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On the summation of divergent, truncated, and underspecified power series via asymptotic approximants
By Barlow, Nathaniel S, Christopher R Stanton, Nicole Hill, Steven J Weinstein, and Allyssa G Cio
Published in The Quarterly Journal of Mechanics and Applied Mathematics 70, 21 (Sunday, January 1, 2017)


A compact and accurate solution method is provided for problems whose infinite power series solution diverges and/or whose series coefficients are only known up to a finite order. The method only requires that either the power series solution or some truncation of the power series solution be available and that some asymptotic behavior of the solution is known away from the series' expansion point. Here, we formalize the method of asymptotic approximants that has found recent success in its application to thermodynamic virial series where only a few to (at most) a dozen series coefficients are typically known. We demonstrate how asymptotic approximants may be constructed using simple recurrence relations, obtained through the use of a few known rules of series manipulation. The result is an approximant that bridges two asymptotic regions of the unknown exact solution, while maintaining accuracy in-between. A general algorithm is provided to construct such approximants. To demonstrate the versatility of the 1 method, approximants are constructed for three nonlinear problems relevant to mathematical physics: the Sakiadis boundary layer, the Blasius boundary layer, and the Flierl-Petviashvili monopole. The power series solution to each of these problems is underspecified since, in the absence of numerical simulation, one lower-order coefficient is not known; consequently, higher-order coefficients that depend recur-sively on this coefficient are also unknown. The constructed approx-imants are capable of predicting this unknown coefficient as well as other important properties inherent to each problem. The approxi-mants lead to new benchmark values for the Sakiadis boundary layer and agree with recent numerical values for properties of the Blasius boundary layer and Flierl-Petviashvili monopole.
(PDF) On the summation of divergent, truncated,.... Available from: https://www.researchgate.net/publication/312233747_On_the_summation_of_divergent_truncated_and_underspecified_power_series_via_asymptotic_approximants [accessed Jul 23 2018].