APP MTH 4123 - Partial Differential Equations and Waves - Honours

North Terrace Campus - Semester 2 - 2022

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 4123
    Course Partial Differential Equations and Waves - Honours
    Coordinating Unit School of Mathematical Sciences
    Term Semester 2
    Level Undergraduate
    Location/s North Terrace Campus
    Units 3
    Contact Up to 3 hours per week
    Available for Study Abroad and Exchange Y
    Prerequisites MATHS 2102 or MATHS 2106 or MATHS 2201
    Assumed Knowledge (MATHS 2104 or MATHS 2107) and (MATHS 2101 or MATHS 2202 or ELEC ENG 2106)
    Restrictions Honours students only
    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 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)

    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.

    all

    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.

    all

    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.

    all

    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.

    all

    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.

    all

    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.

    all
  • Learning Resources
    Required Resources
    Access to the internet.
    Recommended Resources
    1. Olver, P.J. (2014), Introduction to Partial Differential Equations, Springer. [Available online from the Barr Smith 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 on line from the Barr Smith Library]
    3. Iserles, A. (2009), A first course in the numerical analysis of differential equations, Cambridge University Press. [Available online from the Barr Smith Library]
    4. Ockendon, J.R. et al (2003) Applied Partial Differential Equations, Oxford University Press.
    5. Billingham, J. and  King, A.C. (2000) Wave motion, Cambridge University Press.
    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: https://myuni.adelaide.edu.au/
  • 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 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.
    Workload

    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 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.

    1. Conservation of mass and momentum: Car traffic has waves; Conservation of fluid; Momentum PDE for ideal gases; The wave equation.
    2. Traffic flow and the method of characteristics: Non-uniqueness; Shocks.
    3. Separation of variables: Linearity empowers analysis; Separation of variables generates boundary value problems.
    4. Wonderful Sturm–Liouville boundary value problems: Self-adjoint operators form Sturm-Liouville problems; Eigenfunctions expand inhomogeneous solutions.
    5. Discretising 1D space: Lagrange’s theorem underpins the method of lines; Find equilibria; Numerical linearisation characterises solution dynamics; PDE-free patch dynamics.
    6. Modelling shallow water waves: Conservation derives the PDEs; Small amplitude waves; Compute seiches in 1D.
    7. 6 PDEs with at least three independent variables: Vibration of a rectangular membrane; The self-adjoint Sturm-Liouville nature of Helmholtz-like PDEs.
    8. 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.
    9. General wave dynamics: Water waves in finite depth; The dispersion relation of waves; Energy travels at the group velocity; Wave propagation in multi-dimensions.
    10. Shocking classification of PDEs: Change of variables transforms the PDE; Reduction to the hyperbolic canonical form; Elliptic and parabolic canonical form.
  • 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.
    Submission
    No information currently available.
    Course Grading

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

    M11 (Honours Mark Scheme)
    GradeGrade reflects following criteria for allocation of gradeReported on Official Transcript
    Fail A mark between 1-49 F
    Third Class A mark between 50-59 3
    Second Class Div B A mark between 60-69 2B
    Second Class Div A A mark between 70-79 2A
    First Class A mark between 80-100 1
    Result Pending An interim result RP
    Continuing Continuing CN

    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

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