14.2 The Diffusion Equation

The differential equation that describes the one-dimensional diffusion problem is given by

where u( x) is the unknown function, c( x) > 0 a given material characteristic function and f( x) a given source term. This differential equation is defined on a onedimensional domain ? that spans the x-axis between x = a and x = b while the boundary, which is formally denoted by ?, is located at x = a and x = b.

Eq. (14.1) is an adapted form of the diffusion equation Eq. (13.43), introduced in the previous chapter. Different symbols for the unknown ( u instead of ?) and coefficient ( c instead of D) are used to emphasize the general character of the equation, applicable to different kinds of problems (see below). Furthermore, the coefficient c can be a function of x and a source term f( x) is introduced.

Two types of boundary conditions can be discerned. Firstly, the essential boundary condition, which must be specified in terms of u. For the derivations that follow the boundary at x = a is chosen, to specify this type of boundary condition:

where ? _u denotes the boundary of the domain ? at x = a. Secondly, a natural boundary condition may be specified. Here the boundary at x