Green's Functions, Integral Equations and Applications

 Unit code: MATH34032 Credit Rating: 10 Unit level: Level 3 Teaching period(s): Semester 2 Offered by School of Mathematics Available as a free choice unit?: N

Requisites

Prerequisite

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.

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").

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.

Assessment methods

• Other - 40%
• Written exam - 60%

Assessment Further Information

• First coursework test worth 20%
• Second coursework test worth 20%
• End of semester 1.5 hour Examination 60%

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]

• 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

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.

Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

Teaching staff

Sean Holman - Unit coordinator