Perturbation th & calc of variations (840G1)
Perturbation theory and calculus of variations
Module 840G1
Module details for 2022/23.
15 credits
FHEQ Level 6
Module Outline
The aim of this course is to introduce the student to a variety of techniques, such as those used in perturbation theory and calculus of variations, primarily involving ordinary differential equations that have applications in various branches of applied mathematics. No particular application is emphasized.
The syllabus of the course is as follows:
1. Dimensional analysis and scaling:
a. physical quantities and their measurement;
b. dimensions;
c. change of units;
d. physical laws;
e. Buckingham Pi Theorem;
f. scaling.
2. Regular perturbation methods:
a. direct method applied to algebraic equations and initial value problems (IVP);
b. Poincar method for periodic solutions;
c. validity of approximations.
3. Singular perturbation methods:
a. finding approximate solutions to algebraic solutions;
b. finding approximate solutions to boundary value problems (BVP) including boundary layers and matching.
4. Calculus of Variations:
a. necessary conditions for a function to be an extremal of a fixed or free end point problem involving a functional of integral form;
b. isoperimetric problems.
Module learning outcomes
Systematic understanding of the concept of dimensions of physical quantities and how to express problems involving them in a dimensionless form using appropriate scaling.
Ability to apply perturbation methods and be able to handle problems that generate secular terms.
Ability to tackle singular perturbation problems using scaling to obtain the inner solution valid in the boundary layer.
Systematic understanding of the calculus of variations and its use in solving simple extremal problems.
| Type | Timing | Weighting |
|---|---|---|
| Computer Based Exam | Semester 2 Assessment | 80.00% |
| Coursework | 20.00% | |
| Coursework components. Weighted as shown below. | ||
| Portfolio | T2 Week 11 | 40.00% |
| Problem Set | T2 Week 11 | 15.00% |
| Problem Set | T2 Week 7 | 15.00% |
| Problem Set | T2 Week 4 | 15.00% |
| Problem Set | T2 Week 9 | 15.00% |
Timing
Submission deadlines may vary for different types of assignment/groups of students.
Weighting
Coursework components (if listed) total 100% of the overall coursework weighting value.
| Term | Method | Duration | Week pattern |
|---|---|---|---|
| Spring Semester | Lecture | 2 hours | 11111111111 |
| Spring Semester | Lecture | 1 hour | 11111111111 |
How to read the week pattern
The numbers indicate the weeks of the term and how many events take place each week.
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