Browsing by Author "Masenge, Ralph Peter and Malak, Sospeter Shikulu"

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Finite Difference and Shooting Methods for Two-Point Boundary Value Problems: A Comparative Analysis
(MUST Journal of Research and Development (MJRD), 2020-05-03) Masenge, Ralph Peter and Malak, Sospeter Shikulu
Finite difference and shooting methods are popular with numerical practitioners in solving two-point boundary value problems governed by ordinary differential equations. However, the available literature is silent on which method is the most suitable with respect to accuracy, efficiency, stability, and convergence. In this article, finite difference and shooting methods are applied to solve numerically three types of two-point boundary value problems. One problem is governed by a linear non-stiff differential equation, a second problem is governed by a linear stiff differential equation, and a third problem is governed by a non-linear differential equation. The analytical solution of each problem is given. These solutions are used in assessing the accuracy attained by each of the numerical methods. It is known a prior that finite difference schemes based on central difference quotients in approximating derivative terms are numerically stable. Numerical experiments carried out on the three problems lead to the conclusion that, for linear non- stiff equations, linear shooting gives significantly more accurate results compared to finite difference methods. However, in the case of linear stiff equations, the finite difference method gives very accurate results while the shooting method fails totally, displaying serious instability. As for problems governed by nonlinear equations, although both finite difference and shooting methods converge, the shooting method converges significantly faster than its counterpart. With respect to the attribute of efficiency, finite difference methods are more efficient than shooting methods due to the ease with which the Runge - Kutta initial value problem solver can be applied.