Finite-difference solution by the explicit method

We consider for now the one-dimensional diffusion equation, with constant diffusion coefficient D  

Such an equation is useful for describing the time evolution of the concentration profile of some diffusant across a plane sheet of given thickness L and infinite transverse extension. In order to model a particular experimental arrangement, this equation must be solved in conjunction with certain initial and boundary conditions. We will consider that Eq. (8-1) is subject to the initial condition  

Such boundary conditions, specifying the values of the solution, are known as Dirichlet boundary conditions. The so-called Neumann boundary conditions, which define the derivative of the solution on the boundaries, form another important category, considered among others later in this chapter. 

Here M represents the number of spatial gridpoints and the spatial mesh constant 

Taking the linear approximation and expressing the first order time derivative, one obtains  

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