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Partial Differential Equations

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)

Lecture 46 - Laplace Equation: Fundamental Solution

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First Order Partial Differential Equations

Lecture 01 - Basic Concepts and Nomenclature

Lecture 02 - First Order Partial Differential Equations: How they Arise?

Lecture 03 - Geometry of Quasilinear Equations

Lecture 04 - General Solutions to Linear and Semilinear Equations

Lecture 05 - Lagrange's Method for Quasilinear Equations

Lecture 06 - Relation between Characteristic Curves and Integral Surfaces for Quasilinear Equations

Lecture 07 - Method of Characteristics for Quasilinear Equations 1

Lecture 08 - Method of Characteristics for Quasilinear Equations 2

Lecture 09 - Failure of Transversality Condition

Lecture 10 - Tutorial of Quasilinear Equations

Lecture 11 - General Nonlinear Equations: Search for a Characteristic Direction

Lecture 12 - General Nonlinear Equations: Characteristic Direction and Characteristic Strip

Lecture 13 - General Nonlinear Equations: Finding an Initial Strip

Lecture 14 - General Nonlinear Equations: Local Existence and Uniqueness Theorem

Lecture 15 - Tutorial on General Nonlinear Equations

Lecture 16 - Initial Value Problems for Burgers Equation

Lecture 17 - Conservation Laws with a View towards Global Solutions to Burgers Equation

Second Order Partial Differential Equations

Lecture 18 - Second Order Partial Differential Equations: Special Curves Associated to a PDE

Lecture 19 - Curves of Discontinuity

Lecture 20 - Classification

Lecture 21 - Canonical Form for an Equation of Hyperbolic Type

Lecture 22 - Canonical Form for an Equation of Parabolic Type

Lecture 23 - Canonical Form for an Equation of Elliptic Type

Lecture 24 - Characteristic Surfaces

Lecture 25 - Canonical Forms for Constant Coefficient PDEs

Second Order Partial Differential Equations: Wave Equation

Lecture 26 - Wave Equation: A Mathematical Model for Vibrating Strings

Lecture 27 - Wave Equation in One Space Dimension: d'Alembert Formula

Lecture 28 - Tutorial on One Dimensional Wave Equation

Lecture 29 - Wave Equation in d Space Dimensions

Lecture 30 - Cauchy Problem for Wave Equation in 3 Space Dimensions: Poisson-Kirchhoff Formulae

Lecture 31 - Cauchy Problem for Wave Equation in 2 Space Dimensions: Hadamard's Method of Descent

Lecture 32 - Nonhomogeneous Wave Equation: Duhamel Principle

Lecture 33 - Wellposedness of Cauchy Problem for Wave Equation

Lecture 34 - Wave Equation on an Interval in R

Lecture 35 - Tutorial on IBVPs for Wave Equation

Lecture 36 - IBVP for Wave Equation: Separation of Variables Method

Lecture 37 - Tutorial on Separation of Variables Method for Wave Equation

Qualitative Analysis of Wave Equation

Lecture 38 - Qualitative Analysis of Wave Equation: Parallelogram Identity

Lecture 39 - Qualitative Analysis of Wave Equation: Domain of Dependence, Domain of Influence

Lecture 40 - Qualitative Analysis of Wave Equation: Causality Principle, Finite Speed of Propagation

Lecture 41 - Qualitative Analysis of Wave Equation: Uniqueness by Energy Method

Lecture 42 - Qualitative Analysis of Wave Equation: Huygens Principle

Lecture 43 - Qualitative Analysis of Wave Equation: Generalized Solution to Wave Equation

Lecture 44 - Qualitative Analysis of Wave Equation: Propagation of Waves

Second Order Partial Differential Equations: Laplace Equation

Lecture 45 - Laplace Equation: Associated Boundary Value Problems

Lecture 46 - Laplace Equation: Fundamental Solution

Lecture 47 - Dirichlet BVP for Laplace Equation: Green's Function and Poisson's Formula

Lecture 48 - Laplace Equation: Weak Maximum Principle and its Applications

Lecture 49 - Laplace Equation: Dirichlet BVP on a Disk in R2 for Laplace Equations

Lecture 50 - Tutorial 1 on Laplace Equation

Lecture 51 - Laplace Equation: Mean Value Property

Lecture 52 - Laplace Equation: More Qualitative Properties

Lecture 53 - Laplace Equation: Strong Maximum Principle and Dirichlet Principle

Lecture 54 - Tutorial 2 on Laplace Equation

Second Order Partial Differential Equations: Heat Equation

Lecture 55 - Cauchy Problem for Heat Equation, Part 1

Lecture 56 - Cauchy Problem for Heat Equation, Part 2

Lecture 57 - IBVP for Heat Equation Subtitle: Method of Separation of Variables

Lecture 58 - Maximum Principle for Heat Equation

Lecture 59 - Tutorial on Heat Equation

Lecture 60 - Heat Equation Subheading: Infinite Speed of Propagation, Energy, Backward Problem