scholarly journals From Asymptotic Series to Self-Similar Approximants

Physics ◽  
2021 ◽  
Vol 3 (4) ◽  
pp. 829-878
Author(s):  
Vyacheslav I. Yukalov ◽  
Elizaveta P. Yukalova
Keyword(s):  
Perturbation Theory ◽  
Approximation Theory ◽  
Good Accuracy ◽  
Asymptotic Series ◽  
Approximate Solutions ◽  
Self Similar ◽  
Similar Approximation ◽  
Physics Problems

The review presents the development of an approach of constructing approximate solutions to complicated physics problems, starting from asymptotic series, through optimized perturbation theory, to self-similar approximation theory. The close interrelation of underlying ideas of these theories is emphasized. Applications of the developed approach are illustrated by typical examples demonstrating that it combines simplicity with good accuracy.

scholarly journals Extrapolation of perturbation-theory expansions by self-similar approximants

2014 ◽  
Vol 25 (5) ◽  
pp. 595-628 ◽  
Author(s):  
S. GLUZMAN ◽  
V.I. YUKALOV
Keyword(s):  
Perturbation Theory ◽  
Approximation Theory ◽  
Asymptotic Expansions ◽  
Applied Mathematics ◽  
Self Similar ◽  
Similar Approximation ◽  
Asymptotic Perturbation

The problem of extrapolating asymptotic perturbation-theory expansions in powers of a small variable to large values of the variable tending to infinity is investigated. The analysis is based on self-similar approximation theory. Several types of self-similar approximants are considered and their use in different problems of applied mathematics is illustrated. Self-similar approximants are shown to constitute a powerful tool for extrapolating asymptotic expansions of different natures.

scholarly journals SELF-SIMILAR EXTRAPOLATION OF ASYMPTOTIC SERIES AND FORECASTING FOR TIME SERIES

2000 ◽  
Vol 14 (22n23) ◽  
pp. 791-800 ◽  
Author(s):  
V. I. YUKALOV
Keyword(s):  
Time Series ◽  
Probability Measure ◽  
Approximation Theory ◽  
Asymptotic Series ◽  
The Self ◽  
Complete Family ◽  
Self Similar ◽  
Similar Approximation

The method of extrapolating asymptotic series, based on the self-similar approximation theory, is developed. Several important questions are answered, which makes the foundation of the method unambiguous and its application straightforward. It is shown how the extrapolation of asymptotic series can be reformulated as forecasting for time series. The probability measure is introduced characterizing the ensemble of forecasted scenarios. The method of choosing the complete family of databases is put forward.

scholarly journals Self-similar extrapolation of nonlinear problems from small-variable to large-variable limit

2020 ◽  
Vol 34 (21) ◽  
pp. 2050208
Author(s):  
V. I. Yukalov ◽  
E. P. Yukalova
Keyword(s):  
Asymptotic Series ◽  
Nonlinear Problems ◽  
Numerical Convergence ◽  
Physical Problems ◽  
Variable Limit ◽  
Small Parameters ◽  
Self Similar ◽  
Similar Approximation ◽  
Similar Factor ◽  
Class Of Functions

Complicated physical problems are usually solved by resorting to perturbation theory leading to solutions in the form of asymptotic series in powers of small parameters. However, finite, and even large values of the parameters, are often of main physical interest. A method is described for predicting the large-variable behavior of solutions to nonlinear problems from the knowledge of only their small-variable expansions. The method is based on self-similar approximation theory resulting in self-similar factor approximants. The latter can well approximate a large class of functions, rational, irrational, and transcendental. The method is illustrated by several examples from statistical and condensed matter physics, where the self-similar predictions can be compared with the available large-variable behavior. It is shown that the method allows for finding the behavior of solutions at large variables when knowing just a few terms of small-variable expansions. Numerical convergence of approximants is demonstrated.

scholarly journals Describing phase transitions in field theory by self-similar approximants

2019 ◽  
Vol 204 ◽  
pp. 02003
Author(s):  
V.I. Yukalov ◽  
E.P. Yukalova
Keyword(s):  
Field Theory ◽  
Phase Transitions ◽  
Coupling Parameter ◽  
Asymptotic Series ◽  
Quantum Field ◽  
Lattice Data ◽  
Borel Summation ◽  
Self Similar ◽  
Similar Approximation ◽  
Good Agreement

Self-similar approximation theory is shown to be a powerful tool for describing phase transitions in quantum field theory. Self-similar approximants present the extrapolation of asymptotic series in powers of small variables to the arbitrary values of the latter, including the variables tending to infinity. The approach is illustrated by considering three problems: (i) The influence of the coupling parameter strength on the critical temperature of the O(N)-symmetric multicomponent field theory. (ii) The calculation of critical exponents for the phase transition in the O(N)-symmetric field theory. (iii) The evaluation of deconfinement temperature in quantum chromodynamics. The results are in good agreement with the available numerical calculations, such as Monte Carlo simulations, Padé-Borel summation, and lattice data.

On nonintegral E corrections in perturbation theory: application to the perturbed Morse oscillator

1994 ◽  
Vol 72 (1-2) ◽  
pp. 80-85 ◽  
Author(s):  
Hafez Kobeissi ◽  
Majida Kobeissi ◽  
Chafia H. Trad
Keyword(s):  
Perturbation Theory ◽  
Good Accuracy ◽  
Morse Oscillator ◽  
Numerical Application ◽  
Correction Formula ◽  
Theory Application ◽  
Nonhomogeneous Differential Equation ◽  
Particular Solution ◽  
New Formulation ◽  
Schrödinger Perturbation

A new formulation of the Rayleigh–Schrödinger perturbation theory is applied to the derivation of the vibrational eigenvalues of the perturbed Morse oscillator (PMO). This formulation avoids the conventional projection of the Ψ corrections on the basis of unperturbed eigenfunctions [Formula: see text], or the projection of the nonhomogeneous Schrödinger equations on [Formula: see text], it gives simple expressions for each E correction [Formula: see text] free of summations and integrals. When the PMO is characterized by the potential U = UM + UP (where UM is the unperturbed Morse potential), the eigenvalue of a vibrational level ν is given by: [Formula: see text]. According to the new formulation the correction £, [Formula: see text] is given by [Formula: see text], where σp(r) is a particular solution of the nonhomogeneous differential equation y″ + f y = sp; here [Formula: see text], sp is well known for each p: for p = 0, [Formula: see text]; for [Formula: see text]. For the numerical application one single routine is used, that of integrating y″ + f y = s, where the coefficients are known as well as the initial values. An example is presented for the Huffaker PMO of the (carbon monoxide) CO-X1Σ+ state. The vibrational eigenvalues Eν are obtained to a good accuracy (with p = 4) even for high levels. This result confirms the validity of this new formulation and gives a semianalytic expression for the PMO eigenvalues.

scholarly journals WEAK QUENCHED DISORDER AND CRITICALITY: RESUMMATION OF ASYMPTOTIC(?) SERIES

2002 ◽  
Vol 16 (27) ◽  
pp. 4027-4079 ◽  
Author(s):  
YU. HOLOVATCH ◽  
V. BLAVATS'KA ◽  
M. DUDKA ◽  
C. VON FERBER ◽  
R. FOLK ◽  
...  
Keyword(s):  
Perturbation Theory ◽  
Critical Behavior ◽  
Asymptotic Series ◽  
Analytical Description ◽  
Quenched Disorder ◽  
Site Disorder ◽  
Disordered Models ◽  
Random Site ◽  
Theoretical Results ◽  
Theory Series

In this review paper, we discuss the influence of weak quenched disorder on the critical behavior in condensed matter and give a brief review of the available experimental and theoretical results as well as results of MC simulations of these phenomena. We concentrate on three cases: (i) uncorrelated random-site disorder, (ii) long-range correlated random-site disorder, and (iii) random anisotropy. Today, the standard analytical description of critical behavior is given by renormalization group results refined by resummation of the perturbation theory series. The convergence properties of the series are unknown for most disordered models. The main object of this review is to discuss the peculiarities of the application of resummation techniques to perturbation theory series of disordered models.

scholarly journals Self-Similar Perturbation Theory

1999 ◽  
Vol 277 (2) ◽  
pp. 219-254 ◽  
Author(s):  
V.I. Yukalov ◽  
E.P. Yukalova
Keyword(s):  
Perturbation Theory ◽  
Self Similar
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