Partial Differential Equations

Dive into the world of partial differential equations with this comprehensive 10-hour course. Explore key concepts such as the Laplace equation, its derivation, and applications. Master techniques like convolution and the Dominated Convergence Theorem. Delve into polar coordinates, normal derivatives, and integration by parts. Examine the Poisson equation, Laplace Mean Value Formula, and mollifiers. Investigate Green's Function and Poisson formulas on half-planes and balls. Get a taste of Calculus of Variations and study various equations including heat, transport, and wave equations. Analyze the Euler Poisson Darboux equation, Kirchhoff Formula, and Separation of Variables. Conclude with an exploration of the Porous Medium Equation and the Heat-Wave Equation, providing a solid foundation in partial differential equations.

Laplace equation.
Laplace Derivation.
Laplace Applications.
Convolution.
Dominated Convergence Theorem.
Polar Coordinates Formula.
Normal Derivative.
Integrate by parts like a pro.
Poisson equation.
Laplace Mean Value Formula.
Mollifiers.
Green's Function.
Poisson formula on half plane.
Poisson Formula on a ball.
Calculus of Variations.
A Taste of Calculus of Variations.
Heat equation.
Heat Equation Initial Condition.
What is Convexity?.
Transport equation.
Wave equation.
the Euler Poisson Darboux equation.
Kirchhoff Formula.
Separation of Variables.
Porous Medium Equation .
Heat-Wave Equation.