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

Ak, Turgut; Karakoç, Seydi Battal Gazi; Biswas, Anjan

doi:10.24200/sci.2017.4096

Cited by 19 publications

(13 citation statements)

References 20 publications

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“…xx (x; t)]dx; (21) corresponding to the mass and energy of the shallow water waves, respectively [8]. Susceptibility of the numerical scheme is controlled by both the error norm [37]:…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

A new numerical application of the generalized Rosenau-RLW equation

Karakoç

2018

Scientia Iranica

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This study implemented a collocation nite element method based on septic B-splines as a tool to obtain the numerical solutions of the nonlinear generalized Rosenau-RLW equation. One of the advantages of this method is that when the bases are chosen at a high degree, better numerical solutions are obtained. E ectiveness of the method is demonstrated by solving the equation with various initial and boundary conditions. Further, in order to detect the performance of the method, L 2 and L 1 error norms and two lowest invariants IM and IE were computed. The obtained numerical results were compared with some of those in the literature for similar parameters. This comparison clearly shows that the obtained results are better than and in good conformity with some of the earlier results. Stability analysis demonstrates that the proposed algorithm, based on a Crank Nicolson approximation in time, is unconditionally stable.

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“…xx (x; t)]dx; (21) corresponding to the mass and energy of the shallow water waves, respectively [8]. Susceptibility of the numerical scheme is controlled by both the error norm [37]:…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

A new numerical application of the generalized Rosenau-RLW equation

Karakoç

2018

Scientia Iranica

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“…Lately, collocation scheme based on B-spline nite elements was investigated for solving the Complex Modi ed Korteweg-de Vries (CMKdV), the generalized nonlinear Schrodinger (GNLS) equation, and generalized Burgers-Fisher and Burgers-Huxley equations [35][36][37]. Moreover, Petrov-Galerkin nite element method based on B-splines was presented for the numerical calculation of the modi ed Korteweg-de Vries (mKdV) equation by Ak et al [38].…”

Section: Introductionmentioning

confidence: 99%

A collocation algorithm based on quintic B-splines for the solitary wave simulation of the GRLW equation

Zeybek

Karakoç

2018

Scientia Iranica

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In this article, a collocation algorithm based on quintic B-splines is proposed to nd a numerical solution to the nonlinear Generalized Regularized Long Wave (GRLW) equation. Moreover, to analyze the linear stability of the numerical scheme, the von-Neumann technique is used. The numerical approach to three test examples consisting of a single solitary wave, the collision of two solitary waves, and the growth of an undular bore is discussed. The accuracy of the method is demonstrated by calculating the error in L2 and L 1 norms and the conservative quantities I 1 , I 2 and I 3 . The ndings are compared with those previously reported in the literature. Finally, the motion of solitary waves is graphically plotted according to di erent parameters.

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“…mKdV denkleminin çoklu soliton çözümleri ile (3+1) boyutlu mKdV denkleminin çözümleri Wazwaz tarafından bulunmuştur [23,24]. Ak ve çalışma arkadaşları mKdV denklemine Lumped Petrov-Galerkin and Galerkin yöntemlerini uygulamışlardır [25,26]. Bu çalışmada, mKdV denkleminin sayısal çözümleri septik B-spline kollokasyon sonlu eleman yöntemi kullanılarak elde edildi.…”

Section: Introductionunclassified

NUMERICAL SOLUTIONS OF THE MKdV EQUATION VIA COLLOCATION FINITE ELEMENT METHOD

Karakoç¹

2018

Anadolu Üniversitesi Bilim Ve Teknoloji Dergisi - B Teorik Bilimler

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Bu çalışmada, modifiye edilmiş Korteweg-de Vries (mKdV) denkleminin sayısal çözümleri septik B-spline kollokasyon sonlu eleman yöntemi kullanılarak elde edilmiştir. Önerilen sayısal algoritmanın doğruluğu, tek soliton dalga, iki ve üç soliton dalganın girişimi gibi üç test probleminin uygulanması ile kontrol edilmiştir. Zamana bağlı Crank Nicolson yaklaşımına dayanan sayısal algoritmamız şartsız olarak kararlıdır. Yeni uygulanan yöntemin performansını kontrol etmek için, 2 , ∞ hata normları ile 1 , 2 , 3 ve 4 değişmezlerinin değerleri hesaplanmıştır. Elde edilen sayısal sonuçlar literatürde bulunan diğer sonuçlarla karşılaştırılmıştır.

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

Cited by 19 publications

References 20 publications

A new numerical application of the generalized Rosenau-RLW equation

A new numerical application of the generalized Rosenau-RLW equation

A collocation algorithm based on quintic B-splines for the solitary wave simulation of the GRLW equation

NUMERICAL SOLUTIONS OF THE MKdV EQUATION VIA COLLOCATION FINITE ELEMENT METHOD

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