Partial differential equations

$\frac{∂²u}{∂x²} + \frac{∂²u}{∂y²} = 0$
This describes steady-state phenomena, like temperature distribution in a 2D plate with no heat sources.

$\frac{∂²u}{∂x²} + \frac{∂²u}{∂y²} = f(x, y)$
This is a non-homogeneous version of the Laplace equation, often used in electrostatics (potential due to a charge distribution).

$\frac{∂u}{∂t} = k \frac{∂²u}{∂x²}$ (1D)
$\frac{∂u}{∂t} = k \left( \frac{∂²u}{∂x²} + \frac{∂²u}{∂y²} \right)$ (2D)
This describes how temperature diffuses over time in a material. $k$ is the thermal diffusivity.

$\frac{∂²u}{∂t²} = c² \frac{∂²u}{∂x²}$ (1D)
$\frac{∂²u}{∂t²} = c² \left( \frac{∂²u}{∂x²} + \frac{∂²u}{∂y²} \right)$ (2D)
This describes wave propagation, like vibrations of a string or sound waves. $c$ is the wave speed.

$\frac{∂u}{∂t} + u \frac{∂u}{∂x} = \nu \frac{∂²u}{∂x²}$
This is a simplified model for fluid dynamics, combining convection and diffusion.

$i\hbar \frac{∂Ψ}{∂t} = -\frac{\hbar²}{2m} \nabla²Ψ + VΨ$
This fundamental equation in quantum mechanics describes the time evolution of a quantum system's wave function $Ψ$.

These are a set of non-linear PDEs describing the motion of viscous fluid substances. The momentum equation is a key part:
$\frac{∂\mathbf{u}}{∂t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\frac{1}{ρ} \nabla p + \nu \nabla²\mathbf{u} + \mathbf{f}$
where $\mathbf{u}$ is velocity, $p$ is pressure, $ρ$ is density, $\nu$ is kinematic viscosity, and $\mathbf{f}$ are body forces.

These are a set of four coupled PDEs that describe the behavior of electric and magnetic fields:

• $\nabla \cdot \mathbf{E} = \frac{ρ}{\epsilon₀}$ (Gauss's Law for Electricity)
• $\nabla \cdot \mathbf{B} = 0$ (Gauss's Law for Magnetism)
• $\nabla \times \mathbf{E} = -\frac{∂\mathbf{B}}{∂t}$ (Faraday's Law of Induction)
• $\nabla \times \mathbf{B} = \mu₀\mathbf{J} + \mu₀\epsilon₀ \frac{∂\mathbf{E}}{∂t}$ (Ampère-Maxwell Law)

$\frac{∂V}{∂t} + rS \frac{∂V}{∂S} + \frac{1}{2} σ²S² \frac{∂²V}{∂S²} - rV = 0$
This PDE is used for pricing options. $V$ is the option price, $S$ is the underlying asset price, $t$ is time, $r$ is the risk-free interest rate, and $σ$ is volatility.

$\frac{∂u}{∂t} + 6u \frac{∂u}{∂x} + \frac{∂³u}{∂x³} = 0$
This non-linear PDE models shallow water waves and is famous for its soliton solutions.