Numerical solution of sine-Gordon equation by variational iteration method

Batiha, Belal; Noorani, Mohd Salmi Md; Hashim, Ishak

doi:10.1016/j.physleta.2007.05.087

Cited by 87 publications

(58 citation statements)

References 31 publications

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“…iteration method to obtain approximate analytical solution of the sine-Gordon equation without any discretization has been developed by Batiha et al [10]. Zheng [11] presented a numerical solution of sine-Gordon equation defined on the whole real axis.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Nonlinear Sine-Gordon Equation by Modified Cubic B-Spline Collocation Method

Mittal

Bhatia

2014

International Journal of Partial Differential Equations

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Modified cubic B-spline collocation method is discussed for the numerical solution of one-dimensional nonlinear sine-Gordon equation. The method is based on collocation of modified cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. The given equation is decomposed into a system of equations and modified cubic B-spline basis functions have been used for spatial variable and its derivatives, which gives results in amenable system of ordinary differential equations. The resulting system of equation has subsequently been solved by SSP-RK54 scheme. The efficacy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and are in good agreement with earlier studies.

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Section: Introductionmentioning

confidence: 99%

Numerical Solution of Nonlinear Sine-Gordon Equation by Modified Cubic B-Spline Collocation Method

Mittal

Bhatia

2014

International Journal of Partial Differential Equations

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“…2) and the phase difference across a long Josephson junction carrying a current infinitely under any applied voltage through two superconductors separated by an insulator layer Derks et al (2003). The classical SG equation in normalized units is Kaya (2003); Batiha et al (2007) The classical SG equation description of behaviours occurring in physical systems is considered to be of superficial form; because this equation is somehow an idealization and therefore cannot bear in itself some processes occurring in nature, due to the complexity of real phenomena. This is so, since all quantum systems always interact with their surroundings and hence are open.…”

Section: Afyon Kocatepe üNiversitesi Fen Ve Mühendislik Bilimleri Dermentioning

confidence: 99%

Approximate Soliton Solutions of Real Order Sine- Gordon Equations

Üzar¹

2017

AKU-J. Sci. Eng.

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In this study, the fractional Sine-Gordon (SG) equations (time-fractional, space-fractional and timespace-fractional) are solved using Homotopy Perturbation Method (HPM). The crucial point is the attained remarkable result from these solutions. While the solutions of classical and time-fractional SG equations are kink of type (Although of being the same type, they are different from each other), solution of the space-fractional SG equation is breather of type i.e., different types of soliton solutions are obtained using similar initial conditions for time and space fractional SG equation. Also these results show that some events such as vortex-antivortex couples in a Josephson junction or losses in signal dispersion of fiber optics communication can be modelled by fractional SG equations. In other words, this study may be very important for bringing to light the real behaviour of physical systems which have usually been described by classical SG equation. Because some physical events such as the memory effects of non-Markovian processes, the effects of non-Gaussian distribution, interactions between the systems and environment and some physical losses in the systems which are neglected in classical SG equation can be taken into account with fractional SG equations. Reel Mertebeli Sine-Gordon Denklemlerinin Yaklaşık Soliton Çözümleri Anahtar kelimelerCaputo kesirsel türev operatörü; HPM; SG denklemi; kesirli lineer olmayan denklemler ÖzetBu çalışmada kesirli Sine-Gordon denklemleri (zaman-kesirli, uzay-kesirli ve zaman-uzay-kesirli) homotopy pertürbasyon metodu (HPM) kullanılarak çözülmüştür. Bu çözümlerden dikkat çekici sonuçlar elde edilmiştir. Klasik ve zaman-kesirli SG denklemlerinin çözümü kink tipi iken (aynı tipte olmalarına rağmen, birbirlerinden farklıdır), uzay-kesirli SG denkleminin çözümü breather tiptir; başka bir ifadeyle zaman ve uzay kesirli SG denklemi için benzer başlangıç koşulları kullanıldığında farklı tip soliton çözümleri elde edilmiştir. Ayrıca bu sonuçlar josephson eklemlerindeki vorteks-antivorteks çifleri veya fiber optik iletişimde sinyal dağılımındaki kayıplar gibi bazı olayları kesirli SG denklemleri ile modelleyebileceğini göstermiştir. Diğer bir deyişle, bu çalışma genellikle klasik SG denklemleri ile tanımlanan fiziksel sistemlerin gerçek davranışlarına ışık tutabilir. Çünkü Markovian olmayan süreçlerin bellek etkileri, Gaussian olmayan dağılımların etkileri, sistem ile dış çevre arasındaki etkileşmeler ve klasik SG denkleminin ihmal ettiği sistemler içerisindeki bazı fiziksel kayıplar gibi bazı fiziksel olaylar kesirli SG denklemleri ile hesaba katılabilir.

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“…Numerical solution of partial differential equations is far more demanding than the ordinary ones. Several analytical or numerical methods such as decomposition method [8], variational iteration method [9], He's variational iteration method [10], collocation and radial basis functions [11], auxiliary equation method [12], spectral method [13] [14] [15], wavelet method [16] [17] [18] and the references therein have been proposed for the numerical solution of these types of equations. Among all these method mentioned above, spectral and wavelet method has got more attention of researcher from the last two decades.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Klein/Sine-Gordon Equations by Spectral Method Coupled with Chebyshev Wavelets

Iqbal¹,

Abass²

2016

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The basic aim of this paper is to introduce and describe an efficient numerical scheme based on spectral approach coupled with Chebyshev wavelets for the approximate solutions of Klein-Gordon and Sine-Gordon equations. The main characteristic is that, it converts the given problem into a system of algebraic equations that can be solved easily with any of the usual methods. To show the accuracy and the efficiency of the method, several benchmark problems are implemented and the comparisons are given with other methods existing in the recent literature. The results of numerical tests confirm that the proposed method is superior to other existing ones and is highly accurate.

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Numerical solution of sine-Gordon equation by variational iteration method

Cited by 87 publications

References 31 publications

Numerical Solution of Nonlinear Sine-Gordon Equation by Modified Cubic B-Spline Collocation Method

Numerical Solution of Nonlinear Sine-Gordon Equation by Modified Cubic B-Spline Collocation Method

Approximate Soliton Solutions of Real Order Sine- Gordon Equations

Numerical Solution of Klein/Sine-Gordon Equations by Spectral Method Coupled with Chebyshev Wavelets

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