Partial differential equations

 

A partial differential equation (PDE) is an equation that involves multiple variables and their partial derivatives. PDEs are used to model many physical phenomena, such as fluid flow, heat transfer, and electromagnetic fields.

There are many types of PDEs, each with its own specific form and set of characteristics. Some common types of PDEs include:

The heat equation: This PDE describes the diffusion of heat in a solid, and is given by the equation ∂u/∂t = α ∇²u, where u is the temperature, t is time, and α is a constant.

The wave equation: This PDE describes the motion of waves, such as sound or light, and is given by the equation ∂²u/∂t² = c² ∇²u, where u is the displacement, t is time, and c is the wave speed.

The Laplace equation: This PDE is a special case of the Poisson equation and appears in many areas of physics and engineering. It is given by the equation ∇²u = 0, where u is a scalar function.

The Navier-Stokes equation: This PDE describes the motion of fluid, given by the equations ∂u/∂t + (u.∇)u = -1/ρ ∇p + μ ∇²u. where u is the velocity, t is time, p is the pressure, ρ is the density, and μ is the dynamic viscosity.

Solving a PDE often involves finding a function that satisfies a set of boundary conditions. This can be a difficult task and may not always have an exact solution, therefore numerical methods are employed.