Lecture 1, January 13, 18:00-20:00 (Introduction to PDEs, Well-posedness, Transport Equation, Multi-index Notation)
Lecture 2, January 15, 16:30-18:30 (Norms and C^k Spaces, Classification of PDEs, Well-posedness, First-order PDEs, Cauchy Problem)
Lecture 3, January 20, 16:30-18:30 (Method of Characteristics, Characteristic Curves and Reduction to ODEs, Integral Surfaces and Vector Field Interpretation, Semilinear and Quasilinear First-order PDEs)
Lecture 4, January 24, 18:00-20:00 (Inviscid Burgers Equation, Shock Formation and Blow-up, Limitations of the Method of Characteristics, Lagrange’s Method for First-order PDEs, Nonlinear First-order PDEs, Existence and Uniqueness of the Cauchy Problem) & Assignment 1 (weightage 2.5 points) & Solution
Lecture 5, January 25, 18:00-20:00 (Real Analytic Functions, Power Series and Multi-index Expansions, Characterization of Analyticity via Derivative Bounds, Cauchy–Kowalevski Theorem, Analytic Existence and Uniqueness, Holmgren’s Uniqueness Theorem)
Quiz, January 29, 16:30-18:30 (weightage 10 points) & Solution
Lecture 6, February 7, 17:30-19:00 (Limitations of Classical Solutions, Weak Derivatives and Weak Solutions, Lewy’s Non-solvability Example, Hamilton–Jacobi Equations and Viscosity Solutions, Conservation Laws in Weak Form, Shock Waves and Rankine–Hugoniot Condition)
Lecture 7, February 8, 17:00-19:00 (Shock Solutions for Burgers’ Equation, Rankine–Hugoniot Condition and Shock Speed, Traffic Flow Conservation Laws, Second-order Linear PDEs, Classification into Elliptic–Parabolic–Hyperbolic Types, Canonical Forms of Second-order PDEs)
Lecture 8, February 10, 16:30-18:30 (Elliptic PDEs and Canonical Form, Separation of Variables, Laplace Equation on a Rectangle, Wave and Heat Equations, Green’s Identities and Gauss–Green Theorem, Polar Coordinates and Laplacian, Basic Integral Inequalities) & Assignment 2 (weightage 2.5 points) & Solution
Lecture 9, February 12, 17:30-19:30 (Lebesgue Differentiation Theorem, Mollifiers and Smooth Approximation, Transport Equation with Initial Value Problem)
Lecture 10, February 17, 16:30-18:30 (Laplace and Poisson Equations, Physical Interpretation via Flux and Divergence, Harmonic Functions and Examples, Fundamental Solution of Laplace Equation, Convolution Representation of the Solution of Poisson's equation)
Lecture 11, February 19, 16:30-18:30 (Mean-Value Property for Harmonic Functions, Converse Mean-Value Theorem, Strong Maximum Principle and Uniqueness, Smoothness of Harmonic Functions, Interior Derivative Estimates, Liouville’s Theorem, Representation Formula via Fundamental Solution)
Lecture 12, February 24, 16:30-18:30 (Analyticity of Harmonic Functions, Harnack’s Inequality) & Presentation of Ankit (weightage 15 points)
Lecture 13, February 26, 16:30-18:30 (Motivation for Constructing Green’s Function) & Presentation of Haripriya (weightage 15 points)
Midsem, March 6, 14:00-17:00 (weightage 25 points) & Solution
Lecture 14, March 10, 16:30-18:30 ()
Lecture 15, March 12, 16:30-18:30 () & Assignment 3 (weightage 2.5 points) & Solution
Presentation of Praveena, March 17, 16:30-18:30 (weightage 15 points)
Lecture 16, March 19, 16:30-18:30 ()
Lecture 17, March 24, 16:30-18:30 ()
Lecture 18, March 26, 16:30-18:30 ()
Lecture 19, March 31, 16:30-18:30 ()
Maundy, April 2, No Class
Lecture 20, April 7, 16:30-18:30 ()
Lecture 21, April 9, 16:30-18:30 () & Assignment 4 (weightage 2.5 points) & Solution
Ambedkar Jayanti, April 14, No Class
Lecture 22, April 16, 16:30-18:30 ()
Lecture 23, April 21, 16:30-18:30 ()
Lecture 24, April 23, 16:30-18:30 ()
Endsem, May 6, 14:00-17:00 (weightage 40 points) & Solution