**Course Code: MATH 6192**

Course Title:

Course Title:

**ADVANCED MATHEMATICAL MODELLING**

Level: Graduate

Level: Graduate

**Course**

Type: Elective

Type: Elective

**No. of Credits: 4**

Prerequisites: None

Prerequisites: None

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**Course Rationale**

Mathematical models founded on the basic laws of science are often used to aid in the understanding and prediction of natural phenomena. It is also common practice to create mathematical models for the design and testing of industrial processes. Students taking this course will be given the opportunity to utilize applied mathematics techniques for real-world problems in engineering, industry and integrated science. This is crucial for students working in applied mathematics, as simply learning new techniques for solving problems does not adequately prepare graduate students for research careers. The course will therefore serve as a solid basis for students interested in working on final MSc projects in the field of applied mathematics. It will also be of interest for MPhil and PhD students from the department of Mathematics and Computer Science, as well as for research students of applied science and engineering.

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**Course Description**

In this course, students will be introduced to the process of developing mathematical models as a means for solving real-world problems. Several modelling situations will be addressed. The mathematical fundamentals will be discussed, but with continual reference to their use in finding the solutions to problems. In particular, mathematical model building techniques based on dimensional analysis, perturbation theory and variational principles will be covered. Students taking this course must have prior knowledge of these techniques.

The coursework component (40% research project) of the course is specially designed to achieve this goal. Ample time will be set aside for project consultation during the course of the semester. To complement the theory outlined, discussions of recently published mathematical models and their limitations will be included during lecture hours.

Examples from solid and fluid mechanics, combustion, polymerization, diffusion phenomena and chemically reactive systems will be included. Computer simulation of these models will also be incorporated.

**Learning Outcomes**

On successful completion of this course, students will be able to:

- solve problems based on dimensional analysis, perturbation theory and variational principles
- construct and design numerical solutions and computer simulations to illustrate complicated model behaviours

- apply numerical, perturbative, optimization and stability analysis methods to real-world applications in the natural sciences
- utilize applied mathematics techniques to find solution to problems in engineering, industry and integrated science
- start their own research projects in the field of advanced mathematical modelling.

**Content (48 hours)**

- Models from Newton’s laws of motion: Planetary motion, energy conservation laws, resonance phenomena, surface area and minimal energy configurations (6 hours).
- Lagrangian/Eulerian equations of motion: trajectories of particles, caustics (6 hours).
- Linear stability analysis for oscillating systems, modelling of two-layer fluid systems, Rayleigh-Taylor instability (6 hours).
- Heat flow problems: characteristic time of cooling, chemically reactive systems, convection-diffusion systems (6 hours).
- Particle motion: Probability density functions, predicting particle positions, nearest neighbour interactions (6 hours).
- Theory of Elasticity: Stress-strain relations, elastic and plastic deformation (6 hours).
- Laws of interaction: Forces between charged particles, principle of superposition, electromagnetic forces, Faraday’s law of magnetic induction (6 hours).
- Interfaces and fronts: Modelling explosive systems with thin reaction zone kinetics - SHS, Frontal Polymerization (6 hours).

**Teaching Methodology**

__Lectures__: Three (3) lectures each week (50 minutes each).

__Labs__: One 2 hour computer lab weekly for 10 weeks (illustrating the modelling concepts from the theory)

__Research Project:__ Owing to the practical nature of the course, the coursework marks will be based totally on the submission and presentation of a group research project in mathematical modelling. Students will be asked to select a research project from a list of possible topics at the beginning of the course. Recent journal articles related to the topics covered during lecture hours will be made available for this purpose. Students must demonstrate an understanding of the rationale for the model, and will be encouraged to suggest possible extensions or improvements. A written report will be submitted, and students will give class presentations based on these projects during the final two weeks of lectures.

**Assessment**

Research Project (submission of a written report and oral presentation) – 40%

Final Examination (one 3-hour written paper) – 60%

**Course Calendar**

Week |
Lecture subjects |
Computer Labs |
---|---|---|

1 |
Preliminaries: Models from Newton’s laws of motion: Planetary motion, energy conservation laws, resonance phenomena, surface area and minimal energy configurations |
- |

2 |
Lagrangian/Eulerian equations of motion: trajectories of particles, caustics |
Lab 1 |

3 |
Linear stability analysis for oscillating systems, modelling of two-layer fluid systems |
Lab 2 |

4 |
Rayleigh-Taylor instability and applications |
Lab 3 |

5 |
Heat flow problems: characteristic time of cooling, chemically reactive systems |
Lab 4 |

6 |
Heat flow problems: Convection-diffusion systems |
Lab 5 |

7 |
Particle motion: Probability density functions, predicting particle positions, nearest neighbour interactions |
Lab 6 |

8 |
Theory of Elasticity: Stress-strain relations, elastic and plastic deformation |
Lab 7 |

9 |
Laws of interaction: Forces between charged particles, principle of superposition, electromagnetic forces |
Lab 8 |

10 |
Laws of interaction: Faraday’s law of magnetic induction Interfaces and fronts: Introductory concepts |
Lab 9 |

11 |
Interfaces and fronts: Modelling explosive systems with thin reaction zone kinetics - SHS, Frontal Polymerization |
Lab 10 |

12 |
Project Presentations |
Project Presentations |

13 |
Project Presentations |
Project Presentations |

**Required Reading**

**Reference Texts **

** (No essential textbook. Lecture notes will be prepared)**

- Course of Theoretical Physics: Mechanics, (3rd Edition) L.D. Landau and E. M. Lifshitz, ISBN: 0 750 62896 0.
- Mathematical Models in the Applied Sciences, A.C. Fowler, ISBN 0-521-46703-9