Finite Difference Approach

We will use the Finite Difference approach in solving these equations. In this approach the solution of the equations is sought on an equally space lattice in both space and time.

The Taylor series expansion of a function near x₀ reads

$$f(x_0+\delta) = f(x_0) + \delta \left. \frac{d f}{d x }\right... ...t. \frac{\delta^2}{2!} \frac{d^2 f}{d x^2 } \right\vert _{x_0}$$ (1)

and

$$f(x_0-\delta) = f(x_0) - \delta \left. \frac{d f}{d x }\right... ...t. \frac{\delta^2}{2!} \frac{d^2 f}{d x^2 } \right\vert _{x_0}$$ (2)

at symmetrical distances $x_0 \pm \delta$ about x₀. Solving for the derivative terms yields

$$\frac{d f}{d x} = \frac{f(x_0+\delta) - f(x_0)}{\delta}$$ (3)

$$\frac{d f}{d x} = \frac{f(x_0) - f(x_0-\delta)}{\delta}.$$ (4)

The neglected terms are of order O(h²).

A more accurate form follows by combining the Taylor series expansions with the function at x₀:

$$\frac{d f}{d x} = \frac{f(x_0+\delta) - f(x_0-\delta)}{2 \delta}$$ (5)

and

$$\frac{d^2 f}{d x^2 } = \frac{f(x_0+\delta) - 2 f(x_0) + f(x_0-\delta)} {\delta^2}$$ (6)