An Efficient Approach to Numerical Study of the MRLW Equation with B-Spline Collocation Method

Karakoç, Seydi Battal Gazi; Ak, Turgut; Zeybek, Halil

doi:10.1155/2014/596406

Cited by 21 publications

(11 citation statements)

References 39 publications

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“…The theory of nonlinear evolution equations (NLEEs) is a very popular and fascinating area of research in the eld of applied mathematics and theoretical physics [1][2][3][4][5][6][7][8][9]. A few of the focused areas of research with NLEEs are uid dynamics, nonlinear optics, nuclear physics, plasma physics, mathematical biosciences, and several others.…”

Section: Introductionmentioning

confidence: 99%

Application of Petrov-Galerkin finite element method to shallow water waves model: Modified Korteweg-de Vries equation

Ak¹,

Karakoç²,

Biswas³

2017

Scientia Iranica

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Abstract. In this article, modi ed Korteweg-de Vries (mKdV) equation is solved numerically by using lumped Petrov-Galerkin approach, where weight functions are quadratic and the element shape functions are cubic B-splines. The proposed numerical scheme is tested by applying four test problems including single solitary wave, interaction of two and three solitary waves, and evolution of solitons with the Gaussian initial condition. In order to show the performance of the algorithm, the error norms, L2, L1, and a couple of conserved quantities are computed. For the linear stability analysis of numerical algorithm, Fourier method is also investigated. As a result, the computed results show that the presented numerical scheme is a successful numerical technique for solving the mKdV equation. Therefore, the presented method is preferable to some recent numerical methods.

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Section: Introductionmentioning

confidence: 99%

Application of Petrov-Galerkin finite element method to shallow water waves model: Modified Korteweg-de Vries equation

Ak¹,

Karakoç²,

Biswas³

2017

Scientia Iranica

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“…( 2008 ) and Karakoç et al. ( 2014 ). Collocation method based on quintic B-spline functions with Rubin and Graves linearization technique was investigated for solving the MRLW equation by Karakoç et al.…”

Section: Introductionmentioning

confidence: 95%

A numerical investigation of the GRLW equation using lumped Galerkin approach with cubic B-spline

Zeybek

Karakoç

2016

SpringerPlus

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In this work, we construct the lumped Galerkin approach based on cubic B-splines to obtain the numerical solution of the generalized regularized long wave equation. Applying the von Neumann approximation, it is shown that the linearized algorithm is unconditionally stable. The presented method is implemented to three test problems including single solitary wave, interaction of two solitary waves and development of an undular bore. To prove the performance of the numerical scheme, the error norms and and the conservative quantities , and are computed and the computational data are compared with the earlier works. In addition, the motion of solitary waves is described at different time levels.

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“…A sextic B-spline collocation algorithm have been developed for solving numerically MRLW equation [20]. The septic and quintic B-spline collocation methods, the subdomain finite element method based on quartic B-splines, the cubic B-spline Galerkin and Petrov-Galerkin methods are implemented to find the numerical solution of the MRLW equation [21][22][23][24][25]. A numerical technique including quartic B-splines for solution of the MRLW equation has been produced by Fazal-i-Haq et all [26].…”

Section: Introductionmentioning

confidence: 99%

Değiştirilmiş düzenli uzun dalga denkleminin kübik trigonometrik B-spline fonksiyonları kullanılarak nümerik çözümü

Görgülü¹,

Irk²

2019

Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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In this study, the Modified Regularized Long Wave (MRLW) equation is solved numerically. The method used for the numerical solution of MRLW equation includes the space discretization with the Galerkin finite element method based on cubic trigonometric B-spline, and also the time discretization with the Crank-Nicolson method. We tried to obtain a more accurate method with the help of trigonometric Bspline for the numerical solution of the MRLW equation than the existing numerical methods in the first test problem. Then, the interaction problem of the two positive solitary waves of the MRLW equation is considered, and the conservation constants are compared with the existing ones to see the correctness of the method. Değiştirilmiş düzenli uzun dalga denkleminin kübik trigonometrik B-spline fonksiyonları kullanılarak nümerik çözümü Özet Bu çalışmada, Modified Regularized Long Wave (MRLW) denklemi sayısal olarak çözülmüştür. MRLW denkleminin sayısal çözümü için kullanılan yöntem, kübik trigonometrik B-spline'a dayalı Galerkin sonlu eleman yöntemi ile konum ayrıştırmasını ve ayrıca Crank-Nicolson yöntemiyle zaman ayrıştırmasını içerir. İlk test probleminde MRLW denkleminin sayısal çözümü için trigonometrik B-spline yardımıyla mevcut * Melis ZORŞAHİN GÖRGÜLÜ, mzorsahin@ogu.edu.tr, https://orcid.org/0000-0001-7506-4162 Dursun IRK, dirk@ogu.edu.tr, https://orcid.org/0000-0002-3340-1578 sayısal metotlardan daha doğru bir yöntem elde etmeye çalıştık. Daha sonra, MRLW denkleminin iki pozitif solitary dalganın etkileşimi problemi göz önüne alınmış ve korunum sabitleri, yöntemin doğruluğunu görmek için mevcut çalışmalarla karşılaştırılmıştır.Anahtar kelimeler: Galerkin metodu, kübik trigonometrik B-spline, değiştirilmiş düzenli uzun dalga denklemi.

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An Efficient Approach to Numerical Study of the MRLW Equation with B-Spline Collocation Method

Cited by 21 publications

References 39 publications

Application of Petrov-Galerkin finite element method to shallow water waves model: Modified Korteweg-de Vries equation

Application of Petrov-Galerkin finite element method to shallow water waves model: Modified Korteweg-de Vries equation

A numerical investigation of the GRLW equation using lumped Galerkin approach with cubic B-spline

Değiştirilmiş düzenli uzun dalga denkleminin kübik trigonometrik B-spline fonksiyonları kullanılarak nümerik çözümü

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