An Explicit Single-Step Nonlinear Numerical Method for First Order Initial Value Problems (IVPs)

Abstract

Interest in the construction of efficient methods for solving initial value problems that have some peculiar properties with it or its solution is recently gaining wide popularity. Based on the assumption that the solution is representable by nonlinear trigonometric expressions, this work presents an explicit single-step nonlinear method for solving first order initial value problems whose solution possesses singularity. The stability and convergence properties of the constructed scheme are also presented. Implementation of the new method on some standard test problems compared with those discussed in the literature proved its accuracy and efficiency.

A Sixth-Order Explicit Single-Step Nonlinear Numerical Method

“…Here, the absolute errors of the results obtained by our proposed method are first compared with those of the derivativefree methods proposed in [12] and [16] as shown in Figure 2. A comparison of the maximum absolute error obtained by the proposed method against those respectively produced by the methods of the authors in [14,4,12,10,8,16] is a also presented in Figure 3. The second test problem considered is given as…”

“…The use of rational functions in the local representation of theoretical solutions in the neighbourhood of singularity is the underlying assumption in the construction of nonlinear schemes [18,13,12]. Interestingly, this approach is now gaining popularity as several methods are now being constructed using under this assumption [14,4,1,10].…”

“…Interestingly, this approach is now gaining popularity as several methods are now being constructed using under this assumption [14,4,1,10]. The works of the authors in [12,13,16,18,1] showed that solution around singularity point are well approximated by this approach. In this work, a sixth-order explicit single-step nonlinear method for solving 1 is presented.…”