Applied Mathematical Analysis (MATH7100)

Course overview

ODE's - Systems: variation of constants, fundamental matrix. Laplace transform, transform for systems, transfer function. Stability, asymptotic stability; phase-lane analysis. PDE's - Fourier series. Wave, heat, Laplace's equations. Simple maximum & uniqueness principles. Separation of variables in rectangular & polar coordinates.

This course will cover the following 4 topics.

• Topic 1. Systems of Ordinary Differential Equations (Kreyszig Chapter 4). Solutions to Homogeneous Linear Systems of ODE's with constant coefficients using matrices. The Phase Plane for 2-Dimensional linear and some nonlinear systems. Critical Points, linearization and the stability properties of critical points. Nonhomogeneous Linear Systems.
• Topic 2. Laplace Transforms (Kreyszig Chapter 6).The Laplace Transform and its Inverse Transform. Linearity, Shifting, Convolution and the use of Partial Fractions. Transforms of Derivatives and Integrals. Use of Laplace Transforms to solve Linear Differential Equations including those involving discontinuous functions such as the Step Function and the Dirac Delta function.
• Topic 3. Fourier Series (Brown and Churchill Chapters 1 and 2, Kreyszig Chapter 11). The idea of Fourier analysis, and of orthogonal periodic functions. Formulas for Fourier coefficients. When a Fourier expansion works. Even and odd functions and their expansions. Half-range sine and cosine series. Differentiation of Fourier series. Application of Fourier series to a forced oscillator.
• Topic 4. Partial Differential Equations (Brown and Churchill Chapters 3 to 6, Kreyszig Chapter 12). Concepts of scalar and vector fields. The operator del; grad, div and curl. Flux of a vector field. Idea of a flux integral. Gauss' divergence theorem. Conservation laws. The Laplacian operator. The heat, wave and Laplace's equation in 1, 2, and 3 dimensions. Nature of solutions of PDEs. The superposition principle and method of separation of variables. General solution of wave equation and d'Alembert's solution of the IVP. Examples. Fourier's method for wave equation. Green's function for the 1-d heat equation. Solution of IVP on whole line. Example. The error function and other examples. Heat conduction and diffusion. Temperature waves in the Earth. Kelvin's estimation of the Earth's age. Fourier's method for 1-d heat equation with source term. Short-cut when source is time independent. Revisit 1-d wave equation with forcing. Fourier's method and Laplace's equation in a 2-dimensional rectangular region. Continuation: other BCs. Idea of conjugate harmonic functions. Heat stream function. Point source for Laplace's equation in 2-d. Images for point sources/sinks. Other applications of Laplace's equation: Fluid flow, electrostatics. Potentials and stream functions.

Course requirements

Assumed background

This course will build on foundations that you have obtained in earlier courses: differential and integral calculus, linear algebra, vector analysis,ﾠ and especially differential equations.ﾠ It is your responsibility to fill in any gaps in the assumed knowledge.ﾠ You may need to undertake background reading to understand the lecture material.ﾠﾠ Chaptersﾠ1, 2, 7, 8, 9ﾠand 10 of the set textﾠ (E. Kreyszig, Advanced Engineering Mathematicsﾠ, 9th Edition) cover most of the background material.ﾠﾠ

IMPORTANT: For any administrative questions related to the course (such as switching tutorial/practical groups,ﾠtextbook requirements, etc.), please send an email to math2100@uq.edu.au (this email will be common for MATH2100, MATH2010, MATH2011 and MATH7100). The course coordinator and lecturers will be available to discuss academic questions during consultation hours.

Incompatible

You can't enrol in this course if you've already completed the following:

MATH2100 (co-taught), MATH2010, MATH2011

Course contact

Course staff

Timetable

The timetable for this course is available on the UQ Public Timetable.

Additional timetable information

All classes will be conducted on campus – consult your personal timetable for times and locations. Students are expected to attend these sessions in person unless they have a valid reason for being unable to attend (such as illness).

The lectures are taught during weeks 1-13:ﾠThe first part of the course on ODEs is taught during weeks 1-6. The second part on PDEs is taught during weeks 7-13. Tutorials are taught in weeks 2-13.

Aims and outcomes

This course is built around four mathematical concepts: Systems of ordinary differential equations (ODEs); Laplace transforms; Fourier series; and Partial Differential Equations (PDEs).

Systems of ODEs generalise the idea of an ODE, as you have seen it covered in MATH1052, for example. Now we have several unknown functions of a single independent variable, say the time t, and we have ODEs linking the unknowns together. We deal mostly with systems of two coupled first-order equations to see the sorts of things that can happen. The notion of the phase-plane is introduced, where the ODEs determine the trajectory of a representative point for the system. Basic notions of stability and instability of equilibrium (critical) points of the system are explored. Illustrative applications are described, such as predator-prey systems, an epidemic model, electrical and mechanical oscillators.

The Laplace Transform is a tool still widely used to deal with linear ODEs and PDEs, especially in engineering and biological applications. We introduce the basic concepts, including applications to simple systems of ODEs.

Fourier series were introduced as a tool for solving linear PDEs, but are important in their own right, and contain the germ of the ideas underlying the most advanced forms of linear analysis. The essence of the idea is to expand an arbitrary periodic signal in terms of harmonics. Again, we introduce the basic ideas with illustrative examples rather than detailed theory.

The final topic in the course is an introduction to PDEs. Here we deal with functions (fields) depending on several variables such as x, y, z and t. Many important applications are described by PDEs, and we look at some of these to introduce the three main types of linear PDEs in two independent variables: heat conduction and molecular diffusion, waves on a stretched string, and steady temperature distributions in 2-dimensions. In particular,

Learning outcomes

After successfully completing this course you should be able to:

Assessment

Assessment summary

A hurdle is an assessment requirement that must be satisfied in order to receive a specific grade for the course. Check the assessment details for more information about hurdle requirements.

Assessment details

4 Assignments

Mode

Written

Category

Tutorial/ Problem Set

Weight

40%

Due date

Assignment 1: 16/08/2024 4:15 pm

Assignment 2: 6/09/2024 4:15 pm

Assignment 3: 4/10/2024 4:15 pm

Assignment 4: 25/10/2024 4:15 pm

Learning outcomes

L01, L02, L03, L04, L05, L06, L07, L08

Task description

Submission guidelines

All assessment items should be submitted electronically through Blackboard.

Deferral or extension

You may be able to apply for an extension.

The maximum extension allowed is 7 days. Extensions are given in multiples of 24 hours.

See ADDITIONAL ASSESSMENT INFORMATION for the extension and deferred examination information relating to this assessment item.

Late submission

A penalty of 10% of the maximum possible mark will be deducted per 24 hours from time submission is due for up to 7 days. After 7 days, you will receive a mark of 0.

You are required to submit assessable items on time. If you fail to meet the submission deadline for any assessment item, then the listed penalty will be deducted per day for up to 7 calendar days, at which point any submission will not receive any marks unless an extension has been approved. Each 24-hour block is recorded from the time the submission is due.

Final Examination

• Hurdle

Mode

Written

Category

Examination

Weight

60%

Due date

End of Semester Exam Period

2/11/2024 - 16/11/2024

Learning outcomes

L01, L02, L03, L04, L05, L06, L07, L08

Task description

Hurdle requirements

Exam details

Submission guidelines

Deferral or extension

You may be able to defer this exam.

See ADDITIONAL ASSESSMENT INFORMATION for the extension and deferred examination information relating to this assessment item.

Course grading

Full criteria for each grade is available in the Assessment Procedure.

Supplementary assessment

Supplementary assessment is available for this course.

Should you fail a course with a grade of 3, you may be eligible for supplementary assessment. Refer to my.UQ for information on supplementary assessment and how to apply. 

Supplementary assessment provides an additional opportunity to demonstrate you have achieved all the required learning outcomes for a course.  

If you apply and are granted supplementary assessment, the type of supplementary assessment set will consider which learning outcome(s) have not been met.  

Supplementary assessment in this course will be a 2-hour examination similar in style to the end-of-semester examination. To receive a passing grade of 3S4, you must obtain a mark of 50% or more on the supplementary assessment. 

Additional assessment information

Important note

Tutors will record your assignment marks on Blackboard. It is your responsibility to check that the mark is correctly recorded. No discussion about incorrect or missing assignment marks will be entertained 21 calendar days after marks are released.

Artificial Intelligence

The assessment tasks in this course evaluate students’ abilities, skills and knowledge without the aid of Artificial Intelligence (AI). Students are advised that the use of AI technologies to develop responses is strictly prohibited and may constitute misconduct under the Student Code of Conduct.

Applications for Extensions to Assessment Due Dates

Extension requests are submitted online via my.UQ – applying for an extension. Extension requests received in any other way will not be approved. Additional details associated with extension requests, including acceptable and unacceptable reasons, may be found at my.UQ.

Please note:

• Requests for an extension to an assessment due date must be submitted through your my.UQ portal and you must provide documentation of your circumstances, as soon as it becomes evident that an extension is needed. Your application must be submitted on or before the assessment item's due date and time.
• Applications for extension can take time to be processed so you should continue to work on your assessment item while awaiting a decision. We recommend that you submit any completed work by the due date, and this will be marked if your application is not approved. Should your application be approved, then you will be able to resubmit by the agreed revised due date.
• If an extension is approved, you will be notified via your my.UQ portal and the new date and time for submission provided. It is important that you check the revised date as it may differ from the date that you requested.
• If the basis of the application is a medical condition, applications should be accompanied by a medical certificate dated prior to the assignment due date. If you are unable to provide documentation to support your application by the due date and time you must still submit your application on time and attach a written statement (Word document) outlining why you cannot provide the documentation. You must then upload the documentation to the portal within 24 hours.
• If an extension is being sought on the basis of exceptional circumstances, it must be accompanied by supporting documentation (eg. Statutory declaration).
• For extensions based on a SAP you may be granted a maximum of 7 days (if no earlier maximum date applies). See the Extension or Deferral availability section of each assessment for details. Your SAP is all that is required as documentation to support your application. However, additional extension requests for the assessment item will require the submission of additional supporting documentation e.g., a medical certificate. All extension requests must be received by the assessment due date and time.
• Students may be asked to submit evidence of work completed to date. Lack of adequate progress on your assessment item may result in an extension being denied.
• If you have been ill or unable to attend class for more than 14 days, you are advised to carefully consider whether you are capable of successfully completing your courses this semester. You might be eligible to withdraw without academic penalty - seek advice from the Faculty that administers your program.
• There are no provisions for exemption from an assessment item within UQ rules. If you are unable to submit an assessment piece then, under special circumstances, you may be granted an exemption, but may be required to submit alternative assessment to ensure all learning outcomes are met.

Applications to defer an exam

In certain circumstances you can apply to take a deferred examination for in-semester and end-of-semester exams. You'll need to demonstrate through supporting documentation how unavoidable circumstances prevented you from sitting your exam. If you can’t, you can apply for a one-off discretionary deferred exam.

Deferred Exam requests are submitted online via mySi-net. Requests received in any other way will not be approved. Additional details associated with deferred examinations, including acceptable and unacceptable reasons may be found at my.UQ.

Please note:

• Applications can be submitted no later than 5 calendar days after the date of the original exam.
• There are no provisions to defer a deferred exam. You need to be available to sit your deferred examination.
• Your deferred examination request(s) must have a status of "submitted" in mySI-net to be assessed.
• All applications for deferred in-semester examinations are assessed by the relevant school. Applications for deferred end-of-semester examinations are assessed by the Academic Services Division.
• You’ll receive an email to your student email account when the status of your application is updated.
• If you have a medical condition, mental health condition or disability and require alternative arrangements for your deferred exam you’ll need to complete the online alternative exam arrangements through my.UQ. This is in addition to your deferred examinations request. You need to submit this request on the same day as your request for a deferred exam or supplementary assessment. Contact Student Services if you need assistance completing your alternative exam arrangements request.ﾠ

Learning resources

You'll need the following resources to successfully complete the course. We've indicated below if you need a personal copy of the reading materials or your own item.

Library resources

Find the required and recommended resources for this course on the UQ Library website.

Other course materials

Additional learning resources information

A different workbook will be used for each part of the course. They will be both available from blackboard.

Learning activities

The learning activities for this course are outlined below. Learn more about the learning outcomes that apply to this course.

Policies and procedures

University policies and procedures apply to all aspects of student life. As a UQ student, you must comply with University-wide and program-specific requirements, including the:

• Student Code of Conduct Policy
• Student Integrity and Misconduct Policy and Procedure
• Assessment Procedure
• Examinations Procedure
• Reasonable Adjustments - Students Policy and Procedure

Learn more about UQ policies on my.UQ and the Policy and Procedure Library.