# 2007-08 Session Applied Mathematics Projects

## _____________ MATH206 _____________

AM1  ---   Orthogonal Polynomials

The Classical Orthogonal Polynomials are a set of polynomials which arose in a number of different situations and which share a number of similar properties. The object of this set of projects is to examine a number of these different properties for each of the functions of the set.

GROUP PROJECT

We are going to examine these properties:

·        Differential Equation: Each of these polynomials satisfies a differential equation. The polynomials arise out of series solutions of a differential equation.

·        Orthogonality Relation: If 1/Jn(x) is the nth order polynomial of the particular type: the orthogonality relation takes the form

where m does not equal to n and w(x) is a non negative weighting function. This relation can be used to define the polynomials.

• Rodrigues Formula: Each polynomial can be expressed in terms of a formula of the type

where g (x) is a zeroth, first or second order polynomial.

• Generating Function: The set of polynomials of a particular type satisfy a relation of the form

00

antn(x) = g(x, t).

n=O

This relation comes in useful in proving a number of other properties of the polynomials. Recurrence Relations Each set of polynomials satisfies a relation of the form

• Step up and step down operations:

and

• Properties of zeros: Each polynomial of order n, (x) has n real distinct zeros in the interval a < xi < b, where a and b are the limits on the integral for the orthogonality relation. Also each zero of (x) lies between a consecutive pair of zeros of (x).

INDIVIDUAL PROJECTS

The aim of each project is to digest the material in the text, completing the proofs where necessary, to work out a few of the polynomials and draw their graphs.

I.                    Laguerre Polynomials

II.                 Associated Laguerre Polynomials

III.               Legendre Polynomials

IV.              Chebyshev Polynomials of the First Kind

V.                 Chebyshev Polynomials of the Second Kind

VI.              Chebyshev Polynomials of the Thrd Kind

VII.            Chebyshev Polynomials of the Fourth Kind

VIII.         Gegenbauer Polynomials

IX.               Hermite Polynomials

Notes will be distributed on each topic.

AM2  ---  Integral Theorems of Vector Calculus

This project area is based on MATH225 and includes integral theorems which follow from the Divergence Theorem and Stokes' Theorem, integral expressions for scalar and vector potentials for solenoidal (divF=0) and irrotational (curlF=0) vector fields, Helmholtz's Theorem, derivation of Maxwell's equations from relevant integral theorems of electromagnetism, derivation of heat equation, and finally, discussion of how one can establish whether a solution to a governing partial differential equation is unique. All these topics can be considered as an introduction to the theory of partial differential equations and their applications in mathematical physics.

In “further applications of vector calculus and related topics”, this project area can include the concept of the delta function and conservation laws for fields with sources and/or sinks, the Green's function method which is used to solve inhomogeneous boundary value problems for linear partial differential equations (PDE' s), and the method of images which is used to construct Green's functions for the Laplace operator. We shall also consider Cartesian tensors (and show that the stress tensor introduced in MATH225 is indeed a Cartesian tensor), discuss some elements of the analytic function theory and show how this theory can be used to study fluid flows. We shall also consider second order linear PDE's, describe the method of characteristics and how this method can be used to solve the one-dimensional wave equation, and finally, we shall discuss some elements of the potential theory, prove the mean value theorem and the maximum principle for the Laplace equation, and find out whether it is always possible to find a solution to the Laplace equation which satisfies given boundary conditions.

The notes will guide you through each topic step by step. You should read them carefully and discuss any difficult points with your tutor until you understand them as well as possible. In some cases you are referred to a book which explains certain concepts in more detail. If you have any problems finding the book please contact your tutor.

GROUP PROJECT

Weeks 1-4. Review relevant ideas from :MATH225: Grad, div, curl; line, surface and volume integrals, flux integral; Divergence Theorem; Stokes' Theorem; continuity equation for fluids, stress tensor and equation of motion, incom­pressible fluid flows and associated stream functions, irrotational flows and associated velocity potentials. Distribute notes on core extension material (for group study over next 2 weeks): introduction of the delta function: prop­erties of the delta function; delta function in more dimensions. In “further applications of vector calculus and related topics”, we also review Green's identities; multiply connected regions, Green's Theorem for multiply connected regions.

Week 3 Discuss progress with extension material and help with difficulties. Week 4 Group presentation of core material. Assign individual topics and distribute materials.

INDIVIDUAL PROJECTS

Possible topics:

X.                 Delta Function and Conservation Laws

XI.               Green's Functions

XII.            Cartesian Tensors

XIII.          Analytic Functions and Fluid Flows

XIV.         Method of Characteristics and Wave Equation

XV.            Introduction to Potential Theory

Alternative topics are:

I.                    Divergence Theorem and some related theorems

II.                 Stokes' Theorem and some related theorems

III.               Expressions for scalar and vector potentials. Helmholtz's Theorem

IV.              Integral theorems and Maxwell's equations

V.                 Heat equation

VI.              Uniqueness of solution

Weeks 5-8 Help students with individual topics.

Week 9 Discussion of structure and content of group project, and division of responsibilities.

Weeks 10-11 Help students with project writing.

Week 12 Group presentation.

AM3  ---   Population Dynamics Using Maple

This topic is about the use of mathematics, a calculator and/or the computer algebra system Maple, to analyze a number of different models in population dynamics. To make the most of this topic you will need to be comfortable about using Maple and be prepared to extend your knowledge of its capabilities: to define functions, solve both algebraic and differential equations, manipulate matrices and plot graphs.

These notes guide you through the topic step by step. You should read them carefully and discuss any points of difficulty with your tutor until you understand them as well as possible. When you write your chapter of the group project, you should demonstrate your understanding by using examples which are different from those treated here, and, wherever possible, by presenting the material in your own words.

GROUP PROJECT

1.      Single Species Models - Continuous Case

2.      Single Species Models – Equilibra

3.      Single Species Models – Harvesting and restocking

4.      Single Species Models – Continuous class-structured case

5.      Single Species Models – Discrete case

6.      Single Species Models – Discrete class-structured case

7.      Two Species Models – Continuous case

INDIVIDUAL PROJECTS

Project 1- The SIR model

The SIR model is concerned with the spread of an infectious disease. The full model involves the inter-related time-development of three classes of individuals: Susceptible, infected and recovered/removed - see section 4 of the core notes.

In this project you will investigate two simpler versions of the SIR model - in the spirit of mathematical modeling - and then an application of the full model to discuss the effect of the Great Plague of 1666 on the population of the village of Eyam.

Project 2 - Discrete systems in a tropical paradise

Vegetable Ivory from Colombia. In an effort to halt the deforestation in South America, much attention has been paid to non-timber forest products. One such NTFP is Vegetable Ivory - a hard, cream-coloured, ivory-like polysaccharide that makes up the seed in the palm species Phytelephas. The material, which was of major economic importance for Ecuador and Colombia during the 19th and early zo" century, can be used for button manufacture - see the article "Demography of the vegetable ivory palm Phytelephas seemannii in Colombia, and the impact of seed harvesting" by Rodrigo Bernal in the Journal of Applied Ecology, 1998,35,64-74. This project is based on data from that article.

Project 3 - Predator-prey models

Predator-Prey (PP) models have two populations, X the predator, which uses y the prey, as a food source. In the simplest model, the development of the two population densities is governed by a coupled system of two differential equations - see section 7 of the core notes for generalities of the mathematics of continuous 2-species systems.

In this project you will consider the basic theory of PP models for which each population has a Malthusian growth law in the absence of the other. You will then look at two enhancements of the basic model: the prey population has a carrying capacity, and this will be applied to the real data for a population of reindeer on a South Atlantic island; a model in which a proportion of the prey can hide from the predator.

Project 4: Competition Models

Competition models have two populations, X and Y, which compete for a food source or living space. In the simplest model, the development of the two population densities is governed by a coupled system of two differential equations of the form: x’ = ax- bxy, y’=gy-dxy, where the parameters are assumed to be positive. These are These are sometime called Lotka-Volterra equations, named after the mathematicians who first studied them and other related equations. Comment on the terms in these equations, and describe what would happen to each population if the other were not present. We tend to think of the species involved in such models as being from the animal kingdom - two species of herbivores, for example. Can you think of a situation where one or both of the species could be other than animal? This system of equations is one of the few that can be integrated.

In this project, we shall consider determining the equilibria, and examining the behaviour near to these, for several systems

Project 5: Harvesting and Drug Administration

In the project, consider the following scenario: A population n(t) which satisfies the Malthus law with relaxation time r, having initial population density n0 = n(0) , is reduced by N after each interval of time r, What is the population density at t = r, , immediately before and immediately after the reduction has taken place? Find the relationship between the various parameters which ensures that the population has not become extinct. Assuming that condition holds, allow the population to develop in a Malthusian way.

Several tasks will be carried out.

Project 6: Population Growth and Earth’ Human Carrying Capacity

It is very easy to turn the predictions of a model into reality in one's mind and to forget that the model is only a model, and as such, depends on a number of assumptions, which may be only partially true (or not true at all!) The chief assumption in the Malthus model is that the growth rate constant is actually a constant. This means that no changes in birth rates due to birth control programmes or economic circumstances, no changes in life expectancy and child mortality, and so on. Quite clearly, such assumptions are unreasonable to a greater or lesser degree. More accurate models, as used by the UN, try to take account of these complexities. Nevertheless, the very crude Malthus model is still useful in at least three important ways: It provides a starting point for more sophisticated models; It provides a framework of understanding; Its predictions provide a yardstick for comparison of empirical data and the results from more complex models.

This project investigates the Malthus model and the Verhulst model.

Project 7: The Verhulst Model and Immigration/Harvesting

In the project, your first task is to investigate the Malthus model in its basic form. ﻿Given a table of data for the population of the USA from 1790 to 1930, ﻿Use Maple to plot this data - find out how to do this by seeking help on "pointplot". Can it be modelled by a logistic growth curve? Try several such curves, based on different triples of data points, and show them on the same diagram as the pointplot - seek help on "display" to achieve this.

Whether or not you are satisfied with any of the fitted curves, do any of them give sensible predictions for the population of the USA over the last 70 years?

Your next tasks will to modify the model to allow immigration and harvesting.

Project 8: The Verhulst Model and Immigration/Harvesting

Given a specific 2-species system of two coupled ODEs: x’ = f(x)+ axy, y’=g(y)+bxy, where f, g are quadratic, show that there can be at most four equilibria. Give a mathematical example where there are no equilibria. Is your example biologically feasible? Feasibility and Stability will be our main tasks in this project.

Project 9: Equilibria and Symbiosis

First consider the equations for two species with populations X and Y which interact with one another in a symbiotic way so that each species improves the conditions for growth of the other species. In the simplest model, the development of the two population densities is governed by a coupled system of two differential equations ad the form: x’ = -ax+bxy, y’=-gy+dxy. Compare this equation with those in projects 3 and 4. Solve this equation to give a relationship between y and x. If you tackle this by hand, don't expect to obtain y as an explicit function of x, and in particular it is not easy to draw the trajectories by hand in the phase-plane. However the Maple solution is explicit, involving the Lambert W function - find out about it using the Maple help system.

Further investigate a more realistic model when carrying capacity is allowed.

Project 10: Predator-prey models with Harvesting

The PP model for population densities of two-species is a system of two coupled differential equations. Harvesting can be modeled by adding a new term to one of the equations. For different scenarios, through Maple solutions, describe the general properties of the solutions and think of harvesting strategies which might prevent extinction.

AM4  ---   Topics In Mathematical Economics

This project is concerned with three topics: using Maple to further investigate topics covered in the MATH227 lectures; extending the Theory of Demand as covered in the MATH227 lectures; and an initial investigation of linear models in microeconomics. The use of Maple may well be helpful in the latter two topics as well, to streamline the calculations.

GROUP PROJECT

1. Introduction

The idea of a utility function to quantify a consumer's preferences was introduced in the lectures. As usual we consider two commodities and define U(x,y) to be the utility function where x and y are the amounts of the two commodities. What properties are required of a utility function? The group should consider various possibilities for a utility functions and use Maple to check that the properties are satisfied. (Start with some known examples from the lectures.)

2. Multimarket equilibrium and the Edgeworth Box

Suppose we have two consumers A and B who start out with certain amounts of two commodities. Will they wish to exchange some of these commodities and if so how much?

3. Linear Programming

There are many aspects of microeconomics which behave linearly or approximately so. In multi-market systems, goods, e.g. steel may be outputs of one process and inputs of another. With such systems it is generally useful to describe quantities by a vector in commodity space, not distinguishing between inputs and outputs. We consider n commodities and assemble them into an n-dimensional vector. We also define a {\it primary} commodity to be one which is only an input and never an output; a {\it secondary} commodity is one is an output of some industry. Suppose there are m primary commodities and therefore n-m secondary commodities. We shall consider the first (n-m) components of our vector to represent the amounts of secondary commodities, and the remainder to represent the amounts of primary commodities.

INDIVIDUAL PROJECTS

Project 1. Cost Functions

Suppose a producer uses amounts x and y of two inputs, and that q(x,y) is the amount of product made and c(x,y) is the cost of using these inputs. We wish to minimise c(x,y). Write a Maple program to evaluate the cost function for a given amount of product q(x,y) and c(x,y). (You may be able to adapt your earlier program for the utility function.) Try various possibilities for c(x,y) and q(x,y) and see what results you get for the cost functions. Again, start with examples from lectures or homework to check you get the right answers.

Project 2. Multimarket Equilibrium

Returning to the Edgeworth Box for the consumers A and B, find the locus of the points for which their indifference curves touch, i.e. the relation between y and x for these points. Plot this curve on your Edgeworth Box diagram. There are 8 small tasks to be done in this project involving Maple solution of equations and various investigations.

Project 3. Monetary Equilibrium

This project investigates the optimisation of utility functions under budget constraints. Next it considers some generalisations in which non specific values are used, either in the budget constraint or utility function. Finally it investigates any change in the original scenario that will ultimately affect the prices of the two commodities, as is the case in all real life situations.

Project 4. Wages and Leisure

We can regard income and leisure as a pair of commodities which the worker wishes to exploit to his maximum advantage. Suppose that x is his amount of leisure time and y is his income. Also, suppose p is his rate of pay and T is his available time (e.g. T could be 1 day if he is being paid daily). Then he works for a time T-x and we have y=p(T-x),  or  px+y=pT. This is effectively a budget constraint. We suppose as usual that there is a utility function U(x,y) which the worker will choose x and y to maximise subject to the budget constraint. The analysis is as usual but with q=1 and also B=pT. We will ask whether the employer can make the worker work longer by increasing the pay rate p. This may seem obvious but in fact will depend on the utility function.

Project 5. Linear Programming and Input-Output Analysis

First consider an example from simple linear programming: A printing company prints two weekly magazines, "Hello" and "Eh up" using paper and ink. It takes 5 tons of paper and 1 pint of ink to produce 1 batch of "Hello", and 4 tons of paper and 2 pints of ink to produce 1 batch of "Eh up". Each week the company is supplied with 17 tons of paper and 7 pints of ink. The profit on "Hello" is $10 per batch and that on "Eh up" is$5 per batch. How many batches of each magazine are produced per week and what is the profit?

If there are more than two variables, a graphical method of solution is not possible. It is then necessary to work out the size of the objective function at the vertices of the feasible region. To calculate every vertex and the corresponding size of the objective function would take a long time and is not necessary. The best way is to start at one vertex, usually the origin if it belongs to the feasible region, and then find, if possible, an adjacent vertex which increases the objective function. This is continued until a vertex is obtained such that no adjacent vertex improves the objective function. This vertex is then the optimal solution. Can you prove that any vertex which is not the optimal solution always has an adjacent vertex which improves the objective function?

Project 6. Further Study of Game theory