APP MTH 7107 - Partial Differential Equations and Waves

North Terrace Campus - Semester 2 - 2021

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
    Course Partial Differential Equations and Waves
    Coordinating Unit 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)
    Assessment Ongoing assessment, Examination
    Course Staff

    Course Coordinator: Professor Yvonne Stokes

    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)
    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
    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
    Career and leadership readiness
    • technology savvy
    • professional and, where relevant, fully accredited
    • forward thinking and well informed
    • tested and validated by work based experiences
    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
    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
  • Learning Resources
    Required Resources
    Access to the internet.
    Recommended Resources
    1. Olver, P.J. (2014), Introduction to Partial Differential Equations, Springer. [Available from as pdf from library]
    2. Agarwal, R. P. & O'Regan, D. (2009), Ordinary and Partial Differential Equations With Special Functions, Fourier Series, and Boundary Value Problems, Springer. [Available as pdf from library]
    3. Iserles, A. (2009), A first course in the numerical analysis of differential equations, CUP. [Available online via Library]
    4. Ockendon, J.R. et al (2003) Applied Partial Differential Equations, OUP.
    5. Billingham, J. and  King, A.C. (2000) Wave motion, CUP.
    6. Kreyszig, E. (2011), Advanced engineering mathematics, 10th edn, Wiley.
    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:
  • Learning & Teaching Activities
    Learning & Teaching Modes
    The 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 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 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
    Component Task type Due Weighting Outcomes Assessed
    Assignments Formative and summative

    Weeks 4,6,8,10,12

    20% All
    Quizzes Formative and summative Weekly 5% All
    Mid-semester test Summative Week 9 15% All
    Exam Summative Exam period 60% All
    More details will be announced later.
    Assessment Related Requirements
    No information currently available.
    Assessment Detail
    No information currently available.
    No information currently available.

    Course Grading

    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.

  • 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 ( 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
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