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

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September 23, 2023

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September 23, 2023

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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)

Course Curriculum

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

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