Course Outline
Course Overview
Applied Mathematics has arisen out of the need to describe aspects of the real world in such a way that the behaviour of the real world can be explained and predicted. Mathematical Modelling is the key to production of mathematical equations which describe continuous phenomena and systems. The branch of differential equations are used as a basis to understand and describe these phenomena and systems.
This course is focused on building a foundation in the skills required to build mathematical descriptions of simple real world situations, with the emphasis on systems requiring differential equations to describe them. It introduces multiple mathematical methods needed for solving differential equations, as well as techniques and thought processes for model building. Problem solving skills are enhanced in this course, developing the intellectual self-reliance of the modern student to model scenarios mathematically and find meaningful solutions to problems.
Staff Members
Lecturer: Mr. Abdul Hamid Carrim
Email: AbdulHamid.Carrim@wits.ac.za
Office: T.W. Kambule Mathematical Sciences Building (MSB), Room Number 118
YouTube: Abdul Hamid Carrim
Course Coordinator: Dr Rahab Kgatle-Maseko (Rahab.Kgatle@wits.ac.za)
A Blended Approach to Learning
For this course, we will adopt a blended approach to learning and teaching - this means that there will be increased in-person activities with a mix of complementary online activities. This online book serves as the primary mechanism to drive this approach, and serves as a complement to the planned in-person lectures.
The book will be updated regularly with new chapters including text, videos, and other resources. There will be several quizzes and tests (see the schedule set out below). The quizzes will be made available via Moodle and all submissions should be done on Moodle, and tests will take place in-person as scheduled preliminarily below.
I believe that a very important aspect of learning is interacting with peers/tutors/lecturers, so I encourage you to interact regularly in-person and on Moodle through the various channels and forums.
Academic integrity is of absolute importance. Communication during quizzes and tests, sharing of answers, and all forms of plagiarism are taken very seriously by the University and will result in failing the course (FCM) and/or being reported to the Wits Legal Office. I will be monitoring Moodle logs closely and will be checking all submissions for cheating and plagiarism.
Learning Management System
We will use Moodle exclusively for this course. We will be using Moodle for the general construction and delivery of the course.
It is your responsibility to keep up to date with all changes made on Moodle. Log in with your student number and usual Wits password. You should already be enrolled in the course.
All course-related announcements for Modelling will be done through Moodle.
Timetable
In the timetable, there is a double lecture for Mathematical Methods and Modelling each week, as well as a single tutorial that will alternate every other week with Scientific Computing.
- Wednesdays: 10:15-12:00
- Thursdays: 12:30-13:15
This course will be presented with a blend of a synchronous and asynchronous manner. Prerecorded videos, readings and exercises will be made available to you earlier in the week to complement the in-person mode of teaching. It is vital that you stay up to date with the weekly activities throughout the semester.
The weekly lessons including pre-recorded video lectures and other course-related material will posted every week on a Friday on my Youtube channel and embedded in the online book. During these videos, we will be concentrating on introducing and elucidating concepts, covering course content and material. These concepts will then be expounded upon and consolidated during the live in-person lectures on a Wednesday.
You will be expected to pre-read the relevant material made available to you before engaging with any of the lecture material for that week. You will need to complete all assigned post-class tasks before the next lesson, or by the specified submission date. You will also be required to have attempted the tutorial questions before attending the tutorial, and engage in all tutorial and lab-related activities.
The main synchronous aspect of the course will be the weekly in-person lecture sessions which will take place on-campus on Wednesdays from 10:15-12:00.
Tutorials will begin in Week 2 of the course. You will be assigned a tut group and every second week you must meet your tutor for a live in-person tutorial. While tutors will be able to answer any theory/practical questions that you may have during this session, the tuts are meant to be interactive. These tutorial sessions are compulsory and will contribute to your Satisfactory Performance (see the Course Information Booklet).
Communication
All official communication will be posted to the announcements forum on Moodle. Moodle will send you email digests daily, but it is still your responsibility to ensure you’re receiving these emails. You can change your email preferences on Moodle to receive a single email per post if you prefer that.
Please use the Q&A Forum on Moodle to ask and answer questions. This means that the tutors and I can answer these questions publicly and helps avoid duplication. Please do not email general course-related questions to me unless it only relates to you personally or is sensitive in some way. I am always happy to help where I can, but avoiding duplication is really important in large classes like this one. I hope that you will use this platform to ask and answer questions, engage with peers, post suggestions, and share resources.
In general, please try to use public channels to speak with me and the tutors unless it is about anything that you should not be posting publicly - it is useful for others to be able to see the answers to your questions. I will always respond to Moodle queries before emails. If you have queries directly for me, I would prefer you use the Q&A Forum on Moodle where these persistent answers to course-related questions could benefit others as well.
Consultations
Due to the number of students in this course, it is really important to book time with me if you would like a live one-on-one consultation. You can book 15 minute time slots directly into my calendar using https://calendly.com/abdulhamidcarrim/mmm. I have times available on Mondays between 12:00-14:00, and additional slots may be added later as I clear my calendar. You can book up to 30 days in advance and calendly will send you a calendar invite with an MS Teams link in it. You will need to answer two questions when setting up a meeting. You also need to come to the consult with evidence of attempted work.
If you have bandwidth or data issues, you can tell MS Teams not to receive video. If you don’t have any of these issues, feel free to keep your video on as it is nice to chat face-to-face and still get to see some of my students!
Course Background and Purpose
Applied Mathematics has arisen out of the need to describe aspects of the real world in such a way that the behaviour of the real world can be explained and predicted. Mathematical Modelling is the key to production of mathematical equations which describe phenomena and systems. The branches of differential and difference equations are used as a basis to understand and describe these phenomena and systems.
This course is focused on building a foundation in the skills required to build mathematical descriptions of simple real world situations, with the emphasis on systems requiring differential equations or difference equations to describe them.
Problem solving skills are enhanced in this course, developing the intellectual self-reliance of the modern student to model scenarios mathematically and find meaningful solutions to problems. Collaboration on tutorial problems is also encouraged in order to build skills in communication of the concepts and methods of model building.
Course Content, Goals, and Outcomes
The development of concepts and skills is progressive throughout the course, and a comprehensive grasp of early sections will enhance the students’ ability to cope with later sections. At the end of each of the chapters, the student will be able to:
Chapter 1: An Introduction to Modelling and Differential Equations
- Understand the difference between a model and a mathematical model
- Explain the modelling process
- Understand the mathematical concept of change in terms of a differential equation
Chapter 2: Differential Equations
- Classify a given differential equation by its order, linearity, homogeneity, and coefficients
- Understand the concept of a solution to a differential equation
- Solve a given ordinary differential equation equation using direct integration
- Solve a given ordinary differential equation equation using separation of variables
- Solve a given ordinary differential equation equation using the method of undetermined coefficients
- Identify the appropriate applicable method(s) to solve the given differential equation based on its classification
- Obtain specific solutions to a given ordinary differential equation by using initial or boundary conditions
Chapter 3: Continuous Mathematical Models
- Express continuous dynamical rules as ordinary differential equations
- Write down a system of ordinary differential equations which mathematically capture the dynamics of a system described in words
- Write down the initial and/or boundary conditions for a problem given a description of the problem
- Use the resulting solution to predict behavior of the modelled system
Tentative Class Schedule
Block 3
| Week | Topic |
|---|---|
| 1 (15 - 19 July) | Orientation, Course Outline |
| 2 (22 - 26 July) | Introduction to Modelling and Differential Equations |
| 3 (29 July - 2 August) | Ordinary Differential Equations and Classification |
| 4 (5 - 9 August) | Solutions to Ordinary Differential Equations and Direct Integration |
| 5 (12 - 16 August) | Separation of Variables and Initial Value Problems |
| 6 (19 - 23 August) | Undetermined Coefficients - Homogeneous Linear Equations with Constant Coefficients |
| 7 (26 - 30 August) | Undetermined Coefficients - Superposition Approach |
Block 4
| Week | Topic |
|---|---|
| 8 (9 - 13 September) | Test 1 |
| 9 (16 - 20 September) | The Formal Model Construction Process & First-Order Models: Growth Models |
| 10 (23 - 27 September) | First-Order Models: Decay Models |
| 11 (30 September - 4 October) | Newton’s Law of Cooling/Warming |
| 12 (7 - 11 October) | Higher Order Models |
| 13 (14 - 18 October) | Test 2 |
Assessment
There will be a mix of in-person formal assessments as well as online formative and continuous assessment tasks. All online assessments will take place on the Moodle platform.
Assessments will be set that will require the student to show a thorough knowledge of applying the theoretical material, and to show a conceptual and deep understanding of the course. The relevant topics to be covered in the assessments will be made known to students in a timely manner.
There will be weekly formative assessments in the form of Did You Get It? questions (DYGIT). These do not contribute towards your course mark, but are there for you to check your current conceptions and self-ascertain your understanding of the course.
The course will be assessed through two in-person timed tests (\(60 - 90\) minutes), multiple quizzes (\(30 - 60\) minutes), an assignment, and a final examination (\(120\) minutes).
The structure and procedure of class tests shall be communicated to you well in advance. Both tests are compulsory and contains content that will be examined in the final exam. The table below shows a full tentative and provisional schedule of all assessment tasks scheduled for this course. The assessment will be scheduled within the specified week. Assessment open and closing times will be communicated to you in advance of the assessment.
Examinations will take place in-person during the November examination period. You will have to consult the examination timetable made available online for the relevant information.
Assessment Schedule
| Assessment | Week | Open/Scheduled Date | Submission Date | Venue |
|---|---|---|---|---|
| Quiz 1 | 4 | 6th August | 9th August | Online |
| Quiz 2 | 6 | 20th August | 23rd August | Online |
| Test 1 | 8 | 11th September | — | RSEH* |
| Quiz 3 | 10 | 24th September | 27th September | Online |
| Assignment | 11 | — | 4th October | Submission |
| Quiz 4 | 12 | 8th October | 11th October | Online |
| Test 2 | 13 | 16th October | — | Flower Hall** |
*RSEH referes to Robert Sobukwe Exams Hall which is located on East Campus. The test venue is on the second floor.
**Flower Hall is located on West Campus. The test will take place on both the ground and first floor.
- The structure and format of continuous assessments is such that it will comprise of both problem and conceptual questions, and may comprise of Multiple Choice Questions (MCQ’s), fill in the blanks, matching, short numerical answers, long calculation questions, etc.
- The tests will take place in-person and on-campus.
- The quizzes will take place and be submitted online on Moodle.
- The examination is expected to take place in-person and on-campus during the November examination period. The format of the examination shall be announced before the start of the examination period. You will have to consult the examination timetable made available online for the relevant information.
Calculation of the final mark
The final course mark for APPM1026A/APPM1027A will comprise of an average of your first semester mark and your second semester mark.
Your final second semester course mark will be calculated as follows. It will comprise of your year mark comprising of all continuous assessment tasks and your final November summative assessment mark. The year mark component will be derived from continuous assessment tasks. The final exam mark will be derived from the November summative assessment task.
The final course mark breakdown is as follows:
| Item | Weight |
|---|---|
| Tests | 30% |
| Assignment | 10% |
| Quizzes | 20% |
| November Assessment | 40% |
Textbooks and Recommended Reading
There is no prescribed textbook for this course. The handed out material will cover all required coursework. However, you are encouraged to make use of the Internet for additional resources. The following books are recommended for further reading to supplement lectures and notes:
- Zill, D.G. and Cullen, M.R.: Differential Equations with Boundary-Value Problems
- Banasiak, J.: Modelling with Difference and Differential Equations (UKZN)
Where relevant, I will post links to other resources as well. I encourage you to use Google and YouTube generally. If you find any other good resources, please share these resources with others in the class as well using the Forums.