Undergraduate Programme and Module Handbook 2013-2014 (archived)
Module MATH4041: PARTIAL DIFFERENTIAL EQUATIONS IV
Department: Mathematical Sciences
MATH4041:
PARTIAL DIFFERENTIAL EQUATIONS IV
Type |
Open |
Level |
4 |
Credits |
20 |
Availability |
Available in 2013/14 |
Module Cap |
None. |
Location |
Durham
|
Prerequisites
- Analysis in Many Variables II (MATH2031) AND other
Mathematics modules to the value of 80 credits in Years 2 and 3, with at
least 40 credits at Level 3
Corequisites
Excluded Combination of Modules
- Partial Differential Equations III (MATH3291)
Aims
- To develop a basic understanding of the theory and methods of
solution for Partial Differential Equations.
- To develop a basic understanding of the ideas of approximate
(numerical) solution to certain Partial Differential Equations.
Content
- First order equations and characteristics. Conservation
laws.
- Systems of first-order equations, conservation laws and
Riemann invariants.
- Hyperbolic systems and discontinuous derivatives.
Acceleration waves.
- Classification of general second order quasi-linear
equations and reduction to standard form for each type (elliptic,
parabolic and hyperbolic).
- Energy methods for parabolic equations. Well-posed
problems.
- Maximum principles for parabolic equations.
- Finite difference solution to parabolic and elliptic
equations.
- Stability and convergence for solution to finite
difference equations.
- Iterative methods of solving Ax=b.
- Reading material on one of the following topics: Strict
form of the maximum principle for parabolic equations; Error estimates
for numerical solutions of partial differential equations; Further
aspects of non-linear partial differential equations.
Learning Outcomes
- By the end of the module students will: be able to solve
problems in Partial Differential Equations.
- have an understanding of theoretical mathematics in the field
of Partial Differential Equations.
- have mastered a coherent body of knowledge of these subjects
demonstrated through one or more of the following topic areas:
Solution of hyperbolic equations and systems.
- Classification of second order PDEs, and their
solution.
- Maximum principles for a parabolic equation.
- Energy estimates for a parabolic equation.
- Finite difference methods for PDEs.
- have an appreciation of the techniques used in one of the
following areas: maximum principle for parabolic equations, error
estimates for numerical solutions of partial differential
equations.
- have an advanced understanding in one of the following areas:
Maximum principles for parabolic equations; Error estimates;
Non-linear partial differential equations.
- Students will have highly specialised and advanced
mathematical skills in the following areas: Modelling, Numerical
Mathematics.
- Students will have an appreciation of Partial Differential
Equations in the real world and how to solve them.
- Students will be able to study independently to further their
knowledge of an advanced topic.
Modes of Teaching, Learning and Assessment and how these contribute to
the learning outcomes of the module
- Lectures demonstrate what is required to be learned and the
application of the theory to practical examples.
- Subject material assigned for independent study develops the
ability to acquire knowledge and understanding without dependence on
lectures.
- Assignments for self-study develop problem-solving skills and
enable students to test and develop their knowledge and
understanding.
- Formatively assessed assignments provide practice in the
application of logic and high level of rigour as well as feedback for
the students and the lecturer on students' progress.
- The end-of-year examination assesses the knowledge acquired
and the ability to solve predictable and unpredictable
problems.
Teaching Methods and Learning Hours
Activity |
Number |
Frequency |
Duration |
Total/Hours |
|
Lectures |
40 |
2 per week for 19 weeks and 2 in term 3 |
1 Hour |
40 |
|
Preparation and Reading |
|
|
|
160 |
|
Total |
|
|
|
200 |
|
Summative Assessment
Component: Examination |
Component Weighting: 100% |
Element |
Length / duration |
Element Weighting |
Resit Opportunity |
three hour written examination |
|
100% |
|
Four written assignments to be assessed and
returned. Other assignments are set for self-study and complete solutions
are made available to students.
■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University