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Ordinary Differential Equations and Applications

Ordinary Differential Equations and Applications. Instructors: A. K. Nandakumaran, IISc Bangalore; P. S. Datti, TIFR-CAM, Bangalore; Raju K. George, IIST,Trivandrum.

Motivation and real life examples: an introduction about differential equations and examples.
Preliminaries: basic concepts from linear algebra and some important preliminaries from analysis like uniform convergence, Arzela-Ascoli theorem, fixed point theorems etc.
First and second order linear equations: examples, a systematic procedure to solve first order and development of the concept integrating factor, Second order homogeneous and non-homogeneous equations.
General existence and uniqueness theory: Picard's iteration, Peano's existence theory, Existence via Arzela Ascoli theorem, non-uniqueness, continuous dependence.
Linear systems: understanding linear system via linear algebra, stability of linear systems, explicit phase portrait in 2D linear with constant coefficients.
Qualitative analysis: examples of nonlinear systems, Stability analysis, Lyapunov stability, phase portrait of 2D systems, Poincare-Bendixson theory.
Introduction to two-point boundary value problems: linear equations, Green's function, nonlinear equations, existence and uniqueness. (from nptel.ac.in)

Introduction


Motivation and Real Life Examples
Lecture 01 - General Introduction
Lecture 02 - Examples
Lecture 03 - Examples Continued I
Lecture 04 - Examples Continued II
Preliminaries
Lecture 05 - Linear Algebra
Lecture 06 - Linear Algebra Continued I
Lecture 07 - Linear Algebra Continued II
Lecture 08 - Analysis
Lecture 09 - Analysis Continued
First and Second Order Linear Equations
Lecture 10 - First Order Linear Equations
Lecture 11 - Exact Equations
Lecture 12 - Second Order Linear Equations
Lecture 13 - Second Order Linear Equations Continued I
Lecture 14 - Second Order Linear Equations Continued II
General Existence and Uniqueness Theory
Lecture 15 - Well-posedness and Examples of IVP
Lecture 16 - Gronwall's Lemma
Lecture 17 - Basic Lemma and Uniqueness Theorem
Lecture 18 - Picard's Existence and Uniqueness Theorem
Lecture 19 - Picard's Existence and Uniqueness Theorem Continued
Lecture 20 - Cauchy Peano Existence Theorem
Lecture 21 - Existence using Fixed Point Theorem
Lecture 22 - Continuation of Solutions
Lecture 23 - Series Solution
Linear Systems
Lecture 24 - General System and Diagonalizability
Lecture 25 - 2 by 2 Systems and Phase Plane Analysis
Lecture 26 - 2 by 2 Systems and Phase Plane Analysis Continued
Lecture 27 - General Systems
Lecture 28 - General Systems Continued and Non-homogeneous Systems
Nonlinear Systems
Lecture 29 - Basic Definitions and Examples
Lecture 30 - Stability Equilibrium Points
Lecture 31 - Stability Equilibrium Points Continued I
Lecture 32 - Stability Equilibrium Points Continued II
Lecture 33 - Stability Equilibrium Points Continued III
Lecture 34 - Lyapunov Function
Lecture 35 - Lyapunov Function Continued
Lecture 36 - Periodic Orbits and Poincare-Bendixson Theory
Lecture 37 - Periodic Orbits and Poincare-Bendixson Theory Continued
Two-point Boundary Value Problems
Lecture 38 - Linear Second Order Equations
Lecture 39 - General Second Order Equations
Lecture 40 - General Second Order Equations Continued

References
Ordinary Differential Equations and Applications
Instructors: A. K. Nandakumaran, IISc Bangalore; P. S. Datti, TIFR-CAM, Bangalore; Raju K. George, IIST,Trivandrum. an introduction about differential equations and examples.