An Extended Padé Procedure for Constructing Global Approximations from Asymptotic Expansions: An Explication with Examples

Frost, P. A.; Harper, E. Y.

doi:10.1137/1018003

Cited by 20 publications

(16 citation statements)

References 15 publications

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“…J. P. Boyd [49,51] showed that if the power series coefficients diverge as (n!) r or (rn)!, i.e., lim sup n→∞ log |b n | n log n = r (129) then the Chebyshev coefficients satisfy the inequalities…”

Section: Numerical Methods For Exponential Smallness Or: Poltergeist-mentioning

confidence: 99%

“…Padé approximants have been generalized in several other directions, too [129]. Reinhardt [269] has developed a procedure using double Padé approximants which works well even for monotonic, factorially-diverging series, including the computation of exponentially small (E), although it has been largely displaced by Shafer approximants.…”

Section: Numerical Methods Ii: Sequence Acceleration and Padé And Hementioning

confidence: 99%

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Boyd

1999

Acta Applicandae Mathematicae

203

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Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.Mathematics Subject Classifications (1991): 34E05, 40G99, 41A60, 65B10.

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Section: Numerical Methods For Exponential Smallness Or: Poltergeist-mentioning

confidence: 99%

Section: Numerical Methods Ii: Sequence Acceleration and Padé And Hementioning

confidence: 99%

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Boyd

1999

Acta Applicandae Mathematicae

203

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“…Most of the approximant coefficients calculated here may be obtained recursively using a few simple series relations listed in Appendix A. This is in contrast with Padés and their extensions, which often require the inversion of a linear system [3,4,10]. As shall be apparent, a theme of this work is to find/impose the simplest asymptotic approximant form possible while maintaining the desired accuracy and precision.…”

Section: Limmentioning

confidence: 99%

“…We also record the value S as the magnitude of the singularity, η s , closest to η=0 in the complex η plane for the Sakiadis problem, as predicted by (13) and listed in Table ( 1). This value limits the radius of convergence of the series (10).…”

Section: Simple Asymptotic Approximantmentioning

confidence: 99%

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On the Summation of Divergent, Truncated, and Underspecified Power Series via Asymptotic Approximants

Barlow¹,

Stanton²,

Hill³

et al. 2017

Q J Mechanics Appl Math

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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 recursively on this coefficient are also unknown. The constructed approximants are capable of predicting this unknown coefficient as well as other important properties inherent to each problem. The approximants 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.

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Asymptotic Approaches

2014

Asymptotic Methods in the Theory of Plates With Mixed Boundary Conditions

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An Extended Padé Procedure for Constructing Global Approximations from Asymptotic Expansions: An Explication with Examples

Cited by 20 publications

References 15 publications

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Untitled

On the Summation of Divergent, Truncated, and Underspecified Power Series via Asymptotic Approximants

Asymptotic Approaches

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