Solitary-wave solutions of the GRLW equation using septic B-spline collocation method

Karakoç, Seydi Battal Gazi; Zeybek, Halil

doi:10.1016/j.amc.2016.05.021

Cited by 23 publications

(35 citation statements)

References 30 publications

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“…Many researchers have proposed numerical solution for the nonlinear equations including Burger equations, such as, Galerkin B-Spline-collocation method (Bryan et al, 2017), exponential cubic B-spline differential quadrature method (Korkmaz and Akmaz, 2015), trigonometric cubic B-spline differential quadrature method (Korkmaz and Akmaz, 2018), cubic Bspline collocation method (Sharifi and Rashidinia, 2016), B-spline collocation and self-adapting differential evolution (jDE) algorithm (Luo et al, 2018), fourth-order cubic B-spline collocation method (Rohila and Mittal, 2018), cubic B-spline collocation scheme (Mittal and Arora, 2011), non-polynomial spline method (Ali et al, 2015), collocation method with cubic trigonometric B-spline (Raslan et al, 2016), collocation method with quintic B-spline method (Raslan et al, 2017), generalized differential quadrature method (Mokhtari et al, 2011), exponential cubic B-spline finite element method (Ersoy and Dag, 2015), B-spline Differential Quadrature Method (Bashan et al, 2015), and the Galerkin quadratic Bspline finite element method (Kutluay and Ucar, 2013). The septic B-spline approach has been used to establish approximate solutions for several partial differential equations (Ramadan et al, 2005;El-Danaf, 2008;Soliman and Hussien, 2005;Quarteroni et al, 2007;Karakoc and Zeybek, 2016;Geyikli and Karakoc, 2011).…”

Section: Introductionmentioning

confidence: 99%

Septic B-spline collocation method for numerical solution of the coupled Burgers’ equations

Shallal

Ali

Raslan

et al. 2019

Arab Journal of Basic and Applied Sciences

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In this paper, a numerical solution of the coupled Burgers' equations based on septic Bspline collocation method is presented. The scheme is based on the Crank-Nicolson formulation for time integration and septic B-spline functions for space integration. The method has been showed unconditionally stable by using Von-Neumann technique. The efficiency of this method is demonstrated by applying two test problems. The obtained numerical results are found in a good agreement with the exact solution. This method is efficient, powerful, and economical. It can also applicable to other linear and nonlinear partial differential equations.

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Section: Introductionmentioning

confidence: 99%

Septic B-spline collocation method for numerical solution of the coupled Burgers’ equations

Shallal

Ali

Raslan

et al. 2019

Arab Journal of Basic and Applied Sciences

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“…where m (t) is the time-dependent parameter and will be de ned under the boundary and collocation conditions [30]. 7 [x m 1 ;…”

Section: Collocation Methods With Septic B-splinesmentioning

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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“…Mohammadi [32] obtained a numerical solution to the nonlinear GRLW equation using collocation algorithm based on exponential B-spline nite elements. Zeybek and Karako c used a nite element method with B-splines to solve the GRLW equation [33,34]. 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].…”

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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Solitary-wave solutions of the GRLW equation using septic B-spline collocation method

Cited by 23 publications

References 30 publications

Septic B-spline collocation method for numerical solution of the coupled Burgers’ equations

Septic B-spline collocation method for numerical solution of the coupled Burgers’ equations

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

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