Teaching plan for the course unit

 

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General information

 

Course unit name: Functional Analysis and Partial Differential Equations

Course unit code: 568175

Academic year: 2018-2019

Coordinator: Marķa Jesśs Carro Rossell

Department: Department of Mathematics and Computer Science

Credits: 6

Single program: S

 

 

Estimated learning time

Total number of hours 150

 

Face-to-face learning activities

60

 

-  Lecture

 

30

 

-  Lecture with practical component

 

30

Supervised project

20

Independent learning

70

 

 

Competences to be gained during study

 


— Capacity of understanding the concepts and rigorous proofs of fundamental theorems of Functional Analysis and Partial Differential Equations and transverse areas of mathematics.

— Capacity to apply the results and techniques learned to solve complex problems in different areas of mathematics in academic or professional contexts.

— Ability to prepare and develop logical-mathematical reasoning and identify errors in incorrect reasoning.

— Capacity to know how to construct, interpret, analyze and validate mathematical models developed to simulate real situations.

— Ability to enunciate and to verify statements, and to convey the mathematical knowledge acquired orally or in writing.

— Capacity to choose and use software tools to address problems related to mathematics.

 


— Ability to work in groups.

 

 

 

 

Learning objectives

 

Referring to knowledge


— To learn the basic results on Banach and Hilbert spaces and operators, with special attention to duality.

— To know the theory of distributions and Sobolev spaces, mainly in the context of Hilbert spaces.

— To use the techniques of Functional Analysis in the study of Ordinary and Partial Differential equations.

 

 

Teaching blocks

 

1. Hilbert spaces: orthogonality, duality and elementary spectral theory.

2. Banach and Frechet spaces. Boundedness of linear operators.

3. Fundamental theorems of Functional Analysis

*  Fundamental theorem of Functional Analysis: Baire, Open mapping, Closed graph, Uniform boundedness principle and Hahn-Banach theorems. 

4. Distribution theory.

*   Weak derivatives, convolution, fundamental solutions, temperate distributions and  Fourier transform.  

5. Sobolev spaces: Regularity and compactness

6. Applications to PDE

*  Existence of the fundamental solution, Regularity of the solutions, Application of the spectral theory. 

 

 

Teaching methods and general organization

 


The different teaching blocks will be developed in theoretical classes; For each block, a collection of problems will be distributed and discussed in class; Students will work on all or part of the exercises from the list; After the correction, they will be solved on the blackboard

 

 

Official assessment of learning outcomes

 

This evaluation has 2 parts:

Activity 1: Resolution of exercises and other assignments given in class.

Activity 2: Written exams.

The final grade will be obtained by averaging 20% of  activity 1,  30%   of partial  exams and 50% of the final exam (a minimum of 4/10 points are required on each activity).

 

Examination-based assessment

Final exam with theoretical and practical questions.

In order to be considered for re-evaluation, the student needs to have at least 3.5/10 points as a final grade.