MAT 391E - Advanced Topics in ODE

Course Objectives

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.
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.