10.3.1 Introduction

We can solve the system of four first order ordinary differential equations (10.17) to (10.20) as an initial value problem (IVP), where all four boundary conditions are given at one point, or as a boundary value problem (BVP), where four boundary conditions are specified at two distinct points. Most of the problems that arise in applications of geodesics are not IVP but BVP, which are much more difficult to solve. It is well known that the solution of an IVP is unique, however for a BVP it is possible that the differential equations have many solutions or even no solution [194]. General methods for the solutions of two-point BVPs can be found in [194,117].

It is convenient to write the system of differential equations in vector form, since we can describe the equations for systems in terms of a single vector equation. Let us set

where , are functions and , are constants. Then the general first order vector differential equation for a boundary value problem can be written as:

There are two commonly used approaches to the numerical solution of BVPs. The idea of the first technique is that if all values of are known at , then the problem can be reduced to an IVP. However, can be found only by solving the problem. Therefore an iterative procedure must be used. We assume values at , which are not given as boundary conditions at and compute the solution of the resulting IVP to . The computed values of will not, in general, agree with the corresponding boundary condition at . Consequently, we need to adjust the initial values and try again. The process is repeated until the computed values at the final point agree with the boundary conditions and is referred to as shooting method. The second method is based on a finite difference approximation to on a mesh of points in the interval . This method starts with an initial guess and improves the solution iteratively and is referred to as, direct method, relaxation method or finite difference method. We have implemented both methods and found that the finite difference method is much more reliable than the shooting method. By contrast to the finite difference method, the shooting method is often very sensitive to the unknown initial values at point . First, we briefly discuss the shooting method.