APP MTH 7107 - Partial Differential Equations and Waves
North Terrace Campus - Semester 2 - 2020
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: Michael Chen
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) 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 Career and leadership readiness
- 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
Required ResourcesAccess to the internet.
- Agarwal, R. P. & O'Regan, D. (2009), Ordinary and Partial Differential Equations With Special Functions, Fourier Series, and Boundary Value Problems, Springer.
- Billingham, J. and King, A.C. (2000) Wave motion, CUP.
- Haberman, R. (1987), Elementary applied partial differential equations: with Fourier series and boundary value problems, 2nd edn, Prentice-Hall.
- 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
- Kreyszig, E. (2011), Advanced engineering mathematics, 10th edn, Wiley.
- Roberts, A. J. (1994), A one-dimensional introduction to continuum mechanics, World Sci.
- Roberts, A. J. & Kevrekidis, I. G. (2007), 'General tooth boundary conditions for equation free modelling', SIAM J. Scientific Computing 29(4), 1495--1510.
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/webapps/login/
Learning & Teaching Activities
Learning & Teaching ModesThis 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.
Activity Quantity Workload Hours Lectures/tutorials 36 100 Assignments/assessment 7 56 Total 156
Learning Activities SummaryLecture 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.
- 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
- 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
- Discretise 1D space: Lagrange’s theorem underpins the method of lines; Find equilibria; Numerical linearisation characterises solution dynamics; PDE-free patch dynamics
- Model shallow water waves: Conservation derives the PDEs; Small amplitude waves; Compute seiches in 1D
- 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–Nicolson schemes are reasonably stable and accurate; Invoke sparse matrices for implicit schemes; Crank–Nicolson discretises wave systems; Second order PDEs in 2D
- General wave dynamics: Water waves in finite depth; 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; Traffic flow and the method of characteristics; Loud uni-directional sound
The University's policy on Assessment for Coursework Programs is based on the following four principles:
- Assessment must encourage and reinforce learning.
- Assessment must enable robust and fair judgements about student performance.
- 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 Weighting Objective assessed Assignments 30% all Exam 70% all
Assessment Related RequirementsAn aggregate score of at least 50% is required to pass the course.
Assessment item Distributed Due date Weighting 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%
- 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.
- Late assignments will not be accepted without a medical certificate and a request prior to the due date.
- 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) 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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