AMAT 42383 : Advanced Mathematical Modeling

Course Unit Code : AMAT 42383 

Course Title           : Advanced Mathematical Modeling

Pre-requisites         : AMAT 41363

Learning outcomes:

Upon successful completion of the course unit the student will be able to:

• explain how the general principles arise in the context of mathematical modeling

• analyze existing mathematical models using ordinary differential equations

• formulate simple ODE models for real world problems

• solve system of ordinary differential equations

• analyze the qualitative behavior of mathematical models

• identify the solutions of difference equations

• Solve system of linear difference equations using Putzer algorithm and Jordan form.

• demonstrate biological applications of difference equations.

Course Content:

Introduction to Modeling: 

• Philosophy of modeling, Modeling Methodology, Problem formulation, Mathematical Description, Analysis, Interpretation.

Mathematical Modeling Using Ordinary Differential Equations: 

• Classification of ODE, Equilibrium points, First order Differential Equations: Mixing, chemical reactions, Population models: Logistic growth model, Harvesting models, Traffic Dynamic models: Microscopic and macroscopic models, System of Differential equations: Interacting population models (Predator –Prey models, Competition models), Compartment models (Dynamic of infectious disease, Age structured models, Reaction kinetics), Qualitative analysis of models.

Mathematical Modeling Using Difference Equations: 

• First order difference equations, Equilibrium points, asymptotic stability of equilibrium points, System of linear difference equations: Autonomous systems, Discrete analogue of Putzer algorithm, Jordan form, linear periodic systems, Biological Applications of Difference Equations: Markov chains, Population dynamics, Trade models, Age classes, Business cycle models.

Group Project : Mathematical model formulation for a real world problem

Method of teaching and learning : A combination of lectures and tutorial discussions

Assessment             : Based on Assignments, Group project presentations, Reports and Final Exam.

Recommended Textbook:

1. Kapur, J.N. (2015). Mathematical Modeling, New Age International.

2. Bender, A. (2012). An introduction to Mathematical Modeling, Courier Corporation.

3. Haberman, R. (1998). Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow, SIAM.

4. Allen, L. (2006). An Introduction to Mathematical Biology, Pearson.

5. Elaydi, S. (2005). An Introduction to Difference Equation, Springer.

6. Illner, R., Bohun, C.S., McCollum, S. & Roode, T.V. (2005). Mathematical Modelling: A Case Studies Approach, AMS.