AMAT 41363: Qualitative and Quantitative Behavior of the Solutions of Ordinary Differential Equations

Course Code : AMAT 41363

Title              : Qualitative and Quantitative Behavior of the Solutions of Ordinary Differential Equations

Pre-requisites : AMAT 22292

Learning Outcomes:

At the end of this course, the student should be able to 

1. obtain the numerical solutions of differential equations

2. examine the existence, uniqueness and stability of the solutions

3. analyze the asymptotic behavior of the solutions

4. implement the numerical methods of ordinary differential equations using appropriate software

5. use appropriate techniques for solving real-world problems.

Course Contents:     

Linear Differential Equations: 

• First-Order Linear Differential Equations, Higher-Order Linear Differential Equations, Routh-Hurwitz Criteria, Converting Higher-Order Equations to First-Order Systems, First-Order Linear Systems: Constant Coefficients, Diagonalizations, Methods for Computing the Matrix Exponential, The Fundamental Theorem for Linear Systems, Phase-Plane Analysis, Gershgorin's Theorem, An Example: Pharmacokinetics Model.

Qualitative Theory of Nonlinear Ordinary Differential Equations: 

• Introduction to Nonlinear Ordinary Differential Equations, The Fundamental Existence-Uniqueness Theorem, The Maximal Interval of Existence, Linearization, Stability and Liapunov Functions, Phase Plane Analysis, Stable and Unstable Manifolds, Bifurcations, Periodic Solutions, Poincaré-Bendixson Theorem, Dulac’s Criteria

Numerical Solutions of Ordinary Differential Equations: 

• Review of Numerical Methods, Stability and Convergence Properties of Numerical Schemes, Absolute Stability and Stiff Equations, Implementation of Analytical and Numerical Solutions Using Appropriate Software.                                                                                                                                  

Method of Teaching and Learning : A combination of lectures, computer laboratory sessions, tutorial discussions and presentations.

Assessment : Based on tutorials, tests, presentations and end of course examination.

Recommended Reading  :

1. Chapra, S.C. (4th Ed., 2017). Applied Numerical Methods with MATLAB for Engineers and Scientists, McGraw Hill.

2. Verhulst, F. (2012). Nonlinear Differential Equations and Dynamical Systems, Springer.

3.  Allen, L.J.S. (2007). An Introduction to Mathematical Biology, Pearson.

4. Perko, L. (2001). Differential Equations and Dynamical Systems, Springer.

5. Van Loan, C.F. (2000). Introduction to Scientific Computing: A Matrix-Vector Approach Using MATLAB, Prentice Hall.