Partial Differential Equations

This course develops methods for studying partial differential equations, including Laplace, Poisson, heat, transport, wave, Euler–Poisson–Darboux, and porous medium equations. Topics include Green’s functions, mollifiers, convolution, polar coordinates, separation of variables, and Kirchhoff’s formula.

Syllabus

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.