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MC380 Ordinary Differential Equations

MC380 Ordinary Differential Equations

Credits: 20 Convenor: Dr. M. Marletta Semester: 1

Prerequisites: essential: MC125, MC146, MC240, MC241, MC248 desirable: MC226, MC243
Assessment: Coursework: 10% Three hour exam: 90%

Lectures: 36 Problem Classes: 10
Tutorials: none Private Study: 104
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 150

Explanation of Pre-requisites

Students should already be familiar with the standard elementary methods for solving simple ODEs (MC125). They will require a basic knowledge of real analysis (MC146 and MC248) and some parts of the course will require less work for students who are familiar with the results on completeness given in MC243. For those parts of the course which deal with linear equations, students should be familiar with the idea of linear independence of elements of a vector space (MC241). Finally it may be helpful for some parts of the course that students be familiar with norms of matrices (MC226 or MC243).

Course Description

The theory of ordinary differential equations (often given the designer label `dynamical systems' in recent years) is vast, and so the choice of topics for this course is very much determined by the lecturer's own interests.

The course will start with two basic topics which are required for almost everything else (existence and uniqueness theorems, and linear systems of equations). The remaining two topics (phase space methods for differential equations, and numerical methods for differential equations) will build on the knowledge acquired in the first two sections.

Aims

This course will consider, at an introductory level, four classical topics from the extensive theory of ODEs. By the end of the course the student should know enough to be able to engage in further independent study at postgraduate level in each of the areas covered.

Objectives

By the end of this module a diligent student should be able to

Transferable Skills

Most of the skills acquired on this module are transferable to other branches of pure and applied mathematics where differential equations arise; the course is a `desirable' prerequisite for our Level 4 course on Partial Differential Equations. Additionally, the section on numerical solution of initial value problems studies methods which are widely used in industry - particularly engineering - and in finance to solve differential equations arising in applications.

Syllabus

1.
Existence and Uniqueness Theorems for solutions of initial value problems, including Picard's Theorem and variants thereof.
2.
Linear Systems of ordinary differential equations, including bases for solution spaces. Techniques for solving inhomogeneous equations.
3.
Phase diagram methods for differential equations: analysis of critical points, and criteria for existence of periodic solutions and limit cycles.
4.
Numerical Methods, including linear multistep and Runge-Kutta methods.

Reading list

Recommended:

D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, Oxford University Press.

G. F. Simmons, Differential Equations with Applications and Historical Notes, McGraw-Hill.

J. D. Lambert, Numerical Methods for Ordinary Differential Systems, Wiley.

Details of Assessment

The final assessment of this module will consist of 10% coursework and 90% from a three hour examination during the January exam period. The 10% coursework contribution will be determined by students' solutions to coursework problems. The examination paper will contain 8 questions with full marks on the paper obtainable from 5 complete answers.

Next: MC381 Modelling physical systems Up: Year 3 Previous: MC361 Generalized Linear Models

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Last updated: 10/4/2000
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