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17 pages, 309 KiB

Open AccessArticle

On the Periodic Solutions for the Perturbed Spatial Quantized Hill Problem

by Elbaz I. Abouelmagd, Sawsan Alhowaity, Zouhair Diab, Juan L. G. Guirao and Mahmoud H. Shehata

Mathematics 2022, 10(4), 614; https://doi.org/10.3390/math10040614 - 17 Feb 2022

Cited by 13 | Viewed by 2142

In this work, we investigated the differences and similarities among some perturbation approaches, such as the classical perturbation theory, Poincaré–Lindstedt technique, multiple scales method, the KB averaging method, and averaging theory. The necessary conditions to construct the periodic solutions for the spatial quantized [...] Read more.

In this work, we investigated the differences and similarities among some perturbation approaches, such as the classical perturbation theory, Poincaré–Lindstedt technique, multiple scales method, the KB averaging method, and averaging theory. The necessary conditions to construct the periodic solutions for the spatial quantized Hill problem—in this context, the periodic solutions emerging from the equilibrium points for the spatial Hill problem—were evaluated by using the averaging theory, under the perturbation effect of quantum corrections. This model can be used to develop a Lunar theory and the families of periodic orbits in the frame work for the spatial quantized Hill problem. Thereby, these applications serve to reinforce the obtained results on these periodic solutions and gain its own significance. Full article

(This article belongs to the Special Issue Nonlinear Dynamics Systems with Hysteresis)

17 pages, 1053 KiB

Open AccessArticle

A Simplified Lindstedt-Poincaré Method for Saving Computational Cost to Determine Higher Order Nonlinear Free Vibrations

by Chein-Shan Liu and Yung-Wei Chen

Mathematics 2021, 9(23), 3070; https://doi.org/10.3390/math9233070 - 29 Nov 2021

Cited by 16 | Viewed by 4221

Abstract

In order to improve the Lindstedt-Poincaré method to raise the accuracy and the performance for the application to strongly nonlinear oscillators, a new analytic method by engaging in advance a linearization technique in the nonlinear differential equation is developed, which is realized in [...] Read more.

In order to improve the Lindstedt-Poincaré method to raise the accuracy and the performance for the application to strongly nonlinear oscillators, a new analytic method by engaging in advance a linearization technique in the nonlinear differential equation is developed, which is realized in terms of a weight factor to decompose the nonlinear term into two sides. We expand the constant preceding the displacement in powers of the introduced parameter so that the coefficients can be determined to avoid the appearance of secular solutions. The present linearized Lindstedt-Poincaré method is easily implemented to provide accurate higher order analytic solutions of nonlinear oscillators, such as Duffing and van Der Pol nonlinear oscillators. The accuracy of analytic solutions is evaluated by comparing to the numerical results obtained from the fourth-order Runge-Kotta method. The major novelty is that we can simplify the Lindstedt-Poincaré method to solve strongly a nonlinear oscillator with a large vibration amplitude. Full article

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26 pages, 63790 KiB

Open AccessArticle

Nonlinear EHD Instability of Two-Superposed Walters’ B Fluids Moving through Porous Media

by Ji-Huan He, Galal M. Moatimid and Aya Sayed

Axioms 2021, 10(4), 258; https://doi.org/10.3390/axioms10040258 - 18 Oct 2021

Cited by 23 | Viewed by 2539

Abstract

The current work examines the application of the viscous potential flow to the Kelvin-Helmholtz instability (KHI) of a planar interface between two visco-elastic Walters’ B fluids. The fluids are fully saturated in porous media in the presence of heat and mass transfer across [...] Read more.

The current work examines the application of the viscous potential flow to the Kelvin-Helmholtz instability (KHI) of a planar interface between two visco-elastic Walters’ B fluids. The fluids are fully saturated in porous media in the presence of heat and mass transfer across the interface. Additionally, the structure is pervaded via a uniform, normal electrical field in the absence of superficial charges. The nonlinear scheme basically depends on analyzing the linear principal equation of motion, and then applying the appropriate nonlinear boundary-conditions. The current organization creates a nonlinear characteristic equation describing the amplitude performance of the surface waves. The classical Routh–Hrutwitz theory is employed to judge the linear stability criteria. Once more, the implication of the multiple time scale with the aid of Taylor theory yields a Ginzburg–Landau equation, which controls the nonlinear stability criteria. Furthermore, the Poincaré–Lindstedt technique is implemented to achieve an analytic estimated bounded solution for the surface deflection. Many special cases draw upon appropriate data selections. Finally, all theoretical findings are numerically confirmed in such a way that ensures the effectiveness of various physical parameters. Full article

(This article belongs to the Special Issue Dedicated to Professor Ji-Huan He on the Occasion of His 55th Birthday)

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20 pages, 802 KiB

Open AccessArticle

Periodic Solutions of Nonlinear Relative Motion Satellites

by Ashok Kumar Pal, Elbaz I. Abouelmagd, Juan Luis García Guirao and Dariusz W. Brzeziński

Symmetry 2021, 13(4), 595; https://doi.org/10.3390/sym13040595 - 2 Apr 2021

Cited by 12 | Viewed by 3733

Abstract

The relative motion of an outline of the rendezvous problem has been studied by assuming that the chief satellite is in circular symmetric orbits. The legitimacy of perturbation techniques and nonlinear relative motion are investigated. The deputy satellite equations of motion with respect [...] Read more.

The relative motion of an outline of the rendezvous problem has been studied by assuming that the chief satellite is in circular symmetric orbits. The legitimacy of perturbation techniques and nonlinear relative motion are investigated. The deputy satellite equations of motion with respect to the fixed references at the center of the chief satellite are nonlinear in the general case. We found the periodic solutions of the linear relative motion satellite and for the nonlinear relative motion satellite using the Lindstedt–Poincaré technique. Comparisons among the analytical solutions of linear and nonlinear motions and the obtained solution by the numerical integration of the explicit Euler method for both motions are investigated. We demonstrate that both analytical and numerical solutions of linear motion are symmetric periodic. However, the solutions of nonlinear motion obtained by the Lindstedt–Poincaré technique are periodic and the numerical solutions obtained by integration by using explicit Euler method are non-periodic. Thus, the Lindstedt–Poincaré technique is recommended for designing the periodic solutions. Furthermore, a comparison between linear and nonlinear analytical solutions of relative motion is investigated graphically. Full article

(This article belongs to the Special Issue Recent Advance in Astrophysics)

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11 pages, 300 KiB

Open AccessArticle

Forced Vibrations of Strongly Nonlinear Systems with Multiple Scales Lindstedt Poincaré Method

by M. Pakdemirli, M. M. F. Karahan and H. Boyacı

Math. Comput. Appl. 2011, 16(4), 879-889; https://doi.org/10.3390/mca16040879 - 1 Dec 2011

Cited by 29 | Viewed by 1943

Abstract

Forced vibrations of duffing equation with damping is considered. Recently developed Multiple Scales Lindstedt-Poincare (MSLP) technique for free vibrations is applied for the first time to the forced vibration problem in search of approximate solutions. For the case of weak and strong nonlinearities, [...] Read more.

Forced vibrations of duffing equation with damping is considered. Recently developed Multiple Scales Lindstedt-Poincare (MSLP) technique for free vibrations is applied for the first time to the forced vibration problem in search of approximate solutions. For the case of weak and strong nonlinearities, approximate solutions of the new method are contrasted with the usual Multiple Scales (MS) method and numerical simulations. For weakly nonlinear systems, frequency response curves of both perturbation methods and numerical solutions are in good agreement. For strongly nonlinear systems however, results of MS deviate much from the MSLP method and numerical simulations, the latter two being in good agreement. Full article

14 pages, 231 KiB

Open AccessArticle

A New Perturbation Algorithm With Better Convergence Properties: Multiple Scales Lindstedt Poincare Method

by M. Pakdemirli, M. M. F. Karahan and H. Boyacı

Math. Comput. Appl. 2009, 14(1), 31-44; https://doi.org/10.3390/mca14010031 - 1 Apr 2009

Cited by 46 | Viewed by 2106

Abstract

A new perturbation algorithm combining the Method of Multiple Scales and Lindstedt-Poincare techniques is proposed for the first time. The algorithm combines the advantages of both methods. Convergence to real solutions with large perturbation parameters can be achieved for both constant amplitude and [...] Read more.

A new perturbation algorithm combining the Method of Multiple Scales and Lindstedt-Poincare techniques is proposed for the first time. The algorithm combines the advantages of both methods. Convergence to real solutions with large perturbation parameters can be achieved for both constant amplitude and variable amplitude cases. Three problems are solved: Linear damped vibration equation, classical duffing equation and damped cubic nonlinear equation. Results of Multiple Scales, new method and numerical solutions are contrasted. The proposed new method produces better results for strong nonlinearities. Full article

7 pages, 168 KiB

Open AccessArticle

Numerical Verification and Comparison of Error of Asymptotic Expansion Solution of the Duffing Equation

by Jianping Cai, Shuhui Chen and Cuihong Yang

Math. Comput. Appl. 2008, 13(1), 23-29; https://doi.org/10.3390/mca13010023 - 1 Apr 2008

Cited by 3 | Viewed by 1403

Abstract

A numerical order verification technique is applied to demonstrate that the asymptotic expansions of solutions of the Duffing equation obtained respectively by the Lindstedt-Poincaré(LP) method and the modified Lindstedt-Poincaré(MLP) method are uniformly valid for small parameter values. A numerical comparison of error shows

[...] Read more.

A numerical order verification technique is applied to demonstrate that the asymptotic expansions of solutions of the Duffing equation obtained respectively by the Lindstedt-Poincaré(LP) method and the modified Lindstedt-Poincaré(MLP) method are uniformly valid for small parameter values. A numerical comparison of error shows that the MLP method is valid whereas the LP method is invalid for large parameter values.

Full article

6 pages, 135 KiB

Open AccessArticle

Numerical Verification of the Order of the Asymptotic Solutions of a Nonlinear Differential Equation

by Jianping Cai

Math. Comput. Appl. 2006, 11(1), 85-90; https://doi.org/10.3390/mca11010085 - 1 Apr 2006

Cited by 2 | Viewed by 1141

Abstract

A perturbation method, the Lindstedt-Poincare method, is used to obtain the asymptotic expansions of the solutions of a nonlinear differential equation arising in general relativity. The asymptotic solutions contain no secular term, which overcomes a defect in Khuri’s paper. A technique of numerical [...] Read more.

A perturbation method, the Lindstedt-Poincare method, is used to obtain the asymptotic expansions of the solutions of a nonlinear differential equation arising in general relativity. The asymptotic solutions contain no secular term, which overcomes a defect in Khuri’s paper. A technique of numerical order verification is applied to demonstrate that the asymptotic solutions are uniformly valid for small parameter. Full article

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