Partial differential equation

The Navier–Stokes differential equation, used to model airflow around an obstruction.

A Partial differential equation is a type of differential equation which relates a multivariable function to its partial derivatives. They are very commonly used in engineering and physics.

Examples

• ${\frac {\partial U}{\partial x}}+{\frac {\partial U}{\partial y}}+U=0$
• ${\frac {\partial ^{2}U}{\partial x^{2}}}+{\frac {\partial U}{\partial x}}+xU={\frac {\partial U}{\partial y}}$
• ${\frac {\partial ^{2}U}{\partial x^{2}}}+{\frac {\partial ^{2}U}{\partial y^{2}}}=xy$

The Diffusion Equation

${\frac {\partial u}{\partial t}}=\alpha \bigtriangledown ^{2}u$

The Laplace Equation

$\bigtriangledown ^{2}u=0$

The Wave Equation

${\frac {\partial ^{2}u}{\partial t^{2}}}=\alpha \bigtriangledown ^{2}u$