35231 Differential Equations

Warning: The information on this page is indicative. The subject outline for a particular semester, location and mode of offering is the authoritative source of all information about the subject for that offering. Required texts, recommended texts and references in particular are likely to change. Students will be provided with a subject outline once they enrol in the subject.

Handbook description

Differential equations arise in contexts as diverse as the analysis and pricing of financial options, and the design of novel materials for telecommunications. In this subject students develop familiarity with the theory of differential equations, applications of this theory and some of the main computational techniques used in the solution of differential equations. Topics include existence and uniqueness of solutions; method of Frobenius; variation of parameters; the Taylor and Runge-Kutta methods for initial value problems; Fourier series; solving partial differential equations and boundary value problems by separation of variables, transform methods and finite difference methods.

Subject objectives/outcomes

By the end of this subject, students should be able to demonstrate:

1. proficiency in finding solutions various types of differential equations or systems of differential equations and the ability to judge which method is applicable or most appropriate for finding the solution of a differential equation
2. proficiency in the mathematical techniques which may be needed in solving differential equations. These include using variation of parameters, finding series solutions, separation of variables and Fourier series expansions and calculating and inverting Laplace transforms
3. an understanding of the various theoretical results which justify the use of the above skills
4. the ability to write an exposition on selected topics in the subject
5. the ability to summarise the main strategies of a given proof and conversely to construct a proof from verbal explanations of the methods
6. the ability to critically analyse and comment upon various stages in a given proof
7. the ability to model physical problems in terms of differential equations in other areas of mathematics and its applications
8. an appreciation of the relationship of differential equations to other areas of mathematics and its applications
9. the ability to see the subject as a coherent whole and not merely as a collection of results and techniques.

Contribution to course aims and graduate attributes

This subject is expected to contribute to the following graduate profile attributes
1. Disciplinary knowledge and its appropriate application
An understanding of the nature, practice & application of analysis to practical and theoretical problems which arise in a wide range of fields, from finance to physics,

2. An inquiry-oriented approach
An understanding of the scientific method of knowledge acquisition. Encompasses problem solving, critical thinking and analysis attributes, and the ability to discover new understandings

3. Professional skills and their appropriate application
The ability to acquire, develop, employ and integrate a range of technical, practical and professional skills, in appropriate and ethical ways within a professional context, autonomously and collaboratively and across a range of disciplinary and professional areas.
e.g. Time management skills, personal organisation skills, teamwork skills, computing skills, laboratory skills, data handling, quantitative and graphical literacy skills.

7. Initiative and innovative ability
An ability to think and work creatively, including the capacity for self-starting, and the ability to apply scientific skills to unfamiliar applications.

Teaching and learning strategies

The material will be presented weekly over three hours of lectures. These will be supplemented by a weekly one hour tutorial in which exercises will be carried out by the student to reinforce and broaden their understanding of the materials from lectures. There will also be assignments which will present aspects of the subject which depend on the material presented in lectures, but may require independent work to be fully mastered. Reading the recommended texts and doing exercises from the texts plays an essential part in the study of this subject. It is expected that the student will spend 6 hours per week outside class in the study of the subject.

Content

1. Theory of Ordinary Differential Equations, Wronskians, construction of second solutions from known solutions, series solutions, method of Frobenius, regular singular points, special functions-Bessel functions.
2. Laplace Transform Methods
3. Partial Differential Equations and Fourier Series
4. Applications of partial differential equations to problems in science and finance.
    

Assessment

Assessment item 1: Assignment 1

Assessment item 2: Assignment 2

Assessment item 3: Class Test

Assessment item 4: Final Exam

Minimum requirements

Student must obtain at least 40% of the marks available for the final examination in order to pass this subject. If 40% is not reached, an X grade fail may be awarded for the subject, irrespective of an overall mark greater than 50.

A final overall mark of 50 Percent or more is required to pass the course. The final result is calculated as max(E,A), where E is the exam mark out of 100, and A=0.6E+C+B and C is the class test mark out of 20 and B is the total mark out of 20 for the two assignments.

Note that a supplementary exam will not be offered to any failing student if the student does not obtain a score of at least 40 percent of the possible marks available in the final exam. That is, no supplementary will be offered unless E>= 40.

Recommended texts

The following textbooks are useful, but not required.

• Edwards C H, Penney D E, Differential Equations and Boundary Value Problems, Computing and Modelling (2nd Edn), Prentice Hall, 2000.
• Boyce W E, DiPrima R C, Elementary Differential Equations and Boundary Problems (7th Edn), Wiley, 2001.
• Nagle R K, Saff E B, Fundamantals of Differential Equations and Boundary Value Problems (3rd Edn), Addison-Wesley, 2000.
• Zill D G, A First Course in Differential Equations with Modelling Applications (6th Edn), Brooks/Cole, 1997.
• Zill D G, Cullen M R, Advanced Engineering Mathematics (2nd Edn), Jones and Bartlett, 2000.

The library has dozens of useful textbooks on this subject.

Indicative references

A set of printed lecture notes will be available online as each section is completed in lectures.