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September 23, 2023

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Description

Partial Differential Equations. Instructor: Prof. Sivaji Ganesh, Department of Mathematics, IIT Bombay. Partial Differential Equations (PDEs) appear as mathematical models for many physical phenomena.

Closed-form solutions to most of these PDEs cannot be found. One of the possible ways to understand the models is by studying the qualitative properties exhibited by their solutions. In this course, we study first order nonlinear PDEs, and the properties of the three important types of second order linear PDEs (Wave, Laplace, Heat) would be studied and compared. (from nptel.ac.in)

Course Curriculum

    • Lecture 01 – Basic Concepts and Nomenclature Unlimited
    • Lecture 02 – First Order Partial Differential Equations: How they Arise? Unlimited
    • Lecture 03 – Geometry of Quasilinear Equations Unlimited
    • Lecture 04 – General Solutions to Linear and Semilinear Equations Unlimited
    • Lecture 05 – Lagrange’s Method for Quasilinear Equations Unlimited
    • Lecture 06 – Relation between Characteristic Curves and Integral Surfaces for Quasilinear Equations Unlimited
    • Lecture 07 – Method of Characteristics for Quasilinear Equations 1 Unlimited
    • Lecture 08 – Method of Characteristics for Quasilinear Equations 2 Unlimited
    • Lecture 09 – Failure of Transversality Condition Unlimited
    • Lecture 10 – Tutorial of Quasilinear Equations Unlimited
    • Lecture 11 – General Nonlinear Equations: Search for a Characteristic Direction Unlimited
    • Lecture 12 – General Nonlinear Equations: Characteristic Direction and Characteristic Strip Unlimited
    • Lecture 13 – General Nonlinear Equations: Finding an Initial Strip Unlimited
    • Lecture 14 – General Nonlinear Equations: Local Existence and Uniqueness Theorem Unlimited
    • Lecture 15 – Tutorial on General Nonlinear Equations Unlimited
    • Lecture 16 – Initial Value Problems for Burgers Equation Unlimited
    • Lecture 17 – Conservation Laws with a View towards Global Solutions to Burgers Equation Unlimited
    • Lecture 18 – Second Order Partial Differential Equations: Special Curves Associated to a PDE Unlimited
    • Lecture 19 – Curves of Discontinuity Unlimited
    • Lecture 20 – Classification Unlimited
    • Lecture 21 – Canonical Form for an Equation of Hyperbolic Type Unlimited
    • Lecture 22 – Canonical Form for an Equation of Parabolic Type Unlimited
    • Lecture 23 – Canonical Form for an Equation of Elliptic Type Unlimited
    • Lecture 24 – Characteristic Surfaces Unlimited
    • Lecture 25 – Canonical Forms for Constant Coefficient PDEs Unlimited
    • Lecture 26 – Wave Equation: A Mathematical Model for Vibrating Strings Unlimited
    • Lecture 27 – Wave Equation in One Space Dimension: d’Alembert Formula Unlimited
    • Lecture 28 – Tutorial on One Dimensional Wave Equation Unlimited
    • Lecture 29 – Wave Equation in d Space Dimensions Unlimited
    • Lecture 30 – Cauchy Problem for Wave Equation in 3 Space Dimensions: Poisson-Kirchhoff Formulae Unlimited
    • Lecture 31 – Cauchy Problem for Wave Equation in 2 Space Dimensions: Hadamard’s Method of Descent Unlimited
    • Lecture 32 – Nonhomogeneous Wave Equation: Duhamel Principle Unlimited
    • Lecture 33 – Wellposedness of Cauchy Problem for Wave Equation Unlimited
    • Lecture 34 – Wave Equation on an Interval in R Unlimited
    • Lecture 35 – Tutorial on IBVPs for Wave Equation Unlimited
    • Lecture 36 – IBVP for Wave Equation: Separation of Variables Method Unlimited
    • Lecture 37 – Tutorial on Separation of Variables Method for Wave Equation Unlimited
    • Lecture 38 – Qualitative Analysis of Wave Equation: Parallelogram Identity Unlimited
    • Lecture 39 – Qualitative Analysis of Wave Equation: Domain of Dependence, Domain of Influence Unlimited
    • Lecture 40 – Qualitative Analysis of Wave Equation: Causality Principle, Finite Speed of Propagation Unlimited
    • Lecture 41 – Qualitative Analysis of Wave Equation: Uniqueness by Energy Method Unlimited
    • Lecture 42 – Qualitative Analysis of Wave Equation: Huygens Principle Unlimited
    • Lecture 43 – Qualitative Analysis of Wave Equation: Generalized Solution to Wave Equation Unlimited
    • Lecture 44 – Qualitative Analysis of Wave Equation: Propagation of Waves Unlimited
    • Lecture 45 – Laplace Equation: Associated Boundary Value Problems Unlimited
    • Lecture 46 – Laplace Equation: Fundamental Solution Unlimited
    • Lecture 47 – Dirichlet BVP for Laplace Equation: Green’s Function and Poisson’s Formula Unlimited
    • Lecture 48 – Laplace Equation: Weak Maximum Principle and its Applications Unlimited
    • Lecture 49 – Laplace Equation: Dirichlet BVP on a Disk in R2 for Laplace Equations Unlimited
    • Lecture 50 – Tutorial 1 on Laplace Equation Unlimited
    • Lecture 51 – Laplace Equation: Mean Value Property Unlimited
    • Lecture 52 – Laplace Equation: More Qualitative Properties Unlimited
    • Lecture 53 – Laplace Equation: Strong Maximum Principle and Dirichlet Principle Unlimited
    • Lecture 54 – Tutorial 2 on Laplace Equation Unlimited
    • Lecture 55 – Cauchy Problem for Heat Equation, Part 1 Unlimited
    • Lecture 56 – Cauchy Problem for Heat Equation, Part 2 Unlimited
    • Lecture 57 – IBVP for Heat Equation Subtitle: Method of Separation of Variables Unlimited
    • Lecture 58 – Maximum Principle for Heat Equation Unlimited
    • Lecture 59 – Tutorial on Heat Equation Unlimited
    • Lecture 60 – Heat Equation Subheading: Infinite Speed of Propagation, Energy, Backward Problem Unlimited

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