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

Abstract

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 22/3=1.587 with three function evaluations and over the bound 21/2=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 Dzˇunic´ 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.

Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations

“…The many decimal numbers after this point guarantees that x c is a solution of Equation ( 8), thus satisfying the error tolerance with ε = 10 −15 . Next, we solve Equations ( 1) and ( 2) with F = 3u 2 /2, a = 4, and b = 1, and compare the results with the solution u(y) = 4/(y + 1) 2 . In Equation (18), we take H(t) = q 0 (1 − e −t ), and with a 0 = 2.27615, b 0 = −0.149978, and q 0 = −0.15 through eight iterations, we obtain the slope u ′ (0) = −8.000000000000028, which is very close to the exact one r = −8; 5.33 × 10 −15 is the maximum error obtained for u(y).…”

“…where F(y, u(y), u ′ (y)) : [0, 1] × R 2 → R is a given nonlinear continuous function, and a and b are given constants. We integrate Equation (1), starting from the initial values u(0) = a and u ′ (0) = x, where x is an unknown value determined by u(1) = b in Equation (2), which results in a nonlinear equation:…”

“…is cubically convergent if a 0 = f ′ (r) and b 0 = f ′′ (r)/(2 f ′ (r)), where f (r) = 0 and f ′ (r) ̸ = 0. The parameters a 0 and b 0 can be updated to speed up the convergence by the memory-dependent method [2]. One can refer to [3][4][5][6][7][8][9][10] for more memory-dependent iterative methods.…”

“…By resulting in x 0 and γ 0 and incorporating the memory of x n , Traub's iterative scheme can proceed to find the solution of f (x) = 0 upon convergence. For a recent report on the progress of the memory method with accelerating parameters in the one-step iterative method, one can refer to [2], while for a recent progress report on the memory method with accelerating parameters in the two-step iterative method, one can refer to [11]. One major goal of the paper is the development of multi-step iterative schemes with a new memory method to determine the accelerating parameters by updating the technique with information at the current step.…”

New Memory-Updating Methods in Two-Step Newton’s Variants for Solving Nonlinear Equations with High Efficiency Index

“…In this nonlinear problem, when we apply the NM to solve f (x) = u(1, x) − 1 = 0, we encounter a difficulty to calculate f ′ (x). Recently, Liu et al [9] proposed a single-step memory-dependent method to solve f (x) = 0 by a Lie-symmetry formulation of Equation (18). Consider the following one [10]:…”