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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