# FUNCTIONAL ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS - 2024/5

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 introduce the modern theory of partial differential equations. The module builds on material covered in MAT2004: Real Analysis 2 and MAT2011: Linear PDEs.

### Module provider

Mathematics & Physics

ZELIK Sergey (Maths & Phys)

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

Independent Learning Hours: 66

Lecture Hours: 33

Tutorial Hours: 3

Guided Learning: 15

Captured Content: 33

Semester 1

None.

## Module content

The content of the module will include an introduction to basic concepts of functional analysis applied to partial differential equations. It starts with an introduction to function spaces and then show how these concepts can be applied to various partial differential equation problems.

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

Weak and Strong convergences. Introduce weak compactness of a unit ball in reflexive spaces.

Distributions and Sobolev spaces. Introduce basic concepts of distributions and Sobolev spaces.

Formulations of partial differential equations. Weak formulation of Dirichlet and Neumann problems for the Laplacian. Variational formulation of Dirichlet and Neumann problems for these problems.

Nonlinear partial differential equations. Introduction and application of functional analysis to nonlinear partial differential equations.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test (50 mins) 20
Examination End-of-Semester Examination (2 hours) 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 in-semester test corresponding to Learning Outcomes 1 and 2.

• A synoptic examination corresponding to Learning Outcomes 1 to 3.

Formative assessment
There are two formative unassessed courseworks over an 11 week period, designed to consolidate student learning.

Feedback
Students receive individual written feedback on the formative unassessed coursework and the in-semester test. The feedback is timed so that feedback from the first unassessed coursework assists students with preparation for the in-semester test. The feedback from both unassessed courseworks and the in-semester test assists students with preparation for the end-of-semester examination. This written feedback is complemented by verbal feedback given in lectures. Students also receive verbal feedback in office hours.

## 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 Students will be able to demonstrate knowledge and understanding of basic properties of Hilbert and Banach spaces and associated linear operators. K 002 Students will be able to understand a concept of a distributional solution of a differential equation and give a weak formulation for the Dirichlet and Neumann problems for the Laplace operator. KC 003 Students will 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:

• Three one-hour lectures per week for eleven weeks, with supplementary notes for topics of significant difficulty or special interest to complement the lectures. The lectures provide a structured learning environment with opportunities for students to ask questions and to practice methods taught.

• Three one-hour tutorials to help students prepare for the assessments.

• There are two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these as guidance on their progress and understanding.

• Lectures may be recorded. Lecture recordings are intended to give students the opportunity to review parts of the session that they might not have understood fully and should not be seen as an alternative to attendance at lectures.

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.

Upon accessing the reading list, please search for the module using the module code: MATM022

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourcefulness and Resilience and Sustainability. This module is designed to allow students to develop knowledge, skills, and capabilities in the following areas:

Digital Capabilities: The SurreyLearn page for MATM022 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing.

Employability: Through the module, students cultivate advanced problem-solving skills applicable and valued across diverse industries, such as engineering, finance, and research.

Global and Cultural Capabilities: Student engagement in discussions during lectures naturally cultivates the sharing of the different cultures from which the students originate.

Resourcefulness and Resilience: MATM022 fosters resourcefulness and resilience by immersing students in intricate problem-solving scenarios. Dealing with problems involving the unpredictability of nonlinear systems hones adaptability and perseverance.

Sustainability: The skills gained in understanding the mathematical foundations of partial differential equations is vital in fields such as climate science and renewable energy, enabling innovative solutions for a more sustainable future. For example, understanding ocean flows aids in predicting environmental changes. Likewise, understanding systems of traffic flow aids in informing sustainable practices.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics and Physics MPhys 1 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics and Physics MMath 1 Optional A weighted aggregate mark of 50% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2024/5 academic year.