The slab with uniform initial concentration

When combined with the Fourier expansion of functions, separation of variables is another powerful method of solutions which is particularly useful for systems of finite dimensions. Regardless of boundary conditions, we decompose the solution C(x, t), where the dependence of C on x and t is temporarily emphasized, to the general one-dimensional diffusion equation with constant diffusion coefficient 

Inserting this expression into the diffusion equation gives 

Since the two members in the last equation cannot be functions of the independent variables x and t, they must be equal to a same constant, which suggests using an exponential form for/(f) and a trigonometric form for g(x). The diffusion equation is indeed identically verified for 

Such a sine expansion is generally made possible with the condition of zero concentration at x=0 by using an odd function for the initial concentration distribution. A simple example is for initial uniform concentration C0 between 0 and X for which we can assume a fictitious concentration — C0 between — X and 0. Using the results of Chapter 2, the Fourier expansion of the boxcar function which is C0 between 0 and X, and 0 at x=0 and x = X is 

Formulating the same problem in a symmetrical way, i.e., for a slab with —h x +h may happen to be occasionally convenient. Changing X into 2h and x into (y + h) would give 

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