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PSEM approximations for both branches of Lambert \(W\) function with applications. (English) Zbl 1453.33017

Summary: 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.

MSC:

33F05 Numerical approximation and evaluation of special functions
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[1] Milton, A.; Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1964), Washington, DC, USA: National Bureau of Standards, Washington, DC, USA · Zbl 0171.38503
[2] Whittaker, E. T.; Watson, G. N., A Course of Modern Analysis (1996), Cambridge University Press · Zbl 0951.30002 · doi:10.1017/cbo9780511608759
[3] Olver, F. W. J.; Lozier, D. W.; Boisvert, R. F.; Clark, C. W., NIST Handbook of Mathematical Functions Hardback and CD-ROM (2010), Cambridge University Press · Zbl 1198.00002
[4] Kline, M., Mathematical Thought From Ancient to Modern Times, 3 (1972), New York, NY, USA: Oxford University Press, New York, NY, USA · Zbl 0277.01001
[5] Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Knuth, D. E., On the Lambert W function, Advances in Computational Mathematics, 5, 1, 329-359 (1996) · Zbl 0863.65008 · doi:10.1007/BF02124750
[6] Brito, P. B.; Fabiao, F.; Staubyn, A., Euler, Lambert, and the Lambert W-function today, The Mathematical Scientist, 33, 2, 127-133 (2008) · Zbl 1159.33306
[7] Veberic, D., Having fun with Lambert W(x) function, https://arxiv.org/abs/1003.1628
[8] Veberič, D., Lambert W function for applications in physics, Computer Physics Communications, 183, 12, 2622-2628 (2012) · doi:10.1016/j.cpc.2012.07.008
[9] Johansson, F., Computing the Lambert W function in arbitrary-precision complex interval arithmetic, https://arxiv.org/abs/1705.03266 · Zbl 1477.65047
[10] Chapeau-Blondeau, F.; Monir, A., Numerical evaluation of the Lambert \(W\) function and application to generation of generalized Gaussian noise with exponent 1/2, IEEE Transactions on Signal Processing, 50, 9, 2160-2165 (2002) · Zbl 1369.33022 · doi:10.1109/TSP.2002.801912
[11] Barry, D. A.; Parlange, J.-Y.; Li, L.; Prommer, H.; Cunningham, C. J.; Stagnitti, F., Analytical approximations for real values of the Lambert \(W\)-function, Mathematics and Computers in Simulation, 53, 1-2, 95-103 (2000) · doi:10.1016/S0378-4754(00)00172-5
[12] Iacono, R.; Boyd, J. P., New approximations to the principal real-valued branch of the Lambert \(W\)-function, Advances in Computational Mathematics, 1-34 (2017) · Zbl 1381.33020 · doi:10.1007/s10444-017-9530-3
[13] Jadlovska, I., Application of Lambert W function in oscillation theory, Acta Electrotechnica et Informatica, 14, 1, 9-17 (2014) · doi:10.15546/aeei-2014-0002
[14] Polya, G.; Szego, G., Problems and Theorems in Analysis (1972), Springer-Verlag · Zbl 0236.00003 · doi:10.1007/978-3-642-61905-2
[15] Wright, E. M., Xl the linear difference-differential equation with constant coefficients, Proceedings of the Royal Society of Edinburgh, Section: A Mathematics, 62, 4, 387-393 (1949) · Zbl 0033.12002 · doi:10.1017/S0080454100006804
[16] Wright, E. M., Xii solution of the equation \(z e^z\), Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 65, 2, 193-203 (1959) · Zbl 0093.27204 · doi:10.1017/S0080454100032271
[17] Hayes, N. D., Roots of the transcendental equation associated with a certain difference-differential equation, Journal of the London Mathematical Society, 1, 3, 226-232 (1950) · Zbl 0038.24102
[18] Fritsch, F. N.; Shafer, R. E.; Crowley, W. P., Algorithm 443: solution of the transcendental equation \(w e^W = x\) [c5], Communications of the ACM, 16, 2, 123-124 (1973) · doi:10.1145/361952.361970
[19] Barry, D. A.; Barry, S. J.; Culligan-Hensley, P. J., Algorithm 743: Wapr-a fortran routine for calculating real values of the w-function, ACM Transactions on Mathematical Software (TOMS), 21, 2, 172-181 (1995) · Zbl 0886.65011 · doi:10.1145/203082.203088
[20] Corless, R. M.; Gonnet, G. H.; Hare, D. EG.; Jeffrey, D. J.; Knuth, D., Lambert’s W function in Maple, Maple Technical Newsletter, 9, 1, 12-22 (1993)
[21] Jeffrey, D. J.; Hare, D. E. G.; Corless, R. M., Unwinding the branches of the Lambert \(W\) function, The Mathematical Scientist, 21, 1, 1-7 (1996) · Zbl 0852.33001
[22] Corless, R. M.; Jeffrey, D. J.; Knuth, D. E., A sequence of series for the Lambert \(W\) function, Proceedings of the 1997 international symposium on Symbolic and algebraic computation, ACM · Zbl 0916.65015 · doi:10.1145/258726.258783
[23] Hoorfar, A.; Hassani, M., Inequalities on the Lambert \(W\) function and hyperpower function, Research Report Collection, 10, 2 (2007) · Zbl 1163.33326
[24] Fukushima, T., Precise and fast computation of Lambert \(W\)-functions without transcendental function evaluations, Journal of Computational and Applied Mathematics, 244, 77-89 (2013) · Zbl 1260.65013 · doi:10.1016/j.cam.2012.11.021
[25] Goličnik, M., On the Lambert W function and its utility in biochemical kinetics, Biochemical Engineering Journal, 63, 116-123 (2012) · doi:10.1016/j.bej.2012.01.010
[26] Jeffrey, D. J.; Jankowski, J. E., Branch differences and Lambert W, Proceedings of the 16th international symposium on symbolic and numeric algorithms for scientific computing, SYNASC 2014
[27] Jeffrey, D. J., Branch structure and implementation of Lambert W, Mathematics in Computer Science, 11, 3-4, 341-350 (2017) · Zbl 1425.68465 · doi:10.1007/s11786-017-0320-6
[28] Kheyfits, A. I., Explicit solutions of transcendental equations and the Lambert W function, AMS subject classifications. 33B99, 33F05, 65H05, 30D99, 2003
[29] Dence, T. P., A Brief Look into the Lambert W Function, Applied Mathematics, 4, 6, 887-892 (2013) · doi:10.4236/am.2013.46122
[30] Mezö, I.; Baricz, Á., On the generalization of the Lambert \(W\) function, Transactions of the American Mathematical Society, 369, 11, 7917-7934 (2017) · Zbl 1375.33034 · doi:10.1090/tran/6911
[31] Schnell, S.; Mendoza, C., Closed form solution for time-dependent enzyme kinetics, Journal of Theoretical Biology, 187, 2, 207-212 (1997) · doi:10.1006/jtbi.1997.0425
[32] Jenn, D. C., Applications of the Lambert W function in electromagnetics, IEEE Antennas and Propagation Magazine, 44, 3, 139-142 (2002) · doi:10.1109/MAP.2002.1039394
[33] Ortiz-Conde, A.; García Sánchez, F. J.; Muci, J., Exact analytical solutions of the forward non-ideal diode equation with series and shunt parasitic resistances, Solid-State Electronics, 44, 10, 1861-1864 (2000) · doi:10.1016/S0038-1101(00)00132-5
[34] Shynk, J. J., Mathematical Foundations for Linear Circuits and Systems in Engineering (2016), John Wiley & Sons · Zbl 1330.93001
[35] Tripathy, M. K. M.; Sadhu, P. K., Photovoltaic system using Lambert W function-based technique, Solar Energy, 158, 432-439 (2017)
[36] Casey, M. A., Reduced-rank spectra and minimum entropy priors for generalized sound recognition, Proceedings of the workshop on consistent and reliable cues for sound analysis
[37] Valluri, S. R.; Corless, R. M., Some applications of the Lambert W function to physics, Canadian Journal of Physics, 78, 9, 823-831 (2000)
[38] Bartolini, R.; Botman, J. I. M.; Migliorati, M.; Thomas, C., Longitudinal beam distribution, in a storage ring with pure inductance described by the Lambert W function, Proceedings of the EPAC
[39] Coll, B., A universal law of gravitational deformation for general relativity, Proceedings of the Spanish Relavistic Meeting. EREs
[40] Braun Boni, P.; Briggs, K. M.; Boeni, P., Analytical solution to matthews and blakeslees critical dislocation formation thickness of epitaxially grown thin films, Journal of Crystal Growth, 241, 1-2, 231-234 (2002)
[41] Ambagaspitiya, R. S.; Balakrishnan, N., On the compound generalized poisson distributions, ASTIN Bulletin, 24, 2, 255-263 (1994) · doi:10.2143/AST.24.2.2005069
[42] Ditzel, M.; Serdijn, W. A., Optimal energy assignment for frequency selective fading channels, Proceedings of the 12th International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC 2001), IEEE
[43] Szpankowski, W., On asymptotics of certain recurrences arising in universal coding, Problems of Information Transmission c/C of problemy peredachi, 34, 142-146 (1998) · Zbl 0990.94018
[44] Polya, G., Kombinatorische anzahlbestimmungen fiir gruppen, graphen und chemische verbindungen, Acta Mathematica, 68, 1, 145-254 (1937) · Zbl 0017.23202 · doi:10.1007/BF02546665
[45] Baker, I. N.; Rippon, P. J., A note on complex iteration, The American Mathematical Monthly, 92, 7, 501-504 (1985) · Zbl 0605.30027 · doi:10.2307/2322513
[46] Galidakis, I. N., On an application of Lambert’s W function to infinite exponentials. complex variables, theory and application, An International Journal, 49, 11, 759-780 (2004) · Zbl 1072.30017 · doi:10.1080/02781070412331298796
[47] Flajolet, P.; Salvy, B.; Schaeffer, G., Airy phenomena and analytic combinatorics of connected graphs, Electronic Journal of Combinatorics, 11, 1-30 (2004) · Zbl 1053.05064
[48] Baker, C. T. H.; Paul, C. A. H.; Willé, D. R., A bibliography on the numerical solution of delay differential equations, Numerical Analysis Report, M13 9PL (1995), Manchester, UK: The University of Manchester, Department of Mathematics, Manchester, UK · Zbl 0832.65064 · doi:10.1007/BF03028370
[49] Amann, A.; Schöll, E.; Just, W., Some basic remarks on eigenmode expansions of time-delay dynamics, Physica A: Statistical Mechanics and its Applications, 373, 191-202 (2007) · doi:10.1016/j.physa.2005.12.073
[50] Sun Y, A.; Ulsoy, G.; Patrick, W. N., Analysis of systems of linear delay differential equations using the matrix lambert function and the laplace transformation, Automatica, 1-10 (2006)
[51] Patrick, W. N.; Sun, Y.; Galip, A. U., Time-delay systems: analysis and control using the Lambert W function, World Scientific (2010) · Zbl 1209.93002
[52] Simitev, R. D.; Biktashev, V. N., Asymptotics of conduction velocity restitution in models of electrical excitation in the heart, Bulletin of Mathematical Biology, 73, 1, 72-115 (2011) · Zbl 1209.92004 · doi:10.1007/s11538-010-9523-6
[53] Ehret, A. E.; Böol, M.; Itskov, M., A continuum constitutive model for the active behaviour of skeletal muscle, Journal of the Mechanics and Physics of Solids, 59, 3, 625-636 (2011) · Zbl 1270.74138 · doi:10.1016/j.jmps.2010.12.008
[54] Heße, F.; Radu, F. A.; Thullner, M.; Attinger, S., Upscaling of the advection-diffusion-reaction equation with Monod reaction, Advances in Water Resources, 32, 8, 1336-1351 (2009) · doi:10.1016/j.advwatres.2009.05.009
[55] Gong, R.; Lu, C.; Wu, W.-M.; Cheng, H.; Gu, B.; Watson, D.; Jardine, P. M.; Brooks, S. C.; Criddle, C. S.; Kitanidis, P. K.; Luo, J., Estimating reaction rate coefficients within a travel-time modeling framework, Groundwater, 49, 2, 209-218 (2011) · doi:10.1111/j.1745-6584.2010.00683.x
[56] Brkic, D., Lambert w function in hydraulic problems, Proceedings of the MASSEE International Conference on Mathematics MICOM
[57] Zhang, L.; Xing, D.; Sun, J., Calculating activation energy of amorphous phase with the Lambert W function, Journal of Thermal Analysis and Calorimetry, 100, 1, 3-10 (2010) · doi:10.1007/s10973-009-0264-4
[58] Conrath, M.; Fries, N.; Zhang, M.; Dreyer, M. E., Radial capillary transport from an infinite reservoir, Transport in Porous Media, 84, 1, 109-132 (2010) · doi:10.1007/s11242-009-9488-9
[59] Hadj Belgacem, C.; Fnaiech, M., Solution for the critical thickness models of dislocation generation in epitaxial thin films using the Lambert W function, Journal of Materials Science, 46, 6, 1913-1915 (2011) · doi:10.1007/s10853-010-5026-y
[60] Pohjoranta, A.; Mendelson, A.; Tenno, R., A copper electrolysis cell model including effects of the ohmic potential loss in the cell, Electrochimica Acta, 55, 3, 1001-1012 (2010) · doi:10.1016/j.electacta.2009.09.073
[61] Berthier, J.; Silberzan, P., Microfluidics for Biotechnology (2010), Norwood, Mass, USA: Artech House, Norwood, Mass, USA
[62] Caillol, J.-M., Some applications of the Lambert \(W\) function to classical statistical mechanics, Journal of Physics A: Mathematical and General, 36, 42, 10431-10442 (2003) · Zbl 1039.82021 · doi:10.1088/0305-4470/36/42/001
[63] Kyncheva, V. K.; Yotov, V. V.; Ivanov, S. I., Convergence of Newton, Halley and Chebyshev iterative methods as methods for simultaneous determination of multiple polynomial zeros, Applied Numerical Mathematics, 112, 146-154 (2017) · Zbl 1354.65092 · doi:10.1016/j.apnum.2016.10.013
[64] Osada, N., Chebyshev-Halley methods for analytic functions, Journal of Computational and Applied Mathematics, 216, 2, 585-599 (2008) · Zbl 1146.65043 · doi:10.1016/j.cam.2007.06.020
[65] Yau, L.; Ben-Israel, A., The Newton and Halley methods for complex roots, The American Mathematical Monthly, 105, 9, 806-818 (1998) · Zbl 1002.65059 · doi:10.2307/2589209
[66] Boyd, J. P., Global approximations to the principal real-valued branch of the Lambert \(W\)-function, Applied Mathematics Letters, 11, 6, 27-31 (1998) · Zbl 0940.65018 · doi:10.1016/S0893-9659(98)00097-4
[67] de Brujin, N. G., Asymptotic Methods in Analysis (1961), North-Holland · Zbl 0098.26404
[68] Karamata, J., Sur quelques problemes poses par ramanujan, Journal of the Indian Mathematical Society, 24, 3-4, 343-365 (1960) · Zbl 0217.32101
[69] Vazquez-Leal, H., The enhanced power series method to find exact or approximate solutions of nonlinear differential equations, Applied and Computational Mathematics, 14, 2, 168-179 (2015) · Zbl 1333.34018
[70] Wang, Y.-G.; Lin, W.-H.; Liu, N., A homotopy perturbation-based method for large deflection of a cantilever beam under a terminal follower force, International Journal for Computational Methods in Engineering Science and Mechanics, 13, 3, 197-201 (2012) · doi:10.1080/15502287.2012.660229
[71] Vazquez-Leal, H.; Sarmiento-Reyes, A., Power series extender method for the solution of nonlinear differential equations, Mathematical Problems in Engineering, 2015, 1-2 (2015) · doi:10.1155/2015/717404
[72] Zill, D. G., A First Course in Differential Equations with Modeling Applications (2012), Cenage Learning
[73] Balser, W., Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations (1999), Springer
[74] Fritsch, F. N.; Shafer, R. E.; Crowley, W. P., Solution of the transcendental equation wew = x, Communications of the ACM, 16, 2, 123-124 (1973) · doi:10.1145/361952.361970
[75] Barry, D. A.; Culligan-Hensley, P. J.; Barry, S. J., Real values of the \(W\)-function, ACM Transactions on Mathematical Software, 21, 2, 161-171 (1995) · Zbl 0886.65010 · doi:10.1145/203082.203084
[76] Disney, S. M.; Warburton, R. D. H., On the Lambert W function: Economic Order Quantity applications and pedagogical considerations, International Journal of Production Economics, 140, 2, 756-764 (2012) · doi:10.1016/j.ijpe.2011.02.027
[77] Dubinova, I. D., Application of the Lambert W function in mathematical problems of plasma physics, Plasma Physics Reports, 30, 10, 872-877 (2004) · doi:10.1134/1.1809403
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