# LINEAR PDES - 2025/6

Module code: MAT2011

## Module Overview

The Linear PDEs Module introduces students to linear partial differential equations, mainly in one and two space dimensions. The classical linear PDEs such as the heat, wave and Laplace equation will be analyzed in detail. Questions of existence and uniqueness of solutions will be addressed and their physical and mathematical meaning emphasized.

### Module provider

Mathematics & Physics

NOBILI Camilla (Maths & Phys)

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

Independent Learning Hours: 69

Lecture Hours: 33

Guided Learning: 15

Captured Content: 33

Semester 2

None.

## Module content

The contents of the module will include:

• Linear PDEs: Examples, classification of PDEs and their physical interpretation.

• First-order and Second-order linear PDEs:  Method of characteristics.

• Introduction to Fourier series and Fourier transform. Solution of initial and  boundary-value problems. Method of separation of variables.

• The heat equation. The wave equation, d’Alembert’s solution.  Interpretation of solutions.

• Laplace’s equation: mean-value theorem, maximum principle, Poisson formula.Existence and Uniqueness of solutions for the canonical PDEs. Energy methods.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test (50 minutes) 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 interpret and manipulate various methods for finding the solutions of linear partial differential equations.

• 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 (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1 to 3.

• A synoptic examination (2 hours), worth 80% of the module mark, corresponding to Learning Outcomes 1 to 4.

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 study both qualitative and quantitative aspects of linear PDEs in one and two space dimensions. Students will be introduced to the very important method of characteristics and they will understand and use the method of separation of variables for solving initial-boundary value problems of linear PDEs. The maximum principle will be proved and its power and importance in the analysis of solutions of PDEs conveyed to students.

## Learning outcomes

 Attributes Developed 001 Students will demonstrate understanding by classifying linear PDEs and will choose the appropriate method to solve them. KC 002 Students will solve linear PDEs using the method of characteristics, Fourier transform, and separation of variables. KC 003 Students will interpret solutions and critically relate them to physical settings. KCT 004 Students will understand and use of the maximum principle and energy methods for uniqueness and well-posedness. KC

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:

• Present a detailed introduction to linear partial differential equations and the most common techniques for finding their solutions.

• Give students experience (through demonstration) of the methods used to interpret, understand and solve problems in linear partial differential equations.

The learning and teaching methods include:

• Three one-hour lectures per week for eleven weeks, in which students will be encouraged to take lecture notes to facilitate their learning, and engagement with the module material. These lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.

• 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: MAT2011

## 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 MAT2011 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: Solving complex problems with partial differential equations fosters student resourcefulness, encouraging creative problem-solving. This skillset builds resilience as students navigate the uncertainties of real-world applications.

Sustainability: Students are shown that partial differential equations can be used to model and analyse complex systems such those involved in environmental processes, heat transfer, and wave propagation. This mathematical foundation enables informed decision-making, emphasising the role of partial differential equations in addressing complex challenges for a more sustainable future.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Statistics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MPhys 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 2 Optional A weighted aggregate mark of 40% 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 2025/6 academic year.