# InfoCoBuild

## 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)

 Course Introduction

 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

 References Partial Differential Equations Instructor: Prof. Sivaji Ganesh, Department of Mathematics, IIT Bombay. This course will study first order nonlinear PDEs and the properties of the three important types of second order linear PDEs (Wave, Laplace, Heat).