## 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)

Div, Grad, Curl in Cartesians

Vector identities

Properties of the gradient

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