The phase-integral method in a problem of singular perturbation theory

2017 ◽  
Vol 81 (2) ◽  
pp. 359-390 ◽  
Author(s):  
S A Stepin ◽  
V V Fufaev
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Integral Method ◽  
Singular Perturbation Theory ◽  
Phase Integral

scholarly journals Renormalization group and fractional calculus methods in a complex world: A review

2021 ◽  
Vol 24 (1) ◽  
pp. 5-53
Author(s):  
Lihong Guo ◽  
YangQuan Chen ◽  
Shaoyun Shi ◽  
Bruce J. West
Keyword(s):  
Perturbation Theory ◽  
Renormalization Group ◽  
Fractional Calculus ◽  
Singular Perturbation ◽  
World View ◽  
Singular Perturbation Theory ◽  
Asymptotic Behavior Of Solutions ◽  
Quantum Field ◽  
Self Similarity ◽  
Way Of Thinking

Abstract The concept of the renormalization group (RG) emerged from the renormalization of quantum field variables, which is typically used to deal with the issue of divergences to infinity in quantum field theory. Meanwhile, in the study of phase transitions and critical phenomena, it was found that the self–similarity of systems near critical points can be described using RG methods. Furthermore, since self–similarity is often a defining feature of a complex system, the RG method is also devoted to characterizing complexity. In addition, the RG approach has also proven to be a useful tool to analyze the asymptotic behavior of solutions in the singular perturbation theory. In this review paper, we discuss the origin, development, and application of the RG method in a variety of fields from the physical, social and life sciences, in singular perturbation theory, and reveal the need to connect the RG and the fractional calculus (FC). The FC is another basic mathematical approach for describing complexity. RG and FC entail a potentially new world view, which we present as a way of thinking that differs from the classical Newtonian view. In this new framework, we discuss the essential properties of complex systems from different points of view, as well as, presenting recommendations for future research based on this new way of thinking.

scholarly journals Singular perturbation theory for semibounded operators

1976 ◽  
Vol 82 (2) ◽  
pp. 341-344 ◽  
Author(s):  
W. M. Greenlee
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Singular Perturbation Theory

Periodic solutions for equations with distributed delays

2006 ◽  
Vol 136 (6) ◽  
pp. 1317-1325 ◽  
Author(s):  
Guojian Lin ◽  
Rong Yuan
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Periodic Solutions ◽  
General Theorem ◽  
Linear Chain ◽  
Distributed Delays ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Geometric Singular Perturbation ◽  
Existence Of Periodic Solutions

A general theorem about the existence of periodic solutions for equations with distributed delays is obtained by using the linear chain trick and geometric singular perturbation theory. Two examples are given to illustrate the application of the general the general therom.

scholarly journals Spectral deformation in a problem of singular perturbation theory

2019 ◽  
Vol 484 (4) ◽  
pp. 397-400
Author(s):  
S. A. Stepin ◽  
V. V. Fufaev
Keyword(s):  
Perturbation Theory ◽  
Asymptotic Behavior ◽  
Integral Method ◽  
Deformation Parameter ◽  
Liouville Problem ◽  
Singular Perturbation Theory ◽  
Different Types ◽  
Spectral Deformation ◽  
Sturm Liouville ◽  
Spectral Complex

Quasi-classical asymptotic behavior of the spectrum of a non-self-adjoint Sturm–Liouville problem is studied in the case of a one-parameter family of potentials being third-degree polynomials. For this problem, the phase-integral method is used to derive quantization conditions characterizing the asymptotic distribution of the eigenvalues and their concentration near edges of the limit spectral complex. Topologically different types of limit configurations are described, and critical values of the deformation parameter corresponding to type changes are specified.

On voltage stability from the viewpoint of singular perturbation theory

1994 ◽  
Vol 16 (6) ◽  
pp. 409-417 ◽  
Author(s):  
N. Yorino ◽  
H. Sasaki ◽  
Y. Masuda ◽  
Y. Tamura ◽  
M. Kitagawa ◽  
...  
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Voltage Stability ◽  
Singular Perturbation Theory
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