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New facts concerning the approximation of the inverse Langevin function

Radosław Jedynak
Organization: Kazimierz Pulaski University of Technology and Humanities
Journal / Anthology

Journal of Non-Newtonian Fluid Mechanics
Year: 2017
Volume: 248

The inverse Langevin function is an integral component for statistically-based network models which describe rubber-like materials. This function is very important from theoretical and practical points of view in variety of scientific fields like rheology, theory of magnetism and polymer science. It cannot be expressed in an explicit form and directly used for analytical manipulation and computer simulation. For these reasons, a lot of researchers have considered this function to provide its optimal approximation. We discuss the latest achievements in our work.

We deliver three very important facts about the examined function. We check the hypothesis that an increase in the number of terms of the Taylor series expansion of the inverse Langevin function above 500, does not change its convergence radius. It is true in the light of our detailed analysis. Our achievements are clearly documented in this paper. This analysis is extended up to 3000 terms of the Taylor series expansion, while previous reports include only 500. We verify the statement that the solution based on 115 Taylor series terms shows the best accuracy within the interval [0,0.95]. To be exact, it is true but in a little smaller interval. We find the best new solutions for two intervals [0,0.95] and [0,0.98]. This is the second fact. The last fact is related to rational approximation of this function. We provide a new rational approximation formula, whose maximum relative error is equal to 0.076 %. So far, such a high precision was restricted only to very complex approximation formulas.

We also include a program written in Mathematica to show application of our new formula on the basis of the known three-chain model.

*Science > Physics > Fluid Mechanics