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Last updated:
September 23, 2023
Duration:
Unlimited Duration
FREE
This course includes:
Unlimited Duration
Badge on Completion
Certificate of completion
Unlimited Duration
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.
Closedform 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: PoissonKirchhoff 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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