The periodic heat kernel is simply the periodization of the standard Gaussian heat kernel on the real line. Its explicit form is:
where is the period and is the normalized Gaussian. Given the homogeneous heat equation on a finite interval with homogeneous Dirichlet, Neumann, or mixed boundary conditions, the heat kernel for the problem can be expressed in terms of the periodic heat kernel via the method of reflection. For example, in the case of Dirichlet boundary conditions,
Integrating the resulting kernel in the variable against the initial data provides the solution to the initial value problem. This form of the heat kernel is particularly useful when is small, in which case it approximates a delta function.