Nonlinear initial value ordinary differential equations

Abstract

Ordinary frequently occur as mathematical models in many fields of science, engineering, differential equations (ODEs) and economy. It is rarely that ODEs have closed form analytical solutions, so it is common to seek approximate solutions by means of numerical methods. One of the most useful methods for the solution of ODEs is multi-step methods. Although the existing multi-step methods such as Adams—Moulton are accurate and useful, they also have their own limitations such as instability at large step sizes or weak performance in the case of stiff ODEs. Thus, multi-step methods that show better behavior compared to the existing methods are preferred, because they decrease the computational effort needed to achieve results with the desired order of accuracy. In this chapter, a new multi-step formula is presented for the solution of linear and nonlinear ODEs based on the Bézier curves. Conventional convergence and stability analyses show that the presented formula is convergent and stable. Accuracy and stability of the presented formula are investigated through some numerical examples in comparison with the existing methods.