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!

- 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: $\vec{\nabla}\vec{E}=$ $\nabla^2\phi=0$

- there is still an

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 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.

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:
$$\nabla^2r=a\frac{\partial^2r}{\partial t^2}$$

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.