Fourier Series in Diffusion and Wave Phenomena

Abstract

One of the main fields of application of Fourier series is in finding the solution of processes governed by linear partial differential equations where the space derivative is the Laplacian. In such processes, it is the local curvature of the disturbance which is subject to the time development as determined by the time derivatives. If the latter is a first-order derivative, we have the diffusion equation [Eq. (5.1)], where the rate of change in temperature is proportional to its local curvature. In the wave equation [Eq. (5.15)], it is the acceleration, the second time derivative, which responds linearly to the disturbance curvature. If the boundary conditions are periodic with some period 2L, Fourier series will provide an expansion of the solution in terms of a basis of Laplacian eigenfunctions with exactly these periodicity conditions.