Legendre spectral element method for solving sine-Gordon equation

Lotfi, Mahmoud; Alipanah, Amjad

doi:10.1186/s13662-019-2059-7

Cited by 5 publications

(2 citation statements)

References 33 publications

(41 reference statements)

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“…Albowitz et al [6,7] discussed the equation for the unstable nature using the nonlinear spectrum. Various analytical and numerical approaches have been applied by the researchers to solve this equation for its soliton solutions including the modified decomposition method [8,9] for solving this equation 1D and 2D, the modified Adomian decomposition method [10] to solve the SG equation in (N + 1)-dimensions, homotopy analysis method [11], boundary element and boundary integral approach [12,13], Compact finite difference of order-6 (CFD6) scheme [14], tension spline-based approximation scheme [15], Modified cubic B-spline (MCB) collocation technique [16], localized method of approximate particular solutions [17], Legendre spectral element method [18], virtual element method [19], Barycentric rational interpolation and local radial basis functions [20], fourth-order collocation scheme [21] and rational radial basis function [22].…”

Section: Introductionmentioning

confidence: 99%

Particle Swarm Optimization for Solving Sine-Gordan Equation

Arora¹,

Chauhan²,

Asjad³

et al. 2023

Computer Systems Science and Engineering

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The term 'optimization' refers to the process of maximizing the beneficial attributes of a mathematical function or system while minimizing the unfavorable ones. The majority of real-world situations can be modelled as an optimization problem. The complex nature of models restricts traditional optimization techniques to obtain a global optimal solution and paves the path for global optimization methods. Particle Swarm Optimization is a potential global optimization technique that has been widely used to address problems in a variety of fields. The idea of this research is to use exponential basis functions and the particle swarm optimization technique to find a numerical solution for the Sine-Gordan equation, whose numerical solutions show the soliton form and has diverse applications. The implemented optimization technique is employed to determine the involved parameter in the basis functions, which was previously approximated as a random number in the work reported till now in the literature. The obtained results are comparable with the results obtained in the literature. The work is presented in the form of figures and tables and is found encouraging.

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Section: Introductionmentioning

confidence: 99%

Particle Swarm Optimization for Solving Sine-Gordan Equation

Arora¹,

Chauhan²,

Asjad³

et al. 2023

Computer Systems Science and Engineering

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“…Lu [23] applied modified homotopy perturbation method for the solution of sine-Gordon equation. Many authors applied various approaches to investigate the approximate solution of the Klein-Gordon and sine-Gordon equations [24][25][26].…”

Section: Introductionmentioning

confidence: 99%

A New Iterative Method for the Approximate Solution of Klein-Gordon and Sine-Gordon Equations

Fang

Nadeem

Habib

et al. 2022

Journal of Function Spaces

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This article presents a new iterative method (NIM) for the investigation of the approximate solution of the Klein-Gordon and sine-Gordon equations. This approach is formulated on the combination of the Mohand transform and the homotopy perturbation method. Mohand transform (MT) is capable to handle the linear terms only, thus we introduce homotopy perturbation method (HPM) to tackle the nonlinear terms. This NIM derives the results in the form of a series solution. The proposed method emphasizes the stability of the derived solutions without any linearization, discretization, or hypothesis. Graphical representation and absolute error demonstrate the efficiency and authenticity of this scheme. Some numerical models are illustrated to show the compactness and reliability of this strategy.

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