Analytical approximations to the Lambert W function

Wu, Biao; Zhou, Yixin; Lim, C.W.; Zhong, Huanhuan

doi:10.1016/j.apm.2021.11.024

Cited by 10 publications

(4 citation statements)

References 14 publications

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“…A more generalized solution is The Lambert W function is easily computed using built-in algorithms in Mathematica (i.e ProductLog function), MATLAB ( W function) and Python ( lambertw function) (mathematica code available at ; python code available at ). There are several published analytical approximations to the Lambert W functions that could also be used. , The time dependence of [O 3(b) ] t can then be obtained by replacing [AA (b) ] 0 in eq with [AA (b) ] t (b) from eq ), or more generally, …”

Section: Resultsmentioning

confidence: 99%

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A Kinetic Model for Predicting Trace Gas Uptake and Reaction

2022

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A model is developed to describe trace gas uptake and reaction with applications to aerosols and microdroplets. Gas uptake by the liquid is formulated as a coupled equilibria that links gas, surface, and bulk regions of the droplet or solution. Previously, this framework was used in explicit stochastic reaction–diffusion simulations to predict the reactive uptake kinetics of ozone with droplets containing aqueous aconitic acid, maleic acid, and sodium nitrite. With the use of prior data and simulation results, a new equation for the uptake coefficient is derived, which accounts for both surface and bulk reactions. Lambert W functions are used to obtain closed form solutions to the integrated rate laws for the multiphase kinetics; similar to previous expressions that describe Michaelis–Menten enzyme kinetics. Together these equations couple interface and bulk processes over a wide range of conditions and do not require many of the limiting assumptions needed to apply resistor model formulations to explain trace gas uptake and reaction.

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Section: Resultsmentioning

confidence: 99%

“…There are several published analytical approximations to the Lambert W functions that could also be used. 83,84 The time dependence of [O 3(b) ] t can then be obtained by replacing [AA (b) ] 0 in eq 14 with [AA (b) ] t(b) from eq 22),…”

Section: Definition Of K Transportmentioning

confidence: 99%

A Kinetic Model for Predicting Trace Gas Uptake and Reaction

2022

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“…Lambert W is a z ‐to‐ z ·exp( z ) multivalued inverse function and satisfies (Wu et al., 2022):

\begin{equation}z = W(z)\exp (W(z)).\end{equation}

…”

Section: Methodsmentioning

confidence: 99%

Seismic attribute transformation and porosity prediction of thin water‐rich sandstone based on Lambert W–R model

Li,

Chen,

Yin

et al. 2024

Geophysical Prospecting

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The seismic attributes of water‐rich sandstone contain much information about the rock's physical properties and seismic wave parameters. They are commonly used to predict the rock's physical properties (e.g., porosity). However, the seismic attributes of water‐rich sandstone are affected by porosity, water saturation, and thickness. To eliminate the influence of thickness on the porosity prediction of water‐rich sandstone and improve the accuracy of the porosity prediction, the authors propose a Lambert W‐R transform method to isolate the contribution of thickness and porosity from seismic attributes. Firstly, a rock physical model is used to calculate the equivalent elastic parameters of water‐rich sandstones with different porosity and water saturation. Secondly, the seismic attribute dataset of water‐rich sandstone is established by forward modeling the seismic response of the wedge‐shaped water‐rich sandstone model, and the selection of sensitive physical properties is completed. Then, the transformation parameters (ζAhRs and ηAhRs) are obtained by Lambert W‐R transformation, which is exponentially related to Instantaneous amplitude (Inst‐Amp). ζAhRs and ηAhRs are sensitive to thickness and porosity, respectively. Finally, an interpretative template for porosity prediction of water‐rich sandstone is established by cross‐plot analysis (ζAhRs and ηAhRs) and verified by a practical case. The verification results show that the porosity predicted by the interpretation template is consistent with drilling fluid consumption. However, it is lower than the porosity of logging constrained P‐wave impedance inversion.This article is protected by copyright. All rights reserved

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“…There are several published analytical approximations to the Lambert W functions that could also be used. 84,85 The time dependence of [O3(b)]t can then be obtained by replacing [AA(b)]0 in Eq. ( 19) with [AA(b)]t from Eq.…”

mentioning

confidence: 99%

A Kinetic Model for Predicting Trace Gas Uptake and Reaction

Wilson

Prophet

Willis

2022

Preprint

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A model is developed to describe trace gas uptake and reaction with applications to aerosols and microdroplets. Gas uptake by the liquid is formulated as a coupled equilibria that links gas, surface and bulk regions of the droplet or solution. Previously, this framework was used in explicit stochastic reaction-diffusion simulations to predict the reactive uptake kinetics of ozone with droplets containing aqueous aconitic acid, maleic acid and sodium nitrite. Using prior data and simulation results, a new equation for the uptake coefficient is derived, which accounts for both surface and bulk reactions. Lambert W functions are used to obtain closed form solutions to the integrated rate laws for the multiphase kinetics; similar to previous expressions that describe Michaelis–Menten enzyme kinetics. Together these equations couple interface and bulk processes over a wide range of conditions and don’t require many of the limiting assumptions needed to apply resistor model formulations to explain trace gas uptake and reaction.

show abstract

Analytical approximations to the Lambert W function

Cited by 10 publications

References 14 publications

A Kinetic Model for Predicting Trace Gas Uptake and Reaction

A Kinetic Model for Predicting Trace Gas Uptake and Reaction

Seismic attribute transformation and porosity prediction of thin water‐rich sandstone based on Lambert W–R model

A Kinetic Model for Predicting Trace Gas Uptake and Reaction

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