Module Information

Module Identifier
MA34110
Module Title
Partial Differential Equations
Academic Year
2022/2023
Co-ordinator
Semester
Semester 1
Pre-Requisite
Exclusive (Any Acad Year)
Other Staff

Assessment

Assessment Type Assessment length / details Proportion
Semester Exam 2 Hours   (Written Examination)  100%
Supplementary Exam 2 Hours   (Written Examination)  100%

Learning Outcomes

On successful completion of this module students should be able to:

1. classify partial differential equations and identify appropriate solution techniques;
2. Solve first order linear partial differential equations using the method of characteristics;
3. demonstrate an ability to use the method of separation of variables to solve second order linear partial differential equations on rectangular domains;
4. solve classical second order partial differential equations (wave, heat, Laplace’s equation) in infinite and semi-infinite domains and interpret their solutions;
5. prove results concerning uniqueness of wave equation and heat equation solutions.

Brief description

Many mathematical problems arising in the physical sciences, engineering, and technology, may be formulated in terms of partial differential equations. In attempting to solve such problems, one must be aware of the various types of partial differential equation which exist, and of the different boundary conditions associated with each type. These factors determine which method of solution one should use.

Aims

To teach the student how to recognise the type of a partial differential equation, and how to choose and implement an appropriate method of solution.

Content

1. Fundamentals: definitions and examples, simple partial differential equations.
2. First order equations: the method of characteristics.
3. Boundary conditions: Dirichlet, Neumann, Robin, well-posedness and ill-posedness.
4. Second order equations: classification, reduction to canonical forms.
5. The wave equation: general solution, Cauchy problem, reflection principle, Duhamel principle, bounded string, energy and uniqueness.
6. The heat equation: maximum principle, uniqueness, separation of variables, properties of solutions, the fundamential solution.
7. Green's functions

Module Skills

Skills Type Skills details
Adaptability and resilience Students are expected to develop their own approach to time-management and to use the feedback from marked work to support their learning.
Co-ordinating with others Students will be encouraged to work in groups to solve problems.
Creative Problem Solving The assignments will give the students opportunities to show creativity in finding solutions and develop their problem solving skills.
Digital capability Use of the internet, Blackboard, and mathematical packages will be encouraged to enhance their understanding of the module content and examples of application
Professional communication Students will be expected to submit clearly written solutions to set exercises.
Subject Specific Skills Broadens exposure of students to topics in mathematics, and an area of application that they have not previously encountered.

Notes

This module is at CQFW Level 6