiva 
• MD03. Resolución de problemas y estudio de casos prácticos 
• MD06. Análisis de fuentes y documentos 
• MD07. Realización de trabajos en grupo 
• MD08. Realización de trabajos individuales 

The assessment of the course will be expressed via a numeric mark as follows:

• 40% of the mark will be obtained via several activities proposed by the lecturer, including:
  - Active participation in the lectures and in tutorials as well as exercises and their presentation, which will be made individually or in groups (20%).
  - Written evaluation exercises (20%).
• 60% of the mark will be obtained via a Final Exam.

It is based on a unique Final Exam containing theoretical and practical questions.

It is based on a unique Final Exam containing theoretical and practical questions.