Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem

In this paper, some one-step iterative schemes with memory-accelerating methods are proposed to update three critical values f '(r), f ''(r), and f '''(r) of a nonlinear equation f(x) = 0 with r being its simple root. We can achieve high values of the efficiency index (E.I.) over the bound 2(2/3) = 1.587 with three function evaluations and over the bound 2(1/2) = 1.414 with two function evaluations. The third-degree Newton interpolatory polynomial is derived to update these critical values per iteration. We introduce relaxation factors into the Dzunic method and its variant, which are updated to render fourth-order convergence by the memory-accelerating technique. We developed six types optimal one-step iterative schemes with the memory-accelerating method, rendering a fourth-order convergence or even more, whose original ones are a second-order convergence without memory and without using specific optimal values of the parameters. We evaluated the performance of these one-step iterative schemes by the computed order of convergence (COC) and the E.I. with numerical tests. A Lie symmetry method to solve a second-order nonlinear boundary-value problem with high efficiency and high accuracy was developed.