Finite Difference Method:

For the direct numerical solution of a boundary value problem of class , we introduce the points , where and is an appropriate integer. A scheme is then designed for the determination of numbers which approximate the values of the true solution at the point .

The natural way to obtain such a scheme is to demand that the satisfy at each interior mesh point a difference equation

(10.22)

Again the coefficients are chosen in such a way that the associated difference operator is small for a solution of . One difficulty arises here in the difference scheme. As in any algebraic problem, we need as many equations for the determination of the unknowns as there are unknowns. Since and are determined by the boundary condition, the unknowns in our case are . If the step number of the difference equation , new unknown values such as are introduced for which there is no equation.

This difficulty can be circumvented by suitably modifying the difference equations near the boundary points; it does not arise at all if , the smallest possible value. If in (10.22) and if the associated–difference operator has order p, the difference equation is necessarily proportional to an equation of the form

(10.23)

where . The difference equations most frequently used for boundary value problems are.

(10.24)

and

(10.25)