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 (Green’s Function and Corrector Construction, Representation Formula for Poisson Equation, Symmetry of Green’s Function, Green’s Function for the Half-Space via Reflection Method) & Assignment 3 (weightage 2.5 points) & Solution
Lecture 15, March 12, 16:30-18:30 (Poisson Kernel for the Half-Space and Boundary Convergence, Green’s Function for the Unit Ball via Inversion, Poisson Representation Formula for the Ball, Energy Methods for Poisson’s Equation, Dirichlet Principle and Variational Characterization)
Presentation of Praveena, March 17, 16:30-18:30 (weightage 15 points)
Lecture 16, March 19, 16:30-18:30 (Heat Equation and Physical Derivation, Fundamental Solution, Initial Value Problem, Infinite Propagation Speed, Non-homogeneous Heat Equation)
Lecture 17, March 20, 16:30-18:30 (Non-homogeneous Heat Equation and Verification of Solution, Combined Initial and Source Problem, Parabolic Boundary and Parabolic Cylinder, Heat Balls and Geometry of Heat Equation, Mean-Value Formula for Heat Equation)
Lecture 18, March 24, 16:30-18:30 (Strong Maximum Principle for Heat Equation, Uniqueness for Initial-Boundary Value Problems, Maximum Principle for the Cauchy Problem with Growth Conditions, Uniqueness under Growth Constraints) & Assignment 4 (weightage 2.5 points) & Solution
Lecture 19, March 26, 16:30-18:30 (Regularity and Smoothness of Heat Equation Solutions, Interior Representation via Heat Kernel and Cutoff Functions, Local Derivative Estimates, Energy Methods for Uniqueness, Backward Uniqueness for Heat Equation)
Lecture 20, March 31, 16:30-18:30 (Wave Equation and Physical Interpretation, d’Alembert Formula for 1D Wave Equation, Finite Speed of Propagation and Wave Behavior, Reflection Method for Boundary Problems, Spherical Means Method, Euler–Poisson–Darboux Equation)
Maundy, April 2, No Class
Lecture 21, April 3, 16:30-18:30 (Kirchhoff’s Formula in 3D, Poisson Formula in 2D, General Solution in Odd and Even Dimensions, Finite Propagation Speed and Huygens’ Principle, Duhamel’s Principle for Wave Equation, Energy Methods and Uniqueness, Domain of Dependence)
April 7, No Class
Ambedkar Jayanti, April 14, No Class
Discussion of Doubts, April 21, 16:30-18:30
Discussion of Assignments 3 & 4, April 23, 16:30-18:30