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

@article{Yukalov2020SelfsimilarEO,
  title={Self-similar extrapolation of nonlinear problems from small-variable to large-variable limit},
  author={Vyacheslav I. Yukalov and E. P. Yukalova},
  journal={International Journal of Modern Physics B},
  year={2020}
}
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… 
1 Citations

Tables from this paper

From Asymptotic Series to Self-Similar Approximants
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

References

SHOWING 1-10 OF 96 REFERENCES
Extrapolation of perturbation-theory expansions by self-similar approximants
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 factor approximants.
TLDR
It is proved that the self-similar factor approximants are able to reproduce exactly a wide class of functions, which include a variety of nonalgebraic functions, and provide very accurate approximations, whose accuracy surpasses significantly that of the most accurate Padé approximant.
Summation of power series by self-similar factor approximants
A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a
Interplay between Approximation Theory and Renormalization Group
  • V. I. Yukalov
  • Physics, Mathematics
    Physics of Particles and Nuclei
  • 2019
The review presents general methods for treating complicated problems that cannot be solved exactly and whose solution encounters two major difficulties. First, there are no small parameters allowing
Method of self‐similar approximations
A new method is suggested to approximately find out the limit for a sequence of functions when solely several first terms of a sequence are known. The method seems to be very useful for those
Stability conditions for method of self‐similar approximations
A generalization for the method of self‐similar approximations suggested recently by the author is given. Stability conditions for the method are formulated: first, the mapping multipliers for the
SELF-SIMILAR RENORMALIZATION AS EQUATION OF MOTION
We consider a general approximating sequence generated by some kind of perturbation theory or iterative technique. An arbitrary sequence of this sort can be made fastly convergent, or at least
Calculation of critical exponents by self-similar factor approximants
Abstract.The method of self-similar factor approximants is applied to calculating the critical exponents of the O(N)-symmetric ϕ4 theory and of the Ising glass. It is demonstrated that this method,
Self-similar structures and fractal transforms in approximation theory
Abstract An overview is given of the methods for treating complicated problems without small parameters, when the standard perturbation theory based on the existence of small parameters becomes
SOME ASYMPTOTIC METHODS FOR STRONGLY NONLINEAR EQUATIONS
This paper features a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained
...
1
2
3
4
5
...