FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS

Week 1

Lecture 1 (3 March 2008) (Sections I-IV). Differential equations: ordinary and partial; linear and nonlinear; first and second order. Existence theorem for an autonomous ODE system. Reduction of first-order quasilinear PDE to an equivalent autonomous ODE system. Auxiliary equations. Characteristics. Potential for inconsistent side data. Potential for multiple-valued solutions.

Lecture 2 (5 March 2008) (Sections V-VII). Inhomogeneous problems. A sketch of existence theory. generalizations and specializations.

Lecture 3 (7 March 2008) (Sections VIII-XI). Conservation in one dimension. Genesis of kinematic waves. Constant velocity case. A model for traffic flow.

Week 2

Lecture 4 (10 March 2008) (Sections XII-XIII). Increasing and decreasing initial velocity distributions. Discontinuous data: fans and shocks.

Lecture 5 (12 March 2008) (Section XIV). Hopf's equation. Incomplete version replaced with full version 13/3/08.

Lecture 6 (14 March 2008) (Section XV). More on the traffic flow model.

Week 3

Lecture 7 (17 March 2008) (Sections XVI-XVII). Less naive models; shock structure, the Burgers equation and the Cole-Hopf transformation. Source terms. Old version with a number of errors replaced with corrected version 19/3/08.

Lecture 8 (17 March 2008) (Sections XVIII-XX). Flux and conservation arguments in higher dimensions. Fick's law and convective diffusion leading to a second-order partial differential equation.

SECOND-ORDER LINEAR PARTIAL DIFFERENTIAL EQUATIONS

Week 4

Lecture 9 (31 March 2008) (Sections I-II). Results needed from vector calculus. The canonical linear PDEs of second order.

Lecture 10 (2 April 2008) (Sections III-IV). The (second order) wave equation in one space dimension analysed by characteristics.

Lecture 11 (4 April 2008) (Sections IV-V). The method of images. The forced wave equation.

Week 5

Lecture 12 (7 April 2008) (Sections VI-VIII). The wave equation in higher dimensions. Similarity solutions of the one-dimensional diffusion equation. Harmonic functions: simple examples, statement of a maximum principle.

Lecture 13 (9 April 2008) (Sections VIII-IX). Harmonic functions: proof of a maximum principle and applications. Plane harmonic functions, Cauchy-Riemann relations, connection to holomorphic functions.

Lecture 14 (11 April 2008) (Sections X-XI). General results for Laplace and Poisson equations. Separation of variables. One-dimensional eigenfunctions.

Week 6

Lecture 15 (14 April 2008) (Sections XII-XIII). Corrected version of earlier posting. Fourier series as eigenfunction expansions. Eigenfunctions of the negative Laplacian.

Lecture 16 (16 April 2008) (Section XIII). More about eigenfunctions of the negative Laplacian, solving wave and diffusion equations, forced problems, Poisson's equation.

Lecture 17 (18 April 2008) (Sections XIV-XV). Problems in higher dimensions and polar coordinates. Plane harmonic functions revisited; Poisson's integral formula.

Week 7

Lecture 18 (21 April 2008) (Section XVI). The origin of higher transcendental functions.

There are no other notes for Week 7 due to the mid-semester test on Wednesday 23 April and the Anzac Day holiday on Friday 25 April.

Week 8

Lecture 19 (28 April 2008) (Section XVII). Sturm-Liouville theory. Loaded 12.55 p.m., 28/4/08; replaces earlier version loaded before the lecture.

Lecture 20 (30 April 2008) (Section XVIII). Bessel functions as an illustration of Sturm-Liouville theory.

Lecture 21 (2 May 2008) (Sections XIX-XXI). The linear ODE of second order. Series solution ideas. Power series and complex functions.

Week 9

Lecture 22 (5 May 2008) (Sections XXI-XXII). Power series and complex functions: subtler issues.

Lecture 23 (7 May 2008) (Sections XXIII-XXIV). Gamma function, hypergeometric series. Airy's equation as a first example of series solution methods.

Lecture 24 (9 May 2008) (Sections XXV-XXVI). Ordinary points, Legendre's equation.

Week 10

Lecture 25 (12 May 2008) (Section XXVII). A first look at Bessel's equation.

Lecture 26 (14 May 2008) (Section XXVIII). Regular singular points, Theorem of Frobenius, more on Bessel's equation.

Lecture 27 (16 May 2008) (Section XXIX). Solving problems where Bessel functions arise.

Week 11

Lecture 28 (19 May 2008) (Sections XXX-XXXI). Modified Bessel functions. Series solutions of Laplace's eqaution with inhomogeneous boundary conditions.

Lecture 29 (21 May 2008) (Sections XXXII-XXXIII). Reflecting on the discussion so far. Green function solution for Poisson's equation.

Lecture 30 (23 May 2008) (Section XXXIV). Green function solution for the diffusion equation.

Week 12

Lecture 31 (26 May 2008) (Section XXXV). Green function solution for the wave equation.

Lecture 32 (28 May 2008) (Sections XXXVI-XXXVII) Last lecture containing new material. No scans of handwritten notes for this lecture: see instead the printed notes

Lecture 33 (30 May 2008) Discussion of a past examination, used to review key concepts from the subject. No notes (printed or handwritten) will be issued for this lecture.