A new parameter-free method for Toeplitz systems of weakly 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 approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modified perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings. In this paper the following categories of asymptotic methods are emphasized: (1) variational approaches, (2) parameter-expanding methods, (3) parameterized perturbation method, (4) homotopy perturbation method (5) iteration perturbation method, and ancient Chinese methods. The emphasis of this article is put mainly on the developments in this field in China so the references, therefore, are not exhaustive.

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On the number of positive solutions of some weakly nonlinear equations on annular regions

Mathematische Zeitschrift ◽

10.1007/bf02571362 ◽

1991 ◽

Vol 206 (1) ◽

pp. 551-562 ◽

Cited By ~ 12

Author(s):

E. N. Dancer

Keyword(s):

Nonlinear Equations ◽

Positive Solutions ◽

Weakly Nonlinear

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Nonlinear Wave Modulation in Nanorods Using Nonlocal Elasticity Theory

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2017-0225 ◽

2018 ◽

Vol 19 (7-8) ◽

pp. 709-719 ◽

Cited By ~ 2

Author(s):

Guler Gaygusuzoglu ◽

Metin Aydogdu ◽

Ufuk Gul

Keyword(s):

Elasticity Theory ◽

Nonlinear Equations ◽

Nonlinear Wave ◽

Nonlocal Elasticity ◽

Nonlocal Elasticity Theory ◽

Reductive Perturbation Method ◽

Nls Equation ◽

Dispersive Waves ◽

Weakly Nonlinear ◽

Wave Modulation

AbstractIn this study, nonlinear wave modulation in nanorods is examined on the basis of nonlocal elasticity theory. Eringen's nonlocal elasticity theory is employed to derive nonlinear equations for the motion of nanorods. The analysis of the modulation of axial waves in nonlocal elastic media is performed, and the reductive perturbation method is used for the solution of the nonlinear equations. The propagation of weakly nonlinear and strongly dispersive waves is investigated, and the nonlinear Schrödinger (NLS) equation is acquired as an evolution equation. For the purpose of a numerical investigation of the nonlocal impacts on the NLS equation, it has been investigated whether envelope solitary wave solutions exist by utilizing the physical and geometric features of the carbon nanotubes. Amplitude dependent wave frequencies, phase and group velocities have been obtained and they have compared for the linear local, the linear nonlocal, the nonlinear local and the nonlinear nonlocal cases.

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Limit Cycle for the Brusselator by He's Variational Method

Mathematical Problems in Engineering ◽

10.1155/2007/85145 ◽

2007 ◽

Vol 2007 ◽

pp. 1-8 ◽

Cited By ~ 5

Author(s):

Juan Zhang

Keyword(s):

Variational Method ◽

Limit Cycle ◽

Nonlinear Equations ◽

Limit Cycles ◽

Present Theory ◽

Variational Theory ◽

Weakly Nonlinear ◽

Solution Domain

He's variational method for finding limit cycles is applied to the Brusselator. The technique developed in this paper is similar to Kantorovitch's method in variational theory. The present theory can be applied not only to weakly nonlinear equations, but also to strongly ones, and the obtained results are valid for the whole solution domain.

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