Application of Spectral Techniques in Solving Diffusion Problems

Abstract

The distributed RC line is of great importance; pretty much in all cases where inductance is not relevant, such as low speeds we end up with a distributed RC line. This line is governed by the diffusion equation which is second order in space and first order in time. The diffusion equation is solved by the method of separation of variables, where one part depends on position while the other on time. It is the spatial solution that inherits spectral techniques and is solved for using a Fourier series. To get the final solution we need both initial conditions and boundary conditions. In the chapter we try some examples with variants of ICs and BCs. For each case we extract the solution and compare favorably with SPICE. As expected the more harmonics we add, the better the solution. But really the most interesting case is where the boundary conditions are time-dependent; there we throw in yet another Fourier series, but this time in time! So we have harmonic expansion both in space and time; and as shown in the examples it all adds up and we get excellent results.