APP MTH 7107 - Partial Differential Equations & Waves III

North Terrace Campus - Semester 2 - 2014

Differential equation models describe a wide range of complex problems in biology, engineering, physical sciences, economics and finance. This course focusses on partial differential equation (PDE) models, which will be developed in the context of modelling heat and mass transport and, in particular, wave phenomena, such as sound and water waves. This course develops students' skills in the formulation, solution, understanding and interpretation of PDE models. As well as developing analytic solutions, this course establishes general structures, characterisations, and numerical solutions of PDEs. In particular, computational methods using finite differences are implemented and analysed. Topics covered are: Formulation of PDEs using conservation laws: heat/mass/ wave energy transport; waves on strings and membranes; sound waves; Euler equations and velocity potential for water waves. The structure of solutions to PDEs: separation of variables (space/space, space/time); boundary value problems; SturmLouiville theory; method of characteristics; and classification of PDEs via coordinate transformation. Complex-variable form of waves. Wave dispersion. Group velocity. Finite difference solution of PDEs and BVPs: implicit and explicit methods; programming; consistency, stability and convergence; numerical differentiation.

• General Course Information
Course Details
Course Code APP MTH 7107 Partial Differential Equations & Waves III Applied Mathematics Semester 2 Postgraduate Coursework North Terrace Campus 3 Up to 3 contact hours per week. MATHS 1012 APP MTH 3000, APP MTH 3017 MATHS 2102, MATHS 2101 Ongoing assessment: 30%, Exam 70%
Course Staff

Course Coordinator: Professor Anthony Roberts

Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
Course Learning Outcomes
On successful completion of this course students will be able to:
1. use knowledge of partial differential equations (PDEs), modelling, the general structure of solutions, and analytic and numerical methods for solutions.
2. formulate physical problems as PDEs using conservation laws.
3. understand analogies between mathematical descriptions of different (wave) phenomena in physics and engineering.
4. classify PDEs, apply analytical methods, and physically interpret the solutions.
5. solve practical PDE problems with finite difference methods, implemented in code, and analyse the consistency, stability and convergence properties of such numerical methods.
6. interpret solutions in a physical context, such as identifying travelling waves, standing waves, and shock waves.

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute Course Learning Outcome(s)
Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised. all
The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. all
An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. all
Skills of a high order in interpersonal understanding, teamwork and communication. all
A proficiency in the appropriate use of contemporary technologies. all
A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. all
An awareness of ethical, social and cultural issues within a global context and their importance in the exercise of professional skills and responsibilities. all
• Learning Resources
Recommended Resources
1. Agarwal, R. P. & O'Regan, D. (2009), Ordinary and Partial Differential Equations With Special Functions, Fourier Series, and Boundary Value Problems, Springer.
2. Billingham, J. and  King, A.C. (2000) Wave motion, CUP.
3. Haberman, R. (1987), Elementary applied partial differential equations: with Fourier series and boundary value problems, 2nd edn, Prentice-Hall.
4. Kevrekidis, I. G. & Samaey, G. (2009), Equation-free multiscale computation: Algorithms and applications, Annu. Rev. Phys.  Chem.  60, 321-344.  doi:10.1146/annurev.physchem.59.032607.093610
5. Kreyszig, E. (2011), Advanced engineering mathematics, 10th edn, Wiley.
6. Roberts, A. J. (1994), A one-dimensional introduction to continuum mechanics, World Sci.
Online Learning
This course uses MyUni exclusively for providing electronic resources, such as lecture notes, assignment papers, and sample solutions.  Students should make appropriate use of these resources.  Link to MyUni login page:  https://myuni.adelaide.edu.au/webapps/login/
• Learning & Teaching Activities
Learning & Teaching Modes
This course relies on combined lecture and tutorial classes as the primary learning mechanism for the material.  A sequence of written or online assignments provides assessment opportunities for students to gauge their progress and understanding.

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

Lectures/tutorials 36 100
Assignments/assessment 7 56
Total 156
Learning Activities Summary
Lecture classess will explore the following.  Conservation of mass. Conservation of momentum.  Separation of variables.  Wave properties.  Sturm--Liouville BVPs.  Discretise 1D space.  Model shallow water waves.  Computational integration.  General wave systems.  PDEs in higher dimension.  Classification of PDEs. Shocking characteristics.  Tutorial work is integrated into lecture class times.

In more detail, the course includes material from the following.

1. Conservation of mass:   1D, varying cross-section,  application to gas,  physical dimensions of terms, d'Alembert for age structured populations
2. Conservation of momentum:   1D, varying cross-section, forces and pressure, physical dimensions, ideal gas law closure, d'Alembert waves
3. Separation of variables: constant coeff linear PDE on finite domain, boundary conditions at open\slash closed end of pipe, connect to diffusion PDE, superposition of normal modes, general homogeneous solution, summarise separation of variables
4. Wave properties: frequency,  wavenumber,  dispersion relation
5. Sturm--Liouville BVPs: linear and varying coefficients on finite domain, regular Sturm--Liouville, self-adjoint operators, Eigenfunctions are orthogonal, Eigenvalues are real, Eigenfunctions are usually unique, normal modes of vibration, Eigenfunctions expand inhomogeneous solutions, Fredholm alternative
6. Discretise 1D space:   Taylor's theorem, consistency, varying coefficients, finding equilibria, eigenvalues, symmetric discretisations reflect conservation
7. Model shallow water waves: water depth and mean velocity, hydrostatic closure, dispersion relation, varying depth, numerical seiches in 1D
8. Computational integration: method of lines, explicit time steps, consistency and stability, Lax equivalence theorem, Crank-Nicholson scheme, sparse matrices, numerical dispersion
9. General wave systems: deep irrotational water waves, phase and group velocity, energy propagation other dispersion relations, phase and group velocity, wave guide refraction
10. PDEs in higher dimension: shallow water in lakes, separation of variables, Sturm-Liouville nature of Helmholtz-like PDEs, finite differences in nD, consistency, anisotropic dispersion relation
11. Classification of PDEs: revisit d'Alembert, characteristics for uni-directional wave, xy-coordinate transform seeks characteristics, hyperbolic, parabolic, elliptic PDEs
12. Shocking characteristics: N-wave in 1D gas flow, bore in shallow water with Chezy law, roll wave instability
• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

Assessment Summary

ComponentWeightingObjective assessed
Assignments 30% all
Exam 70% all
Assessment Related Requirements
An aggregate score of at least 50% is required to pass the course.
Assessment Detail

Assessment itemDistributedDue dateWeighting
Continuous assessment TBA TBA 6%
Assignment 1 week 1 week 2 4%
Assignment 2 week 3 week 4 4%
Assignment 3 week 5 week 6 4%
Assignment 4 week 7 week 8 4%
Assignment 5 week 9 week 10 4%
Assignment 6 week 11 week 12 4%
Submission

1. All written assignments are to be submitted to the designated hand-in boxes in the School of Mathematical Sciences with a signed cover sheet attached, or submitted as pdf via MyUni.
2. Late assignments will not be accepted without a medical certificate.
3. Assignments normally have a two week turn-around time for feedback to students.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M10 (Coursework Mark Scheme)
FNS   Fail No Submission
F 1-49 Fail
P 50-64 Pass
C 65-74 Credit
D 75-84 Distinction
HD 85-100 High Distinction
CN   Continuing
NFE   No Formal Examination
RP   Result Pending

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through Access Adelaide.

• Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
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