Henri Poincaré: On the Partial Differential Equations of Mathematical Physics – Law of cooling of an isolated solid body in space

Suppose we are looking for the law of cooling of an isolated solid body in space. It will be a question of finding a function V satisfying the equation

dV/dt = kΔV,

and which moreover is given for t = 0. Finally at the surface of the body the ratio of V at dV/dn is given.

In the problems of optics, there are three unknown functions u, v, w, and four equations:

d²u/dt² = kΔu , d²v/dt² = kΔv, d²w/dt² = kΔw, du/dx + dv/dy + dw/dz = 0

The boundary conditions vary according to the problems; but in questions of diffraction principally they are not without analogy with those which we have encountered hitherto.

It is again the same equations, with analogous boundary conditions, although different, that one encounters in the problem of the viscosity of liquids, treated according to the ideas of Navier. The recent works of M. Oouette have drawn attention to this question, which had fallen into unjust oblivion, despite the beautiful chapter that Kirchholf had devoted to it in his Mathematical Physics.

Source: Henri Poincaré, Sur les Equations aux Dérivées Partielles de la Physique Mathématique, American Journal of Mathematics, Mar., 1890, Vol. 12, No. 3 (Mar., 1890), pp. 211-294. Translation by Nicolae Sfetcu. © 2022 Nicolae Sfetcu