BSc Mathematics

Year of entry: 2022

Course unit details:Green's Functions, Integral Equations and Applications

Unit code MATH34032 10 Level 3 Semester 2 Department of Mathematics No

Overview

This is a methods based course, focusing on the theory and application of Green's functions in applied mathematics. Green's functions enable the solution to a variety of interesting and important problems. In particular one can set up solutions to ordinary and partial differential equations of general type in integral form by the use of a Green's function. In more difficult problems, such as scattering from an object, integral equations result. In this course we will show how Green's functions are defined, why they are important and then show their application to various problems in applied mathematics. In particular we will exhibit how they may be used to understand wave propagation on a string, potential flow on bounded domains, wave propagation and scattering from an object and how one can construct an acoustic "cloak" in order to render regions invisible from acoustic waves (and thus construct theoretical "domains of silence").

Pre/co-requisites

Unit title Unit code Requirement type Description
Partial Differential Equations and Vector Calculus A MATH20401 Pre-Requisite Compulsory
Partial Differential Equations and Vector Calculus B MATH20411 Pre-Requisite Compulsory
MATH34032 pre-requisites

Students must have taken MATH20401 OR MATH20411

Aims

To introduce students to Green's functions and integral equations (and how they are linked). To explain how Green's functions may be used in practice, with applications to a variety of ordinary and partial differential equations and physical applied problems such as potential flow, wave propagation and scattering.

Learning outcomes

On successful completion of this course students will:

• Construct Green’s functions for one dimensional boundary value problems from fundamental solutions, and use these Green’s functions to express solutions to such problems.
• Recognise regular Sturm-Liouville boundary value and eigenvalue problems, prove basic properties of such problems, calculate the eigenvalues and eigenfunctions in simple cases, and express the Green’s function using an eigenfunction expansion.
• Apply the Fredholm alternative to one dimensional boundary value problems to determine whether solutions exist, and whether they are unique.
• Express the solution of boundary value problems for the Laplacian in two and three dimensions using Green’s functions
• Prove that the Laplacian of the free-space Green’s functions in two and three dimensions equals the Dirac delta function, and apply the free-space Green’s functions and the method of images to solve boundary value problems.
• Distinguish between the different types of integral equations, relate these integral equations to corresponding initial or boundary value problems and solve Fredholm integral equations of the second kind with degenerate kernel.
• Formulate boundary value problems modelling waves on a string, and apply the Neumann series solution for a corresponding Fredholm integral equation of the second kind to analyze scattering of waves in one dimension.

Syllabus

• Section 1: Preliminaries. Dirac Delta function, Heaviside function, Operators, Adjoint operator [1 lecture]
• Section 2: Greens functions in 1D. Construction for constant coefficient ODEs and Sturm Liouville problems. Applications to the steady state heat equation and wave equation. [5 lectures]
• Section 3: Greens functions in 2 and 3D. Steady state heat equation and Potential flow problems (Laplace) and time-harmonic wave equation (Helmholtz). Applications to cloaking. [5 lectures]
• Section 4: Integral equations in 1D. Motivated by 1D scattering problem. Series solution and physical interpretation. General integral equation types. Degenerate (separable) kernels and solution method. Neumann series and iterated kernels. [4 lectures]
• Section 5: Integral equations in 2 and 3D. Greens second identity. Generation of integral equation for Potential Flow problems (Laplace) via Greens functions for bounded domains. Single and double layer potentials. Solution via Boundary Element methods. Extention to integral equation for inhomogeneity in steady state thermal problem and Potential flow. Eshelbys conjecture. Applications to homogenization. [7 lectures]

Assessment methods

Method Weight
Other 20%
Written exam 80%
• Coursework weighting 20%
• End of semester examination 80%

Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• GF Roach, Green's functions, introductory theory with applications, Van Nostrand Reinhold, 1982
• I Stakgold, MJ Holst, Green's Functions and Boundary Value Problems, John Wiley and Sons, 2011
• D Porter and DSG Stirling, Integral Equations: A Practical Treatment, Cambridge University Press, 1990
• E Zauderer, Partial Differential Equations of Applied Mathematics, John Wiley and Sons, 1983

Study hours

Scheduled activity hours
Lectures 12
Tutorials 12
Independent study hours
Independent study 76

Teaching staff

Staff member Role
Sean Holman Unit coordinator

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

•         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently
•         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week
•         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises
•         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.
The remaining independent study time comprises revision for and taking the end-of-semester assessment.

The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.