Finite-difference expressions of derivatives

As we have seen, diffusion equations are differential equations. These can be discretised into a series of finite-difference equations and these solved by simple arithmetic. The idea behind this can be simply illustrated as in Fig. 3.1. Suppose we have a curve y(t) for which we know only the first derivative dy/dt. 

Suppose we have already generated the curve up to the point P at t and now wish to find the next point Q at t2 = t + 5t. If we know dy/dt at P (the tangent drawn) we may simply move along the tangent to t2 and call this Q or y(t2). If St is sufficiently small, this will be a reasonably good approximation. The procedure just described is called the Euler method. 

A better approximation would result if we could somehow obtain the slope of the chord joining P and Q, and move along it instead. As will be shown, something like this can in fact be done. The two approaches are, essentially, what constitutes the finite-difference solution of diffusion equations. The different means of arriving at the working expressions will now be examined. 

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