Partial differential equations in Physics :: Maths for Physicists and vice versa

It is surprising how many problems in physics can be solved with a fairly small number of fairly straightforward PDE. Four of them are introduced on this page and subsequently solved on the next pages. The same equation often comes up in different contexts - but of course with different symbols used. It is important to recognise the type of equation irrespective of what letters appear!

Electrostatics

in a charge-free region:
- there is still an electric potential defined: $\phi$
- the corresponding electrical field isn't necessarily zero: $\vec{E}=\left(\frac{\partial\phi}{\partial x},\frac{\partial\phi}{\partial y},\frac{\partial\phi}{\partial z}\right)$
- but it is constant across space: $\vec{\nabla}\vec{E}=$ $\nabla^2\phi=0$

The highlighted PDE is called Laplace's equation.
The del operator (the upside-down triangle) is the derivative along the steepest gradient.

in a region containing charges:
- potential: $\phi$
- electric field: $\vec{E}=\vec{\nabla}\phi$
- which changes due to presence of charges: $\vec{\nabla}\vec{E}=$ $\nabla^2\phi=f(x,y,z)$

The highlighted PDE is called Poisson's equation.

The same equations apply also

in mechanics: replace electric potential with gravitational potential
in thermodynamics: replace electric potential with temperature

Diffusion

Diffusion is the change of a concentration, $c$, profile over time, $t$: $$\nabla^2c=a\frac{\partial c}{\partial t}$$

This PDE is called the diffusion equation.

The same equation is useful in thermodynamics, where it describes the flow of heat over time. The time-dependent Schrödinger equation in quantum mechanics follows the same pattern also.

Acoustics

The vibration of a membrane follows an oscillating pattern both in space and in time. Generally, in mechanics, the motion of a pendulum and of a vibrating string follow the same pattern. In electrodynamics, the exchange of energy between the electric and the magnetic field can be described in the same way.

Whenever energy is periodically transferred back and forth between two different forms (e.g. potential - kinetic or electric - magnetic), the wave equation is used to describe this process: $$\nabla^2r=a\frac{\partial^2r}{\partial t^2}$$

Summary

All of the above are 2nd order PDE. The left-hand side is always a second spatial derivative, but the right-hand side differs:

Laplace's eq.	$\nabla^2u=0$	homogeneous
Poisson's eq.	$\nabla^2u=f(x,y,z)$	heterogeneous, but no differentials in the heterogeneous term
diffusion eq.	$\nabla^2u=a\frac{\partial u}{\partial t}$	1st order time differential in heterogeneous term
wave eq.	$\nabla^2u=a\frac{\partial^2u}{\partial t^2}$	2nd order time differential in heterogeneous term

Before solving Laplace's equation, we'll review the del operator in the next section.