11 - Heat Equation

Summary

The fact that the space of square integrable functions on a finite interval is quite the same as the space of square integrable sequences provides a way to solve the heat equation, one of the fundamental equations of mathematical physics (and of the theory of stochastic processes). As originally posed in the former space, the equation seems to be rather difficult. But the isomorphism between these spaces transforms the equation into a series of ordinary differential equations with constant coefficients, and these can be solved explicitly. On the level of calculations, we are simply using the well-known method of separation of variables of the theory of partial differential equations; more intrinsically, however, we are looking at the method from a proper perspective, the perspective of Hilbert spaces.