One-dimensional diffusion problems general

Let us consider the diffusion of a homogeneously distributed substance out of a slab with unit thickness in the direction x and infinite size in the other two dimensions. C(x, t) being the concentration at time t at a distance x from the origin, the diffusion equation is 

The first step is to discretize the (x, t) domain using a uniform mesh 

From this definition, the second-order difference is 

In the finite difference schemes, the derivatives are replaced by difference operators, e.g., the first derivative 

Rewriting the diffusion equation into a partial difference equation leads to 

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