APP MTH 7107 - Partial Differential Equations and Waves
North Terrace Campus - Semester 2 - 2022
General Course Information
Course Code APP MTH 7107 Course Partial Differential Equations and Waves Coordinating Unit School of Mathematical Sciences Term Semester 2 Level Postgraduate Coursework Location/s North Terrace Campus Units 3 Contact Up to 3 contact hours per week. Available for Study Abroad and Exchange Y Assumed Knowledge (MATHS 2102 or MATHS 2106 or MATHS 2201) and (MATHS 2104 or MATHS 2107) Course Description 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.
Course Coordinator: Professor Yvonne Stokes
The full timetable of all activities for this course can be accessed from Course Planner.
Course Learning OutcomesOn successful completion of this course students will be able to:
- use knowledge of partial differential equations (PDEs), modelling, the general structure of solutions, and analytic and numerical methods for solutions.
- formulate physical problems as PDEs using conservation laws.
- understand analogies between mathematical descriptions of different (wave) phenomena in physics and engineering.
- classify PDEs, apply analytical methods, and physically interpret the solutions.
- solve practical PDE problems with finite difference methods, implemented in code, and analyse the consistency, stability and convergence properties of such numerical methods.
- 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)
Attribute 1: Deep discipline knowledge and intellectual breadth
Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.
Attribute 2: Creative and critical thinking, and problem solving
Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.
Attribute 3: Teamwork and communication skills
Graduates convey ideas and information effectively to a range of audiences for a variety of purposes and contribute in a positive and collaborative manner to achieving common goals.
Attribute 4: Professionalism and leadership readiness
Graduates engage in professional behaviour and have the potential to be entrepreneurial and take leadership roles in their chosen occupations or careers and communities.
Attribute 5: Intercultural and ethical competency
Graduates are responsible and effective global citizens whose personal values and practices are consistent with their roles as responsible members of society.
Attribute 8: Self-awareness and emotional intelligence
Graduates are self-aware and reflective; they are flexible and resilient and have the capacity to accept and give constructive feedback; they act with integrity and take responsibility for their actions.
Required ResourcesAccess to the internet.
- Olver, P.J. (2014), Introduction to Partial Differential Equations, Springer. [Available online from the Barr Smith Library]
- Agarwal, R. P. & O'Regan, D. (2009), Ordinary and Partial Differential Equations With Special Functions, Fourier Series, and Boundary Value Problems, Springer. [Available on line from the Barr Smith Library]
- Iserles, A. (2009), A first course in the numerical analysis of differential equations, Cambridge University Press. [Available online from the Barr Smith Library]
- Ockendon, J.R. et al (2003) Applied Partial Differential Equations, Oxford University Press.
- Billingham, J. and King, A.C. (2000) Wave motion, Cambridge University Press.
- Kreyszig, E. (2011), Advanced engineering mathematics, 10th edn, Wiley.
Online LearningThis 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/
Learning & Teaching Activities
Learning & Teaching ModesThe course material is presented via a number of sources that complement each other: course notes and lecture videos that are posted on MyUni, as well as a weekly lecture. Having studied the material from all sources, students test their initial understanding with online quizzes.
Students deepen their understanding of the material by working on tutorial exercises and attending a tutorial (face to face or online). Assignments and short projects provide students with further opportunities to get feedback on their understanding. Students interact with the lecturer and with each other on the Piazza discussion platform. In addition, the lecturer offers weekly consulting.
The information below is provided as a guide to assist students in engaging appropriately with the course requirements.The information below is provided as a guide to assist students in engaging appropriately with the course requirements.
Activity Quantity Workload Hours Videos/tutorials TBA 100 Assessment tasks TBA 56 Total 156
Learning Activities SummaryLecture classes will explore the following. Conservation of mass and momentum. Method of characteristics, non-uniqueness and shocks. Separation of variables. Sturm-Liouville BVPs. Classification of PDEs, characteristic curves. Discretisation of 1D space. Modelling shallow water waves. PDEs in higher dimension. Computational integration. General wave systems.
Tutorial work is integrated into lecture class times.
In more detail, the course includes material from the following.
- Conservation of mass and momentum: Car traffic has waves; Conservation of fluid; Momentum PDE for ideal gases; The wave equation.
- Traffic flow and the method of characteristics: Non-uniqueness; Shocks.
- Separation of variables: Linearity empowers analysis; Separation of variables generates boundary value problems.
- Wonderful Sturm–Liouville boundary value problems: Self-adjoint operators form Sturm-Liouville problems; Eigenfunctions expand inhomogeneous solutions.
- Discretising 1D space: Lagrange’s theorem underpins the method of lines; Find equilibria; Numerical linearisation characterises solution dynamics; PDE-free patch dynamics.
- Modelling 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.
- Computational integration: 1D heat/diffusion PDE raises fundamental issues; Crank–Nicholson schemes are reasonably stable and accurate; Invoke sparse matrices for implicit schemes; Crank–Nicholson discretises wave systems; Second order PDEs in 2D.
- General wave dynamics: Water waves in finite depth; The dispersion relation of waves; Energy travels at the group velocity; Wave propagation in multi-dimensions.
- Shocking classification of PDEs: Change of variables transforms the PDE; Reduction to the hyperbolic canonical form; Elliptic and parabolic canonical form.
The University's policy on Assessment for Coursework Programs is based on the following four principles:
- Assessment must encourage and reinforce learning.
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- Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
- Assessment must maintain academic standards.
Component Task type Due Weighting Outcomes Assessed Assignments Formative and summative
20% All Quizzes Formative and summative Weekly 5% All Mid-semester test Summative Week 9 15% All Exam Summative Exam period 60% All
Assessment Related RequirementsNo information currently available.
Assessment DetailNo information currently available.
SubmissionNo information currently available.
Grades for your performance in this course will be awarded in accordance with the following scheme:
M10 (Coursework Mark Scheme) Grade Mark Description 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.
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