A Survey in Mathematics for Industry: Two-timing and matched asymptotic expansions for singular perturbation problems

2011 ◽  
Vol 22 (6) ◽  
pp. 613-629 ◽  
Author(s):  
R. E. O'MALLEY ◽  
E. KIRKINIS
Keyword(s):  
Asymptotic Expansions ◽  
Multiple Scale ◽  
Singularly Perturbed ◽  
Matched Asymptotic Expansions ◽  
Amplitude Equations ◽  
Singularly Perturbed Problems ◽  
Multi Scale ◽  
Scale Method ◽  
Perturbation Problems ◽  
Appl Math

Following the derivation of amplitude equations through a new two-time-scale method [O'Malley, R. E., Jr. & Kirkinis, E (2010) A combined renormalization group-multiple scale method for singularly perturbed problems.Stud. Appl. Math.124, 383–410], we show that a multi-scale method may often be preferable for solving singularly perturbed problems than the method of matched asymptotic expansions. We illustrate this approach with 10 singularly perturbed ordinary and partial differential equations.

scholarly journals Identification of the Essential Features of the Transient Amplification of Mistuned Systems

2020 ◽  
Author(s):  
Luigi Carassale ◽  
Vincent Denoël ◽  
Carlos Martel ◽  
Lars Panning-von Scheidt
Keyword(s):  
Linear System ◽  
Numerical Simulations ◽  
Mechanical System ◽  
Asymptotic Expansions ◽  
Damping Ratio ◽  
Multiple Scale ◽  
Analytic Solutions ◽  
Sweep Rate ◽  
Scale Method ◽  
Complex Manner

Abstract The dynamic behavior of bladed disks in resonance crossing has been intensively investigated in the community of turbomachinery, addressing the attention to (1) the transienttype response that appear when the resonance is crossed with a finite sweep rate and (2) the localization of the vibration in the disk due to the blade mistuning. In real conditions, the two mentioned effects coexist and can interact in a complex manner. This paper investigates the problem by means of analytic solutions obtained through asymptotic expansions, as well as numerical simulations. The mechanical system is assumed as simple as possible: a 2-dof linear system defined through the three parameters: damping ratio ξ, frequency mistuning Δ, rotor acceleration Ω˙. The analytic solutions are calculated through the multiple-scale method.

scholarly journals Key Features of the Transient Amplification of Mistuned Systems

2021 ◽  
Author(s):  
Luigi Carassale ◽  
Vincent Denoel ◽  
Carlos Martel ◽  
Lars Panning-von Scheidt
Keyword(s):  
Linear System ◽  
Asymptotic Expansions ◽  
Damping Ratio ◽  
Multiple Scale ◽  
Analytic Solutions ◽  
Sweep Rate ◽  
Scale Method ◽  
Key Features ◽  
Type Response ◽  
Complex Manner

Abstract The dynamic behavior of bladed disks in resonance crossing has been intensively investigated in the community of turbomachinery, addressing the attention to (1) the transient-type response that appear when the resonance is crossed with a finite sweep rate and (2) the localization of the vibration in the disk due to the blade mistuning. In real conditions, the two mentioned effects coexist and can interact in a complex manner. This paper investigates the problem by means of analytic solutions obtained through asymptotic expansions, as well as numerical simulations. The mechanical system is assumed as simple as possible: a 2-dof linear system defined through the three parameters: damping ratio ?, frequency mistuning ?, rotor acceleration . The analytic solutions are calculated through the multiple-scale method.

Three-Wave Parametric Simultons in Nonlinear Lattices

2003 ◽  
Vol 17 (22n24) ◽  
pp. 4215-4221
Author(s):  
Guoxiang Huang ◽  
Bambi Hu ◽  
Jacob Szeftel
Keyword(s):  
Phonon Spectrum ◽  
Resonance Condition ◽  
Amplitude Equations ◽  
Multi Scale ◽  
Scale Method ◽  
Nonlinear Excitations ◽  
Wave Resonance ◽  
Nonlinear Lattices ◽  
New Type

A new type of nonlinear excitations, i.e. three simultaneous lattice solitons (simultons), in a nonlinear diatomic lattice is predicted. We show that three-wave resonance condition can be fulfilled in the diatomic lattice. Using a quasi-discrete multi-scale method we derive nonlinear amplitude equations for the three-wave resonance with the dispersion of the system taken into account. We provide several types of exact lattice simultons solutions and show that the lattice simultons can be non propagating and their oscillating frequencies may be within the gap of phonon spectrum bands.

Turing Patterns of a Lotka–Volterra Competitive System with Nonlocal Delay

2018 ◽  
Vol 28 (07) ◽  
pp. 1830021 ◽  
Author(s):  
Bang-Sheng Han ◽  
Zhi-Cheng Wang
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Dynamical Behavior ◽  
Multiple Scale ◽  
Multiple Scale Method ◽  
Turing Patterns ◽  
Amplitude Equations ◽  
Competitive System ◽  
Scale Method ◽  
Nonlocal Delay

This paper focuses on the dynamical behavior of a Lotka–Volterra competitive system with nonlocal delay. We first establish the conditions of Turing bifurcation occurring in the system. According to it and by using multiple scale method, the amplitude equations of the different Turing patterns are obtained. Then, we observe when these patterns (spots pattern and stripes pattern) arise in the Lotka–Volterra competitive system. Finally, some numerical simulations are given to verify our theoretical analysis.

On the method of matched asymptotic expansions

1969 ◽  
Vol 65 (1) ◽  
pp. 209-231 ◽  
Author(s):  
L. E. Fraenkel
Keyword(s):  
Singular Perturbation ◽  
Asymptotic Expansions ◽  
Unknown Function ◽  
Singular Perturbation Problems ◽  
Matched Asymptotic Expansions ◽  
Uniform Approximations ◽  
Perturbation Problems ◽  
Asymptotic Matching

AbstractThe method of matched (or of ‘inner and outer’) asymptotic expansions is reviewed, with particular reference to two general techniques which have been proposed for ‘matching’; that is, for establishing a relationship between the inner and outer expansions, to finite numbers of terms, of an unknown function. It is shown that the first technique, which uses the idea of overlapping of the two expansions, can be difficult and laborious in some applications; while the second, which is the ‘asymptotic matching principle’ in the form stated by Van Dyke(13) can be incorrect. Two different sets of conditions sufficient for the validity of the asymptotic matching principle are then established, on the basis of assumptions about the structure of expressions which approximate to the desired function f(x,∈) for all relevant values of x. Finally, it is noted that in four classes of singular-perturbation problems for which complete and rigorous asymptotic theories exist, uniform approximations to the solutions have a structure which is a particular form of the general one assumed in this paper.

Singularly perturbed problems with a turning point: The non-compatible case

2014 ◽  
Vol 12 (03) ◽  
pp. 293-321 ◽  
Author(s):  
Chang-Yeol Jung ◽  
Roger Temam
Keyword(s):  
Asymptotic Expansions ◽  
Turning Point ◽  
Limit Problem ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problems ◽  
Compatibility Conditions ◽  
Compactly Supported ◽  
The Given

The singularly perturbed problems with a turning point were discussed in [21]. The case where the limit problem is compatible with the given data was fully resolved. However, with limited compatibility conditions on the data, the asymptotic expansions were constructed only up to the order of the level of compatibilities. In this paper, using a smooth cut-off function compactly supported around the turning point we resolve the difficulties incurred from the non-compatible data and finally provide the full asymptotic expansions up to any order.

Regularization of Integral Operators in Singularly Perturbed Problems Using Normal Forms

Vestnik MEI ◽  
2019 ◽  
Vol 6 ◽  
pp. 131-137
Author(s):  
Abdukhafiz A. Bobodzhanova ◽  
◽  
Valeriy F. Safonov ◽  
Keyword(s):  
Normal Forms ◽  
Integral Operators ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problems
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