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.
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