MAT 391E - Advanced Topics in ODE

Course Objectives

1. Teach the basics on stability of dynamical systems.
2. To develop a basic understanding of occurrence of two point boundary value problems, their classification and related problems; such as, initial value, boundary value and initial-boundary value problems in the real world.
3. To develop a basic understanding of the theory and methods of solutions for these problems.

Course Description

Nonlinear Differential Equations and Stability: The Phase Plane-Linear Systems, Autonomous Systems and Stability, Locally Linear Systems, Competing Species, Predator-Prey Equations, Liapunov’s Second Method, Periodic Solutions and Limit Cycles, Chaos and Strange Attractors: The Lorenz Equations. Two-point boundary-value problems; definition, examples, existence and uniqueness of solutions. Linear homogeneous boundary-value problems; eigenvalues and eigenvectors. Sturm-Liouville boundary-value problems; Lagrange identity, orthogonality of eigenfunctions, self-adjoint problems. Nonhomogeneous boundary-value problems; non-homogeneous Sturm-Liouville problems, non-homogeneous heat conduction problems. Singular Sturm-Liouville problems; definition, continuous spectrum, vibration of a circular elastic membrane, Series of orthogonal functions; convergence and completeness. Techniques of Green`s function; generalised functions, Green`s function, modified Green`s function.