Perhaps the simplest boundary value problem can be represented by the conditions

(10.15)

where and A and B are given constants. It is theoretically always possible to reduce the solution of a boundary value problem to the solution of a sequence of initial value problems.

Let denote the solution of the initial value problem resulting from the above problem by replacing the condition for by the condition , where is a parameter.

The above boundary value problem is then equivalent to solving the (in general non linear) equation for . This can be effected by one of the standard methods such as Newton's method.

Each evaluation of the function requires the solution of an initial value problem. The above ‘shooting' technique nevertheless may represent a feasible procedure. However, if the systems of differential equations are involved or if the initial value problems show signs of instability, other direct procedures may be preferable.

Even for the simple boundary value problem considered above it may happen that there are infinitely many solutions-as in the problem

, for which is a solution for arbitrary or that there is no solution, as in the problem

We now discuss a direct method based on implicit difference approximations for solving a class of non-linear boundary value problems of the second order.