## 1. MATHEMATICAL MODELS(17 lectures) Prof. E.R. Priest

### (i) Mathematical Models involving Differential Equations

Introduction to mathematical modelling

Cooling/Heating problems

Population models

Falling objects

### (ii) Models involving Differential Equations

Introduction

Population Models

Two-Species growth models

Competing Species models

## (iii) Dynamics

### Kinematics:

Position, velocity, acceleration. Relative motion with examples

### Force and Acceleration:

Units and dimensions; force equals mass times acceleration, unidirectional motion with constant force. Constant gravitational acceleration.

### Projectiles:

The equations in two-dimensions; motion without air resistance. Parabolic trajectory. Range and time of flight on horizontal and inclined planes. Examples.

### Projectiles with air resistance:

Resistance proportional to speed [and (if time permits) proportional to square of speed].

### Work and Energy:

Definitions, examples of one-dimensional motion. Conservative forces and their potential. The energy equation with conservative forces. Motion in potential well; examples.

### Newton's Laws:

Newton's three laws of motion; Newton's Law of Gravitation, inertial frames.

### Motion under inverse-square force:

Unidirectional motion: escape from Earth of constant mass. Motion in a circular orbit: communications satellites.

## 2. NUMERICAL METHODS(8 lectures) Prof. A.W. Hood

### (i) Solution of Equations (5 lectures, Kreyszig chap 18)

Revision of Taylor Series (handout)

Approximate Numerical Methods

Newton-Raphson Method, stopping criteria, error analysis

Other Iteration Schemes : Logistic equation, Bifurcation

Bisection Method (if time)

Error Analysis for Bisection Method (if time)

### (ii) Numerical Solutions to ODE's (3 lectures, Kreyszig chap 20)

Euler's algorithm

An error analysis for Euler's algorithm

Heun's method

Error anlayis for Heun's method

Extension to systems of equations

## 3. VECTOR CALCULUS(17 lectures) Prof D.G. Dritschel

### (i) Introduction and Revision

Motivation and Brief History

Products and triple products

Polar co-ordinate systems

Differentiation of vectors

Visualising vector fields

### (ii) Differential Operators (Grad, Div and Curl)

Vector identities

Properties of the curl

Properties of the divergence

Helmholtz' Theorem

Div, Grad and Curl in polar co-ordinates

### (iii) Line Integrals and Conservative Fields

Line Integrals

Conservative vector fields

### (iv) Integration of Fields; Integral Theorems

Direct Integration of Vectors

Surface Integrals

Green's Theorem

Stokes's Theorem

Divergence Theorem of Gauss

## 4. PARTIAL DIFFERENTIAL EQUATIONS(9 lectures) Prof. A.W. Hood

### (i) First Order Partial Differential Equations

Revision of partial differentiation, verification of solutions to PDEs. The method of characteristics.

### (ii) Three important equations

Wave equation, heat equation, Laplace's equation. Derivation of wave equation for string and d'Alembert's solution (handout). Solution of initial-value problems for infinite string; examples.

### (iii) Waves on strings with fixed end points

Separation of variables; application to vibrating string; musical harmonics. The general solution by superposition. Initial-value problems for vibrating strings; determination of the Fourier coefficients.

[The Heat equation similarly, if time available.]

AWH Aug 03