Single Partial Differential Equations: Their Origin

There are many phenomena in nature, which, even though occuring over finite regions of space and time, can be described in terms of properties that prevail at each point of space and time separately. This description originated with Newton, who with the aid of his differential calculus showed us how to grasp a global phenomenon, for example, the elliptic orbit of a planet, by means of a locally applied law, for example $F=ma$ .

This manner of making nature comprehensible has been extended from the motion of single point particles to the behavior of other forms of matter and energy, be it in the form of gasses, fluids, light, heat, electricity, signals traveling along optical fibers and neurons, or even gravitation.

This extension consists of formulating or stating a partial differential equation governing the phenomenon, and then solving that differential equation for the purpose of predicting measurable properties of the phenomenon.

There exist many partial differential equations, but from the view point of mathematics, there are basically only three types of partial differential equations.

They are exemplified by

1. Laplaces equation
   $$\frac{\partial^2\psi}{\partial x^2} +\frac{\partial^2\psi}{\partial y^2} +\frac{\partial^2\psi}{\partial z^2} = 0\,,$$
   which governs electrostatic and magnetic fields as well as the velocity potential of an incompressible fluid, by
2. the wave equation
   $$\nabla^2\psi -\frac{1}{c^2}~\frac{\partial^2\psi}{\partial t^2} = 0$$
   for electromagnetic or sound vibrations, and by
   $$\frac{\partial^2\psi}{\partial x^2} -\frac{1}{c^2}~\frac{\partial^2\psi} {\partial t^2} = 0$$
   for the vibrations of a simple string, and by
3. the diffusion equation
   $$\nabla^2\psi -\frac{1}{k}~\frac{\partial\psi}{\partial t} = 0$$
   for the temperature in three dimensional space and in time, or by
   $$\frac{\partial^2\psi}{\partial x^2} -\frac {1}{k}~\frac{\partial\psi} {\partial t} = 0$$
   for the temperature along a uniform rod.