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Mathematical Methods I, 2024–2025
Mark Opmeer
Contents
Introduction
1 The heat equation on an interval
1.1 The role of the boundary conditions
2 Eigenvalues and eigenvectors/eigenfunctions
2.1 Revision of MA20220
2.2 The connection
2.3 Background: the Spectral Theorem
3 The heat equation on a rectangle
4 The heat equation on a general domain
4.1 The heat equation on a disc
4.2 The inner-product
5 The heat equation on a disc
6 Sturm–Liouville problems
7 Quasi-regular boundary points
8 The Wronskian
9 Boundary conditions
10 The power series method
11 Frobenius’ method
12 The heat equation on a disc revisited
12.1 The radial eigenvalue problem for the disc
12.2 The heat equation on the disc: solution
13 The heat equation on the whole line
14 The Fourier transform
15 Solving PDEs using the Fourier transform
16 Distribution theory
17 The Fourier transform of a distribution
Table of Fourier transforms
18 The method of characteristics: first order PDEs I
19 The method of characteristics: first order PDEs II
20 The method of characteristics: second order PDEs
A Problem Sheet 1 (Lectures 1-3)
A.1 Neumann heat equation on an interval
A.2 Wave equation on an interval
A.3 Wave equation on a rectangle
B Problem Sheet 2 (Lectures 4-5)
B.1 Heat equation in a cylinder 1
B.2 Heat equation in a ball 1
C Problem Sheet 3 (Lectures 6-7)
C.1 Legendre 1
C.2 Legendre 2
C.3 Spherical Bessel 1
C.4 Chebyshev 1
C.5 Chebyshev 2
C.6 Laguerre 1
C.7 Hermite 1
C.8 Not Sturm–Liouville problems
C.9 Legendre 3
C.10 Spherical Bessel 2
C.11 Chebyshev 3
C.12 Laguerre 2
C.13 Hermite 2
D Problem Sheet 4 (Lectures 8-9)
D.1 Legendre 4
D.2 Legendre 5
D.3 Spherical Bessel 3
D.4 Chebyshev 4
D.5 Chebyshev 5
D.6 Laguerre 3
D.7 Hermite 3
E Problem Sheet 5 (Lectures 10-11)
E.1 Legendre 6
E.2 Legendre 7
E.3 Chebyshev 6
E.4 Chebyshev 7
E.5 Laguerre 4
E.6 Laguerre 5
E.7 Laguerre 6
E.8 Hermite 4
E.9 Hermite 5
E.10 Spherical Bessel 4
E.11 Spherical Bessel 5
E.12 Spherical Bessel 6
F Problem Sheet 6 (Lecture 12)
F.1 Wave equation on a disc
F.2 Heat equation in a cylinder 2
F.3 Heat equation in a ball 2
G Problem Sheet 7 (Lectures 13-15)
G.1 Fourier transform of exponential-like functions
G.2 The wave equation on the whole line
G.3 The heat equation on a half-line
G.4 Using Plancherel’s Theorem
H Problem Sheet 8 (Lectures 16-17)
H.1 Dilation formula
H.2 Fourier transform of general Gaussian
H.3 Fourier transform of sine
H.4 Distributional dilation formula
I Problem Sheet 9 (Lectures 18-19)
I.1
I.2
I.3
I.4
I.5
I.6
J Problem Sheet 10 (Lecture 20)
J.1
J.2
J.3
J.4
J.5
Further reading
Introduction
In this module we will explicitly solve partial differential equations. The solution formulas will typically involve infinite sums and integrals and will usually be obtained by solving a sequence of ordinary differential equations. Hence the units MA20220
Ordinary Differential Equations and Control and MA20223 Vector Calculus and Partial Differential Equations are essential prerequisites (the ODE and PDE parts of those units more specifically).
We will be solving some standard partial differential equations of mathematical physics (Laplace, heat, wave) which you may encounter in applied or physics units.
To justify why the solution methods work, we will borrow some results from analysis units (without giving any of the proofs); in particular from MA30055 Introduction to Topology, MA30062 Analysis of Nonlinear Ordinary Differential Equations,
MA40042 Measure Theory & Integration, MA40254 Differential and Geometric Analysis and MA40256 Analysis in Hilbert Spaces. There is however no expectation that you have done or are doing those analysis units.