MTH-2F42 : Mathematics of Diffusion

**1. Introduction: **Taught by lectures and seminars. This is an applied mathematics unit for students who have already studied first-year university-level mathematics. The work derives appropriate partial differential equations for the flow of heat or diffusion of matter, and solves these linear equations in one, two or three space dimensions and time. The work uses separation of variables, Fourier series, integral representations of solutions and other main stream parts of applied mathematics.

**2. Timetable Hours, Credits, Assessments:** ** **This is a 10 UCU unit and is given in Semester II by means of 15 hours of lectures and supported by 3 hours of seminars. The assessment is by 20% coursework and 80% examination. The coursework consists of written answers to set exercises, submitted by given deadlines.

**3. Overview: **The contents have been chosen to show how the phenomena of diffusion can be translated into a precise mathematical description or model. This was first done by Joseph Fourier in 1820, in the form of a linear partial differential equation. Fourier invented Fourier analysis to help solve the heat equation on a finite length of conducting bar. We examine these initial-boundary value problems and extend ideas into solutions for infinite and semi-infinite domains, primarily in one and three space dimensions. The mathematical solutions can be applied to a wide range of problems from the spread of industrial contaminants to the movement of perfume in still air.

**4. Recommended Reading:**

(i) E Kreyszig "Advanced Engineering Mathematics", Wiley.

§11.5 and §11.6 introduces the material well.

(ii) J Crank "The Mathematics of Diffusion", Oxford University Press.

(iii) H S Carslaw "Conduction of heat in solids", Oxford University Press.

& J C Jaeger

5. Lecture Contents:

**§1.1 **Introduction. Examples of diffusion. The concept of flux. Derivation of the general unsteady heat equation in 3-space. Importance of the diffusion coefficient. Boundary conditions. **(2 lectures)**

**1.2** Uniqueness of solutions via integral theorems. Conservation of heat (or mass of substance). Boundary conditions related to physical concepts. **(1 lecture)**

**§2.1** 1-space dimension. Steady solutions are linear with constant heat flux. Double glazing and piecewise linear solutions. **(1 lecture)**

**2.2** Unsteady solutions for x Œ [0,L] obtained by separation of variables. Eigenvalues obtained from boundary conditions. **(1 lecture)**

**2.3** Revision of Fourier analysis. Series solutions on x Œ [0,L] for initial value problems. **(2 lectures)**

**2.4** Examples of §2.2 and §2.3. **(1 lecture)**

**2.5 **Solution of initial delta function on x Œ (-•,•) . **(** **lecture)**

**2.6** General solutions for x Œ (-•,•) for initial-value problems. **(** **lecture)**

**2.7 **Semi-Infinite bar with time harmonic boundary condition (diurnal heating). Use of Laplace Transform shown by example. **(1** **lecture)**

**§3.1** Separation of variables in 2 and 3 space dimensions. Steady heat flow and Laplace's equation briefly covered. **(1 lecture)**

**3.2** Spherical polar coordinates. Spherically-symmetric solutions in r and t . Reduction to the 1D heat equation in r and t . Steady temperature distributions. **(1 lectures)**

**3.3** Presence or absence of singularity at r = 0 . Internal and external problems with appropriate conditions at r = 0 , on a sphere's surface and at r = • . **(2 lectures)**