Advanced Theory of Ordinary Differential Equations

Overview

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ABOUT THE COURSE: Differential equations play an important role in applied mathematics. They provide the modelling tools to depict and simulate real world (physical) problems. This course is intended for all undergraduate students such as students from any BTech, BSc and MSc level courses. It will offer a detailed introduction and tools necessary to model, analyze and solve numerically the differential equations (DEs). We will cover both first order and second order ODEs & their subsequent theories.INTENDED AUDIENCE: UG students including all BTech, BSc and MSc students.PREREQUISITES: Differential calculus of one and several variables, Integral calculus, Ordinary differential equations.

Syllabus

Week 1:ODEs: Existence, Uniqueness, and dependence on Parameters: Lipschitz continuity and uniqueness, Local existence
Week 2:Continuation of local solutions, Dependence on initial value and vector field, Regular perturbations linearisation. Stability: Stability definitions, Stability of linear systems,
Week 3:Nonlinear systems, linearization, Nonlinear systems, Lyapunov functions, Global analysis of the phase plane, periodic ODE, Stability of A-equations.
Week 4:Chaotic Systems: Local divergence, Lyapunov exponents, Strange and chaotic attractors.
Week 5:Fractal dimension, Reconstruction, Prediction. Singular Perturbations and Stiff Differential Equations: Singular perturbations,
Week 6:Matched asymptotic expansions, Stiff differential equations, The increment function A-stable, A(α)-stable methods, BDF methods and their implementation
Week 7:Bifurcation Theory: Basic concepts, One dimensional Bifurcations for scalar equations, Hopf Bifurcations for planar systems
Week 8:Mathematical Models with second order equations, Free mechanical oscillations: Undamped and damped free oscillations, Forced mechanical oscillations: Undamped and damped forced oscillations, Electrical vibrations.
Week 9:Higher Order linear equations: Matrices and Determinants of higher order, System of Linear Algebraic Equations, Linear Independence and Wronskian, homogeneous and non homogeneous equations, Method of undetermined coefficients, variation of parameters
Week 10:Series solutions of differential equations, Power series solution of Legendre, Bessel and Laguerre differential equations, Legendre, Bessel and Laguerre polynomials
Week 11:Sturm-Liouville boundary value problems, Eigenvalues and eigenfunctions
Week 12:Laplace transform: Laplace transform and Inverse Laplace transform, some elementary properties and results, periodic functions, Dirac Delta function Convolutions theorem, Solution of initial value problems.