Separation of variables and Fourier series

The computation that we did for the diffusion equation with linear population growth is an example of a Fourier series solution of the diffusion equation. The way that we computed amplitudes of the different modes of cosine is an example of how one finds the Fourier coefficients. The method of Fourier series is an extremely powerful one and is used in many different ways in applied mathematics. A good introductory book on partial differential equations will explain how the method works in general; see, for example, Haberman (1998). In biological systems, we may have different boundary conditions, depending upon the situation (e.g. the size of a cell or a region and the nature of transport across the boundary). Some of my favorite investigations in this area involve the interaction of boundary conditions and the resulting patterns (Keller 2002, Murray 2003).

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