MC323 Ordinary Differential Equations

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MC323 Ordinary Differential Equations

Credits: 20

Convenor: Dr. M. Marletta

Semester: 2

Prerequisites:	essential: MC121, MC146, MC240, MC241, MC248	desirable: MC221 or MC243
Assessment:	Coursework: 10%	Three hour exam: 90%

Lectures:	36	Classes:	none
Tutorials:	12	Private Study:	102
Labs:	none	Seminars:	none
Project:	none	Other:	none
Total:	150

Explanation of Pre-requisites

Students should already be familiar with the standard elementary methods for solving simple ODEs (MC121). They will require a basic knowledge of real analysis (MC146 and MC248) together with the ideas of uniform convergence and the Contraction Mapping Theorem (MC240). 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 (MC221 or MC243).

Course Description

The theory of ordinary differential equations (often given the deigner 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 (oscillation theory and eigenvalue problems 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

analyze initial value problems in order to determine whether or not they have unique solutions, and over what interval the existence of such solutions may be guaranteed;
know the basic properties of fundamental matrices for linear systems of differential equations, and be able to construct such matrices explicitly in simple cases, using them to find solutions of inhomogeneous systems;
solve simple Sturm-Liouville eigenvalue problems and use comparison theorems and oscillation theory to analyze problems which cannot be solved in closed form;
formulate simple methods for numerical solution of initial value problems and prove their convergence under suitable hypotheses.

Transferable Skills

Most of the skills acquired on this module are only transferable to other branches of pure and applied mathematics where differential equations arise. An exception is the section on numerical solution of initial value problems, which studies methods which are widely used in industry and 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.: Basic Oscillation Theory and Sturm-Liouville problems: the concepts of eigenvalue and eigenfunction for a differential equation, and the Prüfer theory for existence of eigenvalues, including various forms of the Sturm Comparison Theorem.
4.: Numerical Methods, including linear multistep and Runge-Kutta methods.

Reading list

Details of Assessment

There will be a three hour midsummer examination which will count towards 90% of the assessment. The remaining 10% will come from fortnightly assignments set by the lecturer during the course.

Next: MC340 Number Theory Up: Year 3 Previous: MC322 Modelling physical systems

Roy L. Crole
10/22/1998