Course Title | ## Ordinary Differential Equations | ||

Course Code | ## MH3110 | ||

Offered | Study Year 3, Semester 2 | ||

Course Coordinator | Fedor Duzhin (Dr) | fduzhin@ntu.edu.sg | 6513 7469 |

Pre-requisites | MH2100, CY1602 | ||

AU | 4 | ||

Contact hours | Tutorials: 39, Technology-enhanced Learning: 13, Lectures: 2 | ||

Approved for delivery from | AY 2018/19 semester 2 | ||

Last revised | 12 Feb 2019, 15:16 |

Building on Calculus, Linear Algebra, and Algorithms and Computing course sequences, this course aims to equip you with technical knowledge of the theory of ordinary differential questions and communication skills required to work as a team on modelling a real life phenomenon with a system of ordinary differential equations.

Upon successfully completing this course, you should be able to:

- Solve certain classes of ordinary differential equations (separable, first order linear, second order linear with constant coefficients, linear systems with constant coefficients).
- Apply basics of qualitative theory of ordinary differential equations, e.g., interpret phase portraits, derive existence and uniqueness of solutions, analyse stability of critical points of linear and nonlinear systems of differential equations without solving them.
- Present mathematical ideas the form of a report written in LaTeX at the appropriate level for the intended audience.
- Work in a team, i.e., effectively communicate on distribution of roles and work tasks and help other students with their tasks.
- Learn independently from research papers and online tutorials.
- Model a real life phenomenon with a system of ordinary differential equations.

First order ordinary differential equations, integrating factors, separable equations, linear homogeneous and non-homogeneous equations, variation of parameters, Principle of superposition

Second order linear differential equations, Wronskian, Abel's formula, variation of parameters.

First order linear systems, Wronskian, systems with constant coefficients, types of critical points at the origin, qualitative analysis of phase portraits.

First order nonlinear systems, initial value problem. Use of systems of ODE in modeling problems.

Component | Course ILOs tested | SPMS-MAS Graduate Attributes tested | Weighting | Team / Individual | Assessment Rubrics |
---|---|---|---|---|---|

Continuous Assessment | |||||

Technology-enhanced Learning | |||||

Activities | 1, 2, 3, 4 | 1. a, b, c, d | 20 | both | See Appendix for rubric |

Tutorials | |||||

Project | 2, 3, 4, 5, 6 | 1. a, b, c, d2. a3. a, b | 40 | both | See Appendix for rubric |

Mid-semester Quiz | |||||

Short Answer Questions 1 | 1, 2 | 1. b, c, d | 20 | individual | See Appendix for rubric |

Short Answer Questions 2 | 1, 2 | 1. b, c, d | 20 | individual | See Appendix for rubric |

Total | 100% |

These are the relevant SPMS-MAS Graduate Attributes.

## 1. Competence

a. Independently process and interpret mathematical theories and methodologies, and apply them to solve problems

b. Formulate mathematical statements precisely using rigorous mathematical language

c. Discover patterns by abstraction from examples

d. Use computer technology to solve problems, and to communicate mathematical ideas

## 2. Creativity

a. Critically assess the applicability of mathematical tools in the workplace

## 3. Communication

a. Present mathematics ideas logically and coherently at the appropriate level for the intended audience

b. Work in teams on complicated projects that require applications of mathematics, and communicate the results verbally and in written form

There will be 5 team application exercises for homework every week. You are supposed to type solutions in Overleaf and submit to the tutor. The tutor will immediately grade your work and provide feedback that you can use to improve and re-submit. You will have unlimited attempts. Note that this is a team activity and while one student will actually type a solution, the rest should verify it, help to phrase it properly, and read the tutor's comments.

• LaTeX tutorials: https://www.sharelatex.com/blog/latex-guides/beginners-tutorial.html

• Sample Homework with feedback: https://www.overleaf.com/read/ttcfptkdksvg

Tutorials (39 hours) | Note that the actual number of tutorials hours is 36. Below are learning and teaching approaches under tutorials: 1) Team-based learning 2) Online homework 3) Peer evaluation 4) Team project |

Technology-enhanced Learning (13 hours) | 5) Online lectures 6) Online forums |

Lectures (2 hours) | Note that the actual number of lecture hours is 2 (one introductory lecture in the first week). Below are learning and teaching approaches that qualify as lectures: |

Textbook: Boyce, William E., Richard C. DiPrima, and Douglas B. Meade. Elementary differential equations and boundary value problems. Vol. 9. New York: Wiley, 1992.

ISBN 0-471-31999-6References:

Amel’kin, Vladimir Vasil’evič. Differential equations in applications. Mir, 1990.

ISBN 5-03-005521-8Zhang, Wei-Bin. Differential equations, bifurcations, and chaos in economics. Vol. 68. World Scientific Publishing Company, 2005.

ISBN 981-256-333-4Shone, Ronald. Economic Dynamics: Phase diagrams and their economic application. Cambridge University Press, 2002.

ISBN-13 978-0-511-07676-3Brock, William A., and Anastasios G. Malliaris. Differential equations, stability and chaos in dynamic economics. No. 90A16 BROd. 1989.

ISBN 0-444-70500-7Cresswell, Clio. Mathematics and sex. Allen & Unwin, 2003.

ISBN 1 74114 159 1Gottman, John Mordechai. The mathematics of marriage: Dynamic nonlinear models. MIT Press, 2005.

ISBN 0-262-07226-2Clark, Colin W. The worldwide crisis in fisheries: economic models and human behavior. Cambridge University Press, 2006.

ISBN-13 978-0-511-33516-7Plus a number of journal papers and student reports from previous years.

(1) General

You are expected to complete all assigned pre-class readings and activities, attend all seminar classes punctually and take all scheduled assignments and tests by due dates. You are expected to take responsibility to follow up with course notes, assignments and course related announcements for seminar sessions they have missed. You are expected to participate in all discussions and activities.

(2) Absenteeism

TBL requires you to be in class to contribute to team work. In-class activities make up a significant portion of your course grade. Absence from class without a valid reason will affect your overall course grade. Valid reasons include falling sick supported by a medical certificate and participation in NTU’s approved activities supported by an excuse letter from the relevant bodies. There will be no make-up opportunities for in-class activities.

If you miss a seminar session, you must inform your team members and the course instructor via email (see course info on NTULearn) prior to the start of the class. If you miss I-RAT or T-RAT without a valid reason, you will earn nothing for that session of absence. For I-RAT and T-RAT scores, we will consider the average of all the sessions that you were present in. This method will take care of instances where you miss classes with valid reasons.

If you miss a midterm test, a make-up test will be arranged for you.

If you miss your team’s project presentation without a valid reason, your teammates will be in their right to reflect this fact in peer evaluation. If you miss the project presentation because of a valid reason, you must warn your teammates and the course instructor via email. In that case, no penalty will be imposed on you.

(3) Online Homework

You are encouraged to submit online homework typed in LaTeX. You have unlimited attempts and each time you submit a solution, your instructor will give you immediate feedback on logic, phrasing, and typesetting of your solution. The score will not be a part of official assessment but we will give “model answers” only to teams who have completely solved at last 3 out of 5 questions. Besides, teams with higher homework scores will get feedback on the draft of their final report earlier and hence will have more time to update their final report according to feedback.

Good academic work depends on honesty and ethical behaviour. The quality of your work as a student relies on adhering to the principles of academic integrity and to the NTU Honour Code, a set of values shared by the whole university community. Truth, Trust and Justice are at the core of NTU’s shared values.

As a student, it is important that you recognize your responsibilities in understanding and applying the principles of academic integrity in all the work you do at NTU. Not knowing what is involved in maintaining academic integrity does not excuse academic dishonesty. You need to actively equip yourself with strategies to avoid all forms of academic dishonesty, including plagiarism, academic fraud, collusion and cheating. If you are uncertain of the definitions of any of these terms, you should go to the Academic Integrity website for more information. Consult your instructor(s) if you need any clarification about the requirements of academic integrity in the course.

Instructor | Office Location | Phone | |
---|---|---|---|

Fedor Duzhin (Dr) | SPMS-MAS-05-23 | 6513 7469 | fduzhin@ntu.edu.sg |

Week | Topic | Course ILO | Readings/ Activities |
---|---|---|---|

1 | Concept of ODE. Direction fields. | 2 | Introductory lecture, textbook chapter 1 |

2 | Explicitly solvable 1st order ODE: separable and linear equations. | 1, 2, 3, 4, 5 | TBL on lecture 1, textbook sections 2.1, 2.2, research (browsing the library) |

3 | Modelling with 1st order ODE | 1, 2, 3, 4, 5 | TBL on lecture 2, textbook sections 2.3, 2.5, research (browsing the library) |

4 | Existence and uniqueness Theorem. | 1, 2, 3, 4, 5 | TBL on lecture 3, textbook sections 2.4, 2.7, 2.8, research (browsing the library) |

5 | 2nd order linear ODE | 1, 2, 3, 4, 5 | TBL on lecture 4, textbook sections 3.2, 3.3 research (browsing the library), midterm test 1 |

6 | 2nd order linear ODE: explicit solution | 1, 2, 3, 4, 5, 6 | TBL on lecture 5, textbook sections 3.1, 3.4, 3.5 research (deciding the topic) |

7 | Modelling with 2nd order ODE | 1, 2, 3, 4, 5, 6 | TBL on lecture 6, textbook sections 3.7, 3.8, 3.9 research (data collection) |

8 | Higher order linear ODE and Laplace transform - just basics. | 1, 2, 3, 4, 5, 6 | TBL on lecture 7, textbook sections 4.1, 4.2, 6.1, 6.2, modelling |

9 | Systems of 1st order ODE | 1, 2, 3, 4 | TBL on lecture 8, textbook sections 7.1, 7.4, 7.5, midterm test 2 |

10 | Systems of 1st order ODE | 1, 2, 3, 4, 5, 6 | TBL on lecture 9, textbook sections 7.6, 9.1, computer simulations |

11 | Nonlinear equations and stability | 1, 2, 3, 4, 5, 6 | TBL on lecture 10, textbook sections 7.7, 9.2, 9.3, 9.4, 9.5, fitting the mode into data |

Unsatisfactory | Poor | Average | Good | Excellent | ||

MAX | 0 | 0.25 | 0.5 | 0.75 | 1 | |

Literature review | 10 | No literature review at all. | Brief overview of the previous work on modelling the problem with ODE. | Short literature review covering 2 out of the following 4 areas: previous work on modelling the problem with ODE, alternative ways to model the problem mathematically, overview of the non-mathematical aspect of the problem, advanced theory of ODE. | Comprehensive literature review covering 3 out of the following 4 areas: previous work on modelling the problem with ODE, alternative ways to model the problem mathematically, overview of the non-mathematical aspect of the problem, advanced theory of ODE. | Comprehensive literature review covering the following 4 areas: previous work on modelling the problem with ODE, alternative ways to model the problem mathematically, overview of the non-mathematical aspect of the problem, advanced theory of ODE. |

Data collection | 10 | No explanation of what the data come from or what they look like at all. | The data are invented out of thin air and not properly visualized. | The data are invented out of thin air and properly visualized. | The data are realistic, but visualized with silly little errors. | The data are realistic and properly visualized. |

Modelling | 10 | The model is one differential equation that can be solved explicitly. | The model may be a system of 2 or 3 linear or non-autonomous ordinary differential equations; the system of more than 3 differential equations; or 2 or 3 independent differential equations. There is no clear logic behind the model. | The model is a system of 2 or 3 nonlinear autonomous differential equations but it is not clearly justified or inconsistent with the data. | The model is a system of 2 or 3 nonlinear autonomous differential equations established in a literature (e.g., SIR model, Lotka-Volterra model, Rinaldi's model etc). Applicability of the model to the problem of the team's choice is clearly justified. | The model is a system of 2 or 3 nonlinear autonomous differential equations, invented by the team as a modification of some established model. The model is properly justified. |

Numeric simulation, i.e., solving the model on a computer and plotting solutions | 10 | Not done at all | Numerical simulations are performed in an online phase portrait plotter with silly little errors. | Numerical simulations are performed in an online phase portrait plotter. | Numerical simulations are done with silly little errors. | Numerical simulations are performed in MATLAB / Octave / R / Python / Mathematica, relevant plots are constructed, and a real-life interpretation has been formulated. |

Model fitting | 10 | Fitting into data is not done at all, i.e., some random values of parameters are taken. | Fitting parameters of the model into data is done by pure speculation and the explanation does not make much sense. | Fitting parameters of the model into data is done by pure speculation (such as in Rinaldi's paper). | Fitting parameters into data is done by PSO but with silly little errors. | Fitting parameters of the model into data is done by an optimization algorithm, such as Particle Swarm Optimization |

Basic math - finding critical points and analysing their stability (material within the course syllabus) | 10 | Not done at all | Analysis is performed with serious mathematical errors and no real-life interpretation of the findings is given. | Analysis is performed with serious mathematical errors or no real-life interpretation of the findings is given. | Analysis is done with silly little errors, i.e., only one point is analyzed if there are more than one feasible equilibrium point. Real-life interpretations are given. | All the equilibrium solutions are found, eigenvalues of the Jacobi matrix calculated, stability analyzed, and real-life interpretations are given. |

Advanced math (learning material beyond the course syllabus) | 10 | Not done at all or done incorrectly. | The team has learned some material that is not in the course syllabus, but the progress was minimal. | The team has stepped beyond the course syllabus by reading more chapters of the textbook and applying results to their project. | The team has stepped beyond the course syllabus by reading research literature on differential equations and applying results to their project. | The team has gone way beyond the course syllabus by reading research literature on differential equations and applying results to their project. The analysis is comprehensive/ |

Presentation | 10 | The team was not able to prepare the presentation on class of week 13. | The number of slides exceeds 8. The presentation is very messy and the speaker doesn't seem to be familiar with the scope of the project. | Not more than 8 slides that may be prepared on Power Point / Google drive. The presentation may be very messy and by far exceed 10 minutes. Some team members may not be familiar with the structure of the project. | Not more than 8 slides professionally typed in LaTeX. The presentation is very clear but may last slightly more than 10 minutes. Students may not be familiar with their teammates' contribution. | Not more than 8 slides professionally typed in LaTeX. The presentation is very clear and lasts not more than 10 minutes. All team members are able to answer questions about the project |

Final report | 10 | There is no report at all, i.e., it is either not submitted or submitted as commented MATLAB code rather than a structured published report. | The report is not written in full English sentences and is hard to follow. There may be serious mistakes in style, grammar, and structure that make understanding the report a challenging task. There may not be an abstract or the reference list and the length of the report may by far exceed 10 pages. | The report is mostly written in full English sentences, but is not very easy to follow due to mistakes in style or grammar. The structure of the report is not completely as required, i.e., the abstract may be in the end or the number of pages exceeds 10. The report may be produced in MS Word or MATLAB. Sources may not be properly cited. | The report is written in full English sentences, is properly structured and is easy to follow. The report has been prepared in LaTeX. The length of the report does not exceed 10 pages. There are mistakes in style or grammar that don't make the report too hard to understand, such as small errors in LaTeX equations, illustrations in a non-vector format or occasional grammar errors. The abstract may exceed 200 words or may not be informative. | The report is written in full English sentences, is properly structured and is easy to follow. The report has been prepared in LaTeX. The length of the report does not exceed 10 pages. All the sources are appropriately cited. There are no mistakes in style or grammar. |

This is a traditional math exam consisting of a mixture of multiple choice and mini-essay questions. It is assessed by a marking scheme rather than a rubric.

Below is an example of a question and a marking scheme:

**Question (15 points)**. Solve the following initial value problem:

**Marking scheme**: 5 points for separating the variables and integrating, 5 points for finding *y* from the resulting expression, and 5 points for finding the constant and solving the initial value problem.

This is a traditional math exam consisting of a mixture of multiple choice and mini-essay questions. It is assessed by a marking scheme rather than a rubric.

Below is an example of a question and a marking scheme:

**Question (15 points)**. Can the function

be a solution of an initial value problem

with coefficients continuous on some interval containing 0?

**Marking scheme**: 15 points for a fully correct answer, 10 for a correct explanation with missing details, and 5 for mentioning a relevant concept.