Modified Padé–Borel Summation

Gluzman, S.

doi:10.3390/axioms12010050

Cited by 5 publications

(19 citation statements)

References 52 publications

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“…Several methods of finding effective sums of the truncated series (1)-(3) exist, based on the ideas of Borel, Mittag-Leffler, and Hardy [9,44,46,[50][51][52][53][54][55][56]. The well-known method is that of Padé [57][58][59]. The hypergeometric approximants [60][61][62][63] can be employed instead of the Padé approximants.…”

Section: Critical Amplitudes From Fractional Iterationsmentioning

confidence: 99%

“…was introduced to ensure the correct asymptotic behavior [58]. Also, G(x) = f (x) K(x) represents the transformed truncated series, which can be approached again with the diagonal Padé approximants.…”

Section: Modified Padé Approximationsmentioning

confidence: 99%

“…Also, G(x) = f (x) K(x) represents the transformed truncated series, which can be approached again with the diagonal Padé approximants. More details on the application of the modified Padé approximants for the Borel summation can be found in [58].…”

Section: Modified Padé Approximationsmentioning

confidence: 99%

“…These are adjusted to the calculations of the amplitudes in Section 4.1. We also employ the iterated roots explained in Sections 2.4, and diagonal Padé approximants for odd and even number of the terms k in the truncations [58].…”

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confidence: 99%

“…The diagonal Padé approximants are routinely extendable to very high orders. Fractional Borel Summation with iterated roots was previously considered in detail [49], while the diagonal, odd, and even Padé approximants are discussed in [58].…”

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confidence: 99%

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Optimized Self-Similar Borel Summation

Gluzman,

Yukalov

2023

Axioms

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The method of Fractional Borel Summation is suggested in conjunction with self-similar factor approximants. The method used for extrapolating asymptotic expansions at small variables to large variables, including the variables tending to infinity, is described. The method is based on the combination of optimized perturbation theory, self-similar approximation theory, and Borel-type transformations. General Borel Fractional transformation of the original series is employed. The transformed series is resummed in order to adhere to the asymptotic power laws. The starting point is the formulation of dynamics in the approximations space by employing the notion of self-similarity. The flow in the approximation space is controlled, and “deep” control is incorporated into the definitions of the self-similar approximants. The class of self-similar approximations, satisfying, by design, the power law behavior, such as the use of self-similar factor approximants, is chosen for the reasons of transparency, explicitness, and convenience. A detailed comparison of different methods is performed on a rather large set of examples, employing self-similar factor approximants, self-similar iterated root approximants, as well as the approximation technique of self-similarly modified Padé–Borel approximations.

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Section: Critical Amplitudes From Fractional Iterationsmentioning

confidence: 99%

Section: Modified Padé Approximationsmentioning

confidence: 99%

Section: Modified Padé Approximationsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Optimized Self-Similar Borel Summation

Gluzman,

Yukalov

2023

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Resolving the problem of multiple control parameters in optimized Borel-type summation

Yukalov,

Gluzman

2024

J Math Chem

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Critical Permeability from Resummation

Gluzman

2024

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Special calculation methods are presented for critical indices and amplitudes for the permeability of thin wavy channels dependent on the waviness. The effective permeability and wetted perimeter of the two-dimensional random percolating media are considered as well. A special mathematical framework is developed to characterize the dependencies on porosities, critical points, and indices. Various approximation techniques are applied without involving popular lubrication approximation in any sense. In particular, the Borel summation technique is applied to the effective polynomial approximations with or without optimization. Minimal difference and minimal derivative optimal conditions are adapted to calculations of critical indices and amplitudes for the effective permeability of thin wavy channels. Critical indices, amplitudes, and thresholds are obtained for the effective permeability and wetted perimeter of the two-dimensional percolating random media. Closed-form expressions for all porosities, critical points, and indices are calculated from the polynomial approximations for the first time.

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Modified Padé–Borel Summation

Cited by 5 publications

References 52 publications

Optimized Self-Similar Borel Summation

Optimized Self-Similar Borel Summation

Resolving the problem of multiple control parameters in optimized Borel-type summation

Critical Permeability from Resummation

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