Next: MC381 Modelling physical systems
Up: Year 3
Previous: MC361 Generalized Linear Models
MC380 Ordinary Differential Equations
| Credits: 20 |
Convenor: Dr. M. Marletta |
Semester: 2 |
| Prerequisites: |
essential: MC121, MC146, MC240, MC241, MC248 |
desirable: MC226, MC243 |
| Assessment: |
Coursework: 10% |
Three hour exam: 90% |
| Lectures: |
36 |
Classes: |
10 |
| Tutorials: |
none |
Private Study: |
104 |
| 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) 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
(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 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.
- 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
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 Summer 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
S. J. Ambler
11/20/1999