Lectures related to Elliptic Partial Differential Equations

Theoretical Study of Elliptic Partial Differential Equations

Covers the theoretical study of Elliptic Partial Differential Equations, including classical and weak solutions.

Covers the Laplace and Poisson equations, the heat equation, and the wave equation in physics.

Properties of Fundamental Solutions: Green's Representation Formula

Covers the properties of fundamental solutions and introduces Green's representation formula for solving partial differential equations.

Poisson & Laplace Equations: Understanding Electrostatics

Explores the derivation of a potential function without knowing the charge distribution and its practical implications in analyzing electrostatic problems.

Fundamental Solutions

Explores fundamental solutions in partial differential equations, highlighting their significance in mathematical applications.

Linear Partial Differential Equations

Explores linear partial differential equations, elliptic PDEs, Laplace equation, boundary conditions, and classical solutions.

Maximum Principle in Harmonic Functions

Explores the maximum principle in harmonic functions and its implications for uniqueness and bounds on solutions.

Introduction to PDES

Covers harmonic functions, Laplacian operator, Dirichlet and Robin problems, and sub-harmonic functions in Partial Differential Equations.

Partial Differential Equations: Basics and Classification

Introduces the basics and classification of partial differential equations, including notation, definitions, and examples.

Finite Difference Methods: Linear Systems and Band Matrices

Covers the application of finite difference methods to solve partial differential equations.

Classification of PDEs: Linear, Semi-linear, Quasi-linear

Explores the classification of PDEs into linear, semi-linear, and quasi-linear types, emphasizing the properties of their solutions.

Partial Differential Equations: Laplace Equation

Explains the Laplace equation in 2D with mixed boundary conditions.

Partial Differential Equations

Covers the basics of Partial Differential Equations, including the Laplace equation, heat equation, and wave equation.

Partial Differential Equations: Characteristics and Solutions

Explores characteristic curves and solutions in partial differential equations, emphasizing uniqueness and existence in various scenarios.

Fourier Analysis and PDEs

Explores Fourier analysis, PDEs, historical context, heat equation, Laplace equation, and periodic boundary conditions.

Ion Crystal and Poisson/Laplace Equations

Explores ion crystals, conductors, and Poisson/Laplace equations in electrostatics, including charge behavior and energy calculations.

Uniqueness Results for Poisson Equation

Explores uniqueness results for the Poisson equation and harmonic functions with different boundary conditions.

Distribution & Interpolation Spaces

Explores distribution and interpolation spaces, differential operators, Fourier transform, Schwartz space, fundamental solutions, Farrier transform, and uniform continuity.

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Numerical Approximation of PDEs

Covers the numerical approximation of PDEs, including Poisson and heat equations, transport phenomena, and incompressible limits.