5M: Advanced Methods in Differential Equations MATHS5041

  • Academic Session: 2018-19
  • School: School of Mathematics and Statistics
  • Credits: 20
  • Level: Level 5 (SCQF level 11)
  • Typically Offered: Either Semester 1 or Semester 2
  • Available to Visiting Students: No
  • Available to Erasmus Students: No

Short Description

The course aims to demonstrate how rigorous techniques from analysis can be applied to ordinary and partial differential equations.

Timetable

2 hours of lectures a week, over 11 weeks.

1 hour tutorial a week over 10 weeks (or equivalent)

Requirements of Entry

Mandatory Entry Requirements

 

4H Partial Differential Equations

 

Recommended Entry Requirements

Assessment

Assessment

 

100% Final Exam

 

 

Reassessment

In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

Main Assessment In: April/May

Are reassessment opportunities available for all summative assessments? No

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below. 

Course Aims

The course aims to demonstrate how rigorous techniques from analysis can be applied to ordinary and partial differential equations.

 

Course Outline:

 

 

1. Ordinary Differential Equations

 

  (a) Local existence and uniqueness theory for Lipschitz & continuous vector fields.

  (b) A priori estimates and global existence.

 

1. Classical Theory of Linear Partial Differential Equations

 

  (a) Laplace and Poisson's equations: Fundamental solution; maximum principle; Liouville's Theorem; energy methods and variational formulation.

  (b) Heat equation: Fundamental solution; maximum principle; smoothing property.

  (c) Wave equation: Recall solution formulas; energy methods; finite speed of propagation.

 

2. Scalar Conservation Laws: Recall the method of characteristics; shocks; weak formulation; Rankine-Hugoniot and entropy conditions.

Intended Learning Outcomes of Course

On completion of this course the student will be expected to know and understand

the main aspects of the theory and should be able to:

 

1. Apply existence theorems to given ordinary differential equations.

 

2. State the fundamental solution and the maximum principles for the Laplace and heat equations.

 

3. Apply energy methods to prove uniqueness for given linear partial differential equations.

 

4. Derive the variational formulation of the Laplace equation.

 

5. Employ the Rankine-Hugoniot and the entropy conditions to solve scalar conservation laws.

Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.