Acknowledgment

Gluzman, S.; Mityushev, Vladimir; Nawalaniec, Wojciech

doi:10.1016/b978-0-12-811046-1.50015-0

Cited by 12 publications

(51 citation statements)

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“…Such modification can be performed by separating the sought modification of the Padé approximants into two factors [10]. The first factor is to be expressed as an iterated root or factor approximant [11,12]. It is specifically designed to take care of the irrational part of the solution.…”

Section: Introductionmentioning

confidence: 99%

“…We arrive thus to the corrected Padé approximants. They appear to be applicable to a larger class of problems, even when the standard Padé technique is not applicable [11].…”

Section: Introductionmentioning

confidence: 99%

“…The idea of combining information coming from the different limits appear to be fruitful and can be exploited for different types of approximants and various forms of asymptotic expansions [11,12,15,16]. Various self-similar approximants also allow extrapolating and interpolating between the small-variable and large-variable asymptotic expansions, as discussed recently in [16].…”

Section: Introductionmentioning

confidence: 99%

“…Various self-similar approximants also allow extrapolating and interpolating between the small-variable and large-variable asymptotic expansions, as discussed recently in [16]. The key to the success is to introduce the so-called control functions to allow "to sew" the two limit-cases together in the form most natural for each concrete problem [11,12,[15][16][17]. The example of such an approach is brought up in Appendix B.…”

Section: Introductionmentioning

confidence: 99%

“…It is based on progressive improvement of quality of approximants with adding new information through the higher-order coefficients, with the approximants becoming more and more complex. It is exemplified in construction of Pad'e and Euler super-exponential approximants [8], factor, root and additive approximants [11,12,16]. The latter "cluster" of approximations was derived based on the ideas of self-similar approximation theory, a close relative of the field-theoretic renormalization group [17].…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear Approximations to Critical and Relaxation Processes

Gluzman¹

2020

Axioms

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We develop nonlinear approximations to critical and relaxation phenomena, complemented by the optimization procedures. In the first part, we discuss general methods for calculation of critical indices and amplitudes from the perturbative expansions. Several important examples of the Stokes flow through 2D channels are brought up. Power series for the permeability derived for small values of amplitude are employed for calculation of various critical exponents in the regime of large amplitudes. Special nonlinear approximations valid for arbitrary values of the wave amplitude are derived from the expansions. In the second part, the technique developed for critical phenomena is applied to relaxation phenomena. The concept of time-translation invariance is discussed, and its spontaneous violation and restoration considered. Emerging probabilistic patterns correspond to a local breakdown of time-translation invariance. Their evolution leads to the time-translation invariance complete (or partial) restoration. We estimate the typical time extent, amplitude and direction for such a restorative process. The new technique is based on explicit introduction of origin in time as an optimization parameter. After some transformations, we arrive at the exponential and generalized exponential-type solutions (Gompertz approximants), with explicit finite time scale, which is only implicit in the initial parameterization with polynomial approximation. The concept of crash as a fast relaxation phenomenon, consisting of time-translation invariance breaking and restoration, is advanced. Several COVID-related crashes in the time series for Shanghai Composite and Dow Jones Industrial are discussed as an illustration.

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Section: Introductionmentioning

confidence: 99%

“…We arrive thus to the corrected Padé approximants. They appear to be applicable to a larger class of problems, even when the standard Padé technique is not applicable [11].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonlinear Approximations to Critical and Relaxation Processes

Gluzman¹

2020

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Exact and “Exact” Formulae in the Theory of Composites

Andrianov

Mityushev

2018

Trends in Mathematics

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The effective properties of composites and review literature on the methods of Rayleigh, Natanzon-Filshtinsky, functional equations and asymptotic approaches are outlined. In connection with the above methods and new recent publications devoted to composites, we discuss the terms analytical formula, approximate solution, closed form solution, asymptotic formula etc... frequently used in applied mathematics and engineering in various contexts. Though mathematicians give rigorous definitions of exact form solution the term "exact solution" continues to be used too loosely and its attributes are lost. In the present paper, we give examples of misleading usage of such a term.

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A simple alternative in approximation and asymptotic expansion by exponential/trigonometric functions

Mironov

2023

Phys. Scr.

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The approximation of functions is a cornerstone of applied mathematics and mathematical physics. The interpolation and extrapolation of functions and examination of their asymptotic properties represent many important tasks in these fields. Various approximation methods have been developed in the past two centuries, but continued efforts perhaps indicate that no method represents a universal recipe, and each has advantages and shortcomings. Most popular is the use of rational Pade approximants, but they work well only for arguments x < 1 and often contain spurious poles. In this report, I propose a novel and simple procedure of exponential/trigonometric approximation. It accurately fits a variety of functions and solutions of second-order ODE, both linear and non-linear, and clearly outperforms Pade approximants. The main advantages of the method are the simplicity of the algorithm, and only four values of trial functions are needed. The examples presented indicate the potential of the method to find a simple and reliable presentation in a multitude of tasks in applied mathematics and mathematical physics.

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Acknowledgment

Cited by 12 publications

References 0 publications

Nonlinear Approximations to Critical and Relaxation Processes

Nonlinear Approximations to Critical and Relaxation Processes

Exact and “Exact” Formulae in the Theory of Composites

A simple alternative in approximation and asymptotic expansion by exponential/trigonometric functions

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