Power series approach to nonlinear oscillators

As’ad, Ata Abu-; Asad, Jihad

doi:10.1177/14613484231188756

Cited by 5 publications

(1 citation statement)

References 45 publications

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“…As a progress works, other perturbed methods as shown in Refs. [35][36][37][38][39][40][41] may be adopted.…”

Section: Discussionmentioning

confidence: 99%

Studying highly nonlinear oscillators using the non-perturbative methodology

Moatimid,

Amer,

Galal

2023

Sci Rep

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Due to the growing concentration in the field of the nonlinear oscillators (NOSs), the present study aims to use the general He's frequency formula (HFF) to examine the analytical representations for particular kinds of strong NOSs. Three real-world examples are demonstrated by a variety of engineering and scientific disciplines. The new approach is evidently simple and requires less computation than the other perturbation techniques used in this field. The new methodology that is termed as the non-perturbative methodology (NPM) refers to this innovatory strategy, which merely transforms the nonlinear ordinary differential equation (ODE) into a linear one. The method yields a new frequency that is equivalent to the linear ODE as well as a new damping term that may be produced. A thorough explanation of the NPM is offered for the reader's convenience. A numerical comparison utilizing the Mathematical Software (MS) is used to verify the theoretical results. The precise numeric and theoretical solutions exhibited excellent consistency. As is commonly recognized, when the restoration forces are in effect, all traditional perturbation procedures employ Taylor expansion to expand these forces and then reduce the complexity of the specified problem. This susceptibility no longer exists in the presence of the non-perturbative solution (NPS). Additionally, with the NPM, which was not achievable with older conventional approaches, one can scrutinize examining the problem's stability. The NPS is therefore a more reliable source when examining approximations of solutions for severe NOSs. In fact, the above two reasons create the novelty of the present approach. The NPS is also readily transferable for additional nonlinear issues, making it a useful tool in the fields of applied science and engineering, especially in the topic of the dynamical systems.

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“…As a progress works, other perturbed methods as shown in Refs. [35][36][37][38][39][40][41] may be adopted.…”

Section: Discussionmentioning

confidence: 99%

Studying highly nonlinear oscillators using the non-perturbative methodology

Moatimid,

Amer,

Galal

2023

Sci Rep

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Pendulum attached to a vibrating point: Semi-analytical solution by optimal and modified homotopy perturbation method

Roy,

Soqi,

Maiti

et al. 2025

Alexandria Engineering Journal

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Hybrid finite element and laplace transform method for efficient numerical solutions of fractional PDEs on graphics processing units

Vivas-Cruz,

González-Calderón,

Taneco-Hernández

et al. 2024

Phys. Scr.

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Fractional Partial Differential Equations (FPDEs) are essential for modeling complex systems across various scientific and engineering areas. However, efficiently solving FPDEs presents significant computational challenges due to their inherent memory effects, often leading to increased execution times for numerical solutions. This study proposes a highly parallelizable hybrid computational approach that combines the Finite Element Method (FEM) for spatial discretization with Numerical Inversion of the Laplace Transform (NILT) for time-domain solutions, optimized for execution on Graphics Processing Units (GPUs). The NILT method’s high parallelizability,stemming from the independence of its series terms, combined with the robust spatial discretization provided by FEM, enables the efficient and accurate solution of FPDEs on GPUs, demonstrating substantial performance improvements over traditional CPU-based implementations. We observe a generalized pattern in execution time behavior that accounts for both the number of nodes andthe number of NILT terms. Specifically, execution time scales quadratically with the number of nodes, while showing only a logarithmic marginal increase with the number of NILT terms. These behaviors not only enables consistent performance assessment but also highlights potential areas for algorithm optimization. Validation against exact solutions of fractional diffusion and wave equations, employing Caputo, modified Caputo-Fabrizio, and modified Atangana-Baleanu derivatives, demonstrates the accuracy and convergence of the hybrid FEM-NILT method. Notably, the exact solutions of wave equation are novel in literature. The results highlight the method’s potential for enabling high-precision, large-scale simulations in fractional calculus applications, thereby advancingcomputational capabilities and efficiency in the field.

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Power series approach to nonlinear oscillators

Cited by 5 publications

References 45 publications

Studying highly nonlinear oscillators using the non-perturbative methodology

Studying highly nonlinear oscillators using the non-perturbative methodology

Pendulum attached to a vibrating point: Semi-analytical solution by optimal and modified homotopy perturbation method

Hybrid finite element and laplace transform method for efficient numerical solutions of fractional PDEs on graphics processing units

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