PSEM Approximations for Both Branches of Lambert W Function with Applications

Abstract

Transcendental functions are a fundamental building block of science and engineering. Among them, a relatively new function denominated as Lambert W is highlighted. The importance of such function relies on the fact that it can perform novel isolation of variables. In this work, we propose two accurate piece-wise approximate solutions, one for the lower branch and another one for the upper branch, respectively. The proposed analytic approximations are obtained by using the power series extender method (PSEM) in combination with asymptotic solutions. In addition, we will compare some published approximations with our proposal, highlighting our advantages in terms of significant digits and speed of evaluation. Furthermore, the approximations are validated by the successful simulation of a problem of economy and other acoustic waves of nonlinear ions.

The novel family of transcendental Leal-functions with applications to science and engineering

“…respectively, where Taylor coefficients are expressed in terms of parameters . Finally, we equate/match the coefficients of power series either in (12) or (13) with the correspondent ones of (9) to obtain the values of and substitute them into (10) or (11), to obtain the PSEM approximation [63,64,65,66].…”

“…In order to solve transcendental equations based on the definition of Leal-functions (see Section 4), we will employ the approximations of Table 4 to provide the initial value 0 for Halley's method. Figs. 14 and 15 present the significant digits (SD) (as reported by [64]) for two iterations of Halley's method. It is important to remark that iteration zero means that we use directly the approximations of Table 4 to provide an initial approximation.…”

“…In fact, for some approximations after two iterations, the number of SD reach more than 40 digits. Therefore, users can increase the number of SD [64] for the approximations in Table 4 by increasing the number of Halley's iterations.…”

“…Such approximate functions have the advantage of being handy and easy computable, providing highly accurate results. These expressions will be obtained by using Power Series Extender Method (PSEM) [63,64,65,66], which is simple to use and does not require either integral calculations or existence of perturbative parameters, among other characteristics.…”

Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function: Reply to Discussion

“…To avoid misinterpretations, here we present the Colebrook equation expressed directly through the Wright ω-function; Equation (3) (while in [1] it was through the Lambert W-function [30]; Equation (2) of [1]; here Equation (1)).…”