Comparative analysis of the shooting method and finite difference method in solving two-point boundary value problems

Two-point boundary value problems (BVPs) governed by linear variable-coefficient or by non-linear differential equations are difficult to solve analytically. Solution of such problems are approximated using numerical methods. There are several numerical methods for solving two points BVPs but the two most popular methods are finite difference and shooting methods. The literature we have accessed on the subject is silent on which of these two methods to use in solving which type of problem with regards to accuracy, efficiency, stability, and convergence. In this study, finite difference and shooting methods are used in approximating the solution of three carefully selected two-point boundary value problems. The first problem is governed by a linear non-stiff differential equation; the second problem is governed by a linear stiff differential equation. The analytical solution of each problem is known, a factor which enables one to assess the degree of accuracy of the particular numerical method applied. The results of the numerical experiments show that finite difference discretization of the yields numerically stable finite difference schemes for solving any type of two-point boundary value problem. The results also show that shooting methods are stable except for problems governed by stiff differential equations. In general, shooting methods are more accurate than finite difference methods, except where the differential equation is stiff, in which case shooting leads to very inaccurate results. With regards to efficiency, finite difference methods are more efficient than shooting methods. In solving problems governed by nonlinear equations the two methods are competitive in that they have same rate of convergence depending on the rate of convergence of the particular technique chosen in the respective iteration processes.