## APP MTH 7107 - Partial Differential Equations & Waves III

### North Terrace Campus - Semester 2 - 2018

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 Mathematical Sciences Semester 2 Postgraduate Coursework North Terrace Campus 3 Up to 3 contact hours per week. Y (MATHS 2101 and MATHS 2102) or (MATHS 2201 and MATHS 2202) APP MTH 3000, APP MTH 3017 MATHS 2104 Ongoing assessment: 30%, Exam 70%
##### Course Staff

Course Coordinator: Dr Michael Chen

##### 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.
##### University Graduate Attributes

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

University Graduate Attribute Course Learning Outcome(s)
Deep discipline knowledge
• informed and infused by cutting edge research, scaffolded throughout their program of studies
• acquired from personal interaction with research active educators, from year 1
• accredited or validated against national or international standards (for relevant programs)
all
Critical thinking and problem solving
• steeped in research methods and rigor
• based on empirical evidence and the scientific approach to knowledge development
• demonstrated through appropriate and relevant assessment
all
Teamwork and communication skills
• developed from, with, and via the SGDE
• honed through assessment and practice throughout the program of studies
• encouraged and valued in all aspects of learning
all
• technology savvy
• professional and, where relevant, fully accredited
• forward thinking and well informed
• tested and validated by work based experiences
all
Intercultural and ethical competency
• adept at operating in other cultures
• comfortable with different nationalities and social contexts
• able to determine and contribute to desirable social outcomes
• demonstrated by study abroad or with an understanding of indigenous knowledges
all
Self-awareness and emotional intelligence
• a capacity for self-reflection and a willingness to engage in self-appraisal
• open to objective and constructive feedback from supervisors and peers
• able to negotiate difficult social situations, defuse conflict and engage positively in purposeful debate
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.
7. Roberts, A. J. & Kevrekidis, I. G. (2007), 'General tooth boundary conditions for equation free modelling', SIAM J. Scientific Computing 29(4), 1495--1510.
##### 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 and momentum. Separation of variables. Sturm--Liouville BVPs. Discretise 1D space. Model shallow water waves. PDEs in higher dimension. Computational integration. General wave systems. Classification of PDEs, characteristics and shocks.

Tutorial work is integrated into lecture class times.

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

1. Conservation of mass and momentum: Car traffic has waves; Conservation of fluid; Momentum PDE for ideal gases; The wave equation; The dispersion relation of waves
2. Separation of variables: Linearity empowers analysis; Separation of variables generates boundary value problems
3. Wonderful Sturm–Liouville boundary value problems: Self-adjoint operators form Sturm--Liouville problems; Eigenfunctions expand inhomogeneous solutions
4. Discretise 1D space: Lagrange’s theorem underpins the method of lines; Find equilibria; Numerical linearisation characterises solution dynamics; PDE-free patch dynamics
5. Model shallow water waves: Conservation derives the PDEs; Small amplitude waves; Compute seiches in 1D
6. PDEs with at least three independent variables: Vibration of a rectangular membrane; The self-adjoint Sturm--Liouville nature of Helmholtz-like PDEs
7. Computational integration: 1D heat/diffusion PDE raises fundamental issues; Crank–Nicolson schemes are reasonably stable and accurate; Invoke sparse matrices for implicit schemes; Crank–Nicolson discretises wave systems; Second order PDEs in 2D
8. General wave dynamics: Water waves in finite depth; Energy travels at the group velocity; Wave propagation in multi-dimensions
9. Shocking classification of PDEs: Change of variables transforms the PDE; Reduction to the hyperbolic canonical form; Elliptic and parabolic canonical form; Traffic flow and the method of characteristics; Loud uni-directional sound
• 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 either submitted to the designated hand in boxes within 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 and a request prior to the due date.
3. Assignments normally have a one 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
• Policies & Guidelines
• Fraud Awareness

Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's studentâ€™s disciplinary procedures.

The University of Adelaide is committed to regular reviews of the courses and programs it offers to students. The University of Adelaide therefore reserves the right to discontinue or vary programs and courses without notice. Please read the important information contained in the disclaimer.

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