Advanced Partial Differential Equations (Dr. Kaushik Bal, IIT Kanpur)| Mathematics

Advanced Partial Differential Equations

Advanced Partial Differential Equations. Instructor: Dr. Kaushik Bal, Department of Mathematics and Statistics, IIT Kanpur. The precise idea to study partial differential equations is to interpret physical phenomena occurring in nature. Most often the systems encountered fail to admit explicit solutions but fortunately qualitative methods were discovered which does provide ample information about the system without explicitly solving it. In this course we will explore the basic ideas of studying first order equations starting with the inner workings of method of characteristics followed by the three fundamental second order PDEs namely Laplace equation, Heat equation and Wave equation. (from nptel.ac.in)

Introduction

Lecture 01 - Basic Definitions and Notations

Lecture 02 - Defining What PDE is, Classification of Linear PDE, Strategies for Studying PDE

Lecture 03 - Method of Characteristics

Lecture 04 - Method of Characteristics (cont.)

Lecture 05 - Method of Characteristics for Fully Nonlinear Equations

Lecture 06 - Introduction to Laplace Equations

Lecture 07 - Laplace Equations: Dirichlet and Neumann Boundary Conditions

Lecture 08 - Mean Value Property of Harmonic Functions

Lecture 09 - Introduction to Convolution and Mollifiers

Lecture 10 - Strong Maximum Principle for Harmonic Functions and Some of its Applications

Lecture 11 - The Regularity Theorem, its Proof and Applications

Lecture 12 - Analyticity of Harmonic Functions, Harnack Inequality

Lecture 13 - Solution of the Poisson Equation

Lecture 14 - Solution of Poisson Equation (cont.)

Lecture 15 - Green's Functions and their Existence

Lecture 16 - Derivation of Green's Functions on a Ball

Lecture 17 - Introduction to the Heat Equation

Lecture 18 - Backward Uniqueness

Lecture 19 - The Existence Theory for Heat Equation and the Fundamental Solution

Lecture 20 - Duhamel's Principle for a System of ODE, Homogeneous Heat Equation

Lecture 21 - Non-Homogeneous Heat Equation

Lecture 22 - Initial and Boundary Value Problems for Heat Equation

Lecture 23 - Maximum Principles, Uniqueness and Regularity of the Solutions of Heat Equation

Lecture 24 - Introduction to Wave Equation

Lecture 25 - Solution of Wave Equations, Especially D'Alembert's Formula

Lecture 26 - Higher Dimensional Wave Equation, Euler-Poisson-Darboux Theorem

Lecture 27 - 2-dimensional Wave Equation, Poisson Formula and Duhamel Principle

Lecture 28 - Wave Equations in Odd Spatial Dimensions

Lecture 29 - Wave Equations in Even Spatial Dimensions

Lecture 30 - Introduction to Conservation Laws

Lecture 31 - Rankine-Hugoniot Jump Condition

Lecture 32 - Discussion on Conservation Laws (cont.)

Lecture 33 - Examples of Conservation Laws

References

Advanced Partial Differential Equations
Instructor: Dr. Kaushik Bal, Department of Mathematics and Statistics, IIT Kanpur. The precise idea to study partial differential equations is to interpret physical phenomena occurring in nature.