Project 1 : Introduction to PDEs and Matlab pde toolbox

Definition of a PDE

A Partial Differential Equation (PDE) is an equation in which

• the unknown $u(x,y,z)$ is a multivariable function
• at least one term in the equation must have some partial derivative of the unknown

Examples

Poisson’s equation [Wiki]

$$\Dt \phi = f$$

Heat Equation, Diffusion Equation (Wiki]

$$\frac{\p u}{\p t} -\alpha \left(\frac{\p^2 u}{\p x^2} + \frac{\p^2 u}{\p y^2} + \frac{\p^2 u}{\p z^2} \right) = 0$$

Convection and diffusion equation [Wiki]

$$\frac{\partial c}{\partial t} = \nabla \cdot (D \nabla c) - \nabla \cdot (\vec{v} c) + R$$

Wave equation [Wiki]

${ \partial^2 u \over \partial t^2 } = c^2 \nabla^2 u$

Equations of Mechanical Engineering [Link]

PDE is air pollution study [Link]

Navier-Stokes Equation [Link]

Typical structure of a PDE problem:

Typically, we will be looking for a function $u$ which

• satisfies a given PDE in the interior of a given domain $\Om$
• and some information about $u$ will be given at the boundary of the domain $\p \Om$.