One-dimensional finite medium and constant D, separation of variables

7 One-dimensional finite medium and constant D, separation of variables 

When the medium is finite, there will be two boundaries in the case of onedimensional diffusion. This finite one-dimensional diffusion medium will also be referred as plate sheet (bounded by two parallel planes) or slab. The standard method of solving for such a diffusion problem is to separate variables x and t when the boundary conditions are zero. This method is called separation of variables. As will be clear later, the method is applicable only when the boundary conditions are zero. 

Starting from Equation 3-10, assume that the function C(x, t) may be separated into the product of a function that depends only on x, (x), and a function that depends only on t, x(t). Then 

Because by assumption the left-hand side is a function of t only (independent of x) and the right-hand side is a function of x only (independent of t), they cannot 

From Equation 3-50f, it can be seen that the constant defined as must be 

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