# FUNCTIONAL ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS - 2023/4

Module code: MATM022

## Module Overview

This module introduces the basic concepts of functional analysis including Hilbert and Banach spaces, the associated spaces of linear functionals, weak convergences, etc. The introduced concepts are then used to give an introduction to the modern theory of partial differential equations.

### Module provider

Mathematics

ZELIK Sergey (Maths & Phys)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 78

Lecture Hours: 36

Captured Content: 36

Semester 1

## Prerequisites / Co-requisites

MAT2004 Real Analysis 2. MAT2011 Linear PDEs.

## Module content

Indicative content includes:

• Hilbert and Banach spaces, linear functionals, dual spaces, reflexivity.

• Weak and strong convergences. Weak compactness of a unit ball in reflexive spaces.

• Introduction to distributions and Sobolev spaces.

• Weak formulation of Dirichlet and Neumann problems for the Laplacian.

• Variational formulation of these problems.

• Introduction to non-linear partial differential equations.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test (50 mins) 20
Examination Exam (2 hrs) 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

·         Understanding of and ability to solve/prove problems in operator and PDE theories.

·         Subject knowledge through the recall of key definitions, theorems and their proofs.

·         Analytical ability through the solution of unseen problems in the test and exam.

Thus, the summative assessment for this module consists of:

• One examination at the end of the semester; worth 80% module mark.

• One in-semester test; worth 20% of the module mark.

Formative assessment and feedback

Students receive written feedback via a number of marked coursework assignments over an 11 week period.  In addition, verbal feedback is provided by lecturer at tutorials.

## Module aims

• The aim of this module is to introduce students to basic concepts and methods of functional analysis with applications to PDEs.

## Learning outcomes

 Attributes Developed 001 Have an understanding of basic properties of Hilbert and Banach spaces and associated linear operators; K 002 Understand a concept of a distributional solution of a differential equation and to be able to give a weak formulation for the Dirichlet and Neumann problems for the Laplace operator; KC 003 Be able to prove the existence and uniqueness of a solution for some classical partial differential equations using the methods of functional analysis. KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

• A detailed introduction to basic concepts and methods of functional analysis with applications to PDEs.

• Experience (through demonstration) of the methods used to interpret, understand and solve/prove problems in functional analysis and PDEs.

The learning and teaching methods include:

• Lectures with blackboard/whiteboard notes to supplement the module lecture notes and Q + A opportunities for students.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.