PDEs You Should Know

Wave Equation

$$ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} $$


Simple Harmonic Motion

$$ m \frac{d^2 x}{d t^2} = -kx $$


Heat Equation

$$ \frac{\partial u}{\partial t}=\nabla^2 u $$


Laplace/Poisson Equation

Poisson: $$ \nabla^2 f = h $$ Laplace: $$ \nabla^2 f = 0 $$


Maxwell's Equations

Gauss's law: $$ \nabla\cdot E = \frac{\rho}{\epsilon_0} $$ No magnetic monopoles: $$ \nabla\cdot B = 0 $$ Faraday equation: $$ \nabla\times E=-\frac{\partial B}{\partial t} $$ Ampere equation: $$ \nabla\times B=\mu_0(J + \epsilon_0\frac{\partial E}{\partial t}) $$


Schrödinger Equation

$$ i \hbar\frac{\partial \Psi}{\partial t} =-\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2}+V\Psi $$


Black Scholes Equation

$$ \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} +rS\frac{\partial V}{\partial S} - rV = 0 $$