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

Zeybek, Halil; Karakoç, Seydi Battal Gazi

doi:10.24200/sci.2018.20781

Cited by 3 publications

(2 citation statements)

References 32 publications

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“…The conservation laws for the model are yet to be extracted. A wide variety of rich mathematical methodologies are available for implementation [12][13][14][15][16][17][18][19][20][21][22][23][24][25]. Thus, there are plentiful issues that need to be addressed with the dynamics of embedded solitons.…”

Section: Discussionmentioning

confidence: 99%

Embedded Solitons With X(2) Nonlinear Susceptibility

et al. 2022

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This paper recovers optical soliton solutions with χ (2) -nonlinear susceptibility. Bright, dark, singular, brightdark combo solitons are recovered. A variety of algorithms are implemented. These include the Riccati equation approach, exp-function expansion method, modified simple equation algorithm, sine-Gordon equation scheme, F -expansion approach, trial function method and functional variable algorithm.

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Section: Discussionmentioning

confidence: 99%

Embedded Solitons With X(2) Nonlinear Susceptibility

et al. 2022

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“…Moreover, as shown in Part I (Ramos and García L opez, 2020), equation (1) reduces to that of the standard (p ¼ 2), modified (p ¼ 3) and generalized (p > 3) inviscid RLW equations when both the relaxation time and the viscosity coefficient are nil. The standard (Saka et al, 2011;Mittal and Rohila, 2018), modified (Karakoç et al, 2013(Karakoç et al, , 2014(Karakoç et al, , 2015 and generalized (Ramos, 2016;García L opez and Ramos, 2015;Ramos and García L opez, 2017;Karakoç and Zeybek, 2016;Zeybek and Karakoç, 2019;Karakoç et al, 2022) inviscid RLW equations, as well as the standard, modified and generalized inviscid equal-width (Onder et al, 2023) equations have been the subject of a large number of numerical studies aimed at understanding solitary wave propagation and interactions between solitary waves and assessing the accuracy of numerical methods by comparing the numerical results with the available analytical solutions for these equations and their finite number of invariants. By way of contrast, few analytical and numerical studies on the viscous equal-width (Ramos, 2006(Ramos, , 2007 and viscous RLW equations (García L opez and Ramos, 2015) which may be obtained from equation (1) by setting t ¼ 0, have been reported.…”

Section: Introductionmentioning

confidence: 99%

Effect of initial conditions on a one-dimensional model of small-amplitude wave propagation in shallow water. II: Blowup for nonsmooth conditions

Ramos,

García López

2023

HFF

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Purpose The purpose of this paper is to analyze numerically the blowup in finite time of the solutions to a one-dimensional, bidirectional, nonlinear wave model equation for the propagation of small-amplitude waves in shallow water, as a function of the relaxation time, linear and nonlinear drift, power of the nonlinear advection flux, viscosity coefficient, viscous attenuation, and amplitude, smoothness and width of three types of initial conditions. Design/methodology/approach An implicit, first-order accurate in time, finite difference method valid for semipositive relaxation times has been used to solve the equation in a truncated domain for three different initial conditions, a first-order time derivative initially equal to zero and several constant wave speeds. Findings The numerical experiments show a very rapid transient from the initial conditions to the formation of a leading propagating wave, whose duration depends strongly on the shape, amplitude and width of the initial data as well as on the coefficients of the bidirectional equation. The blowup times for the triangular conditions have been found to be larger than those for the Gaussian ones, and the latter are larger than those for rectangular conditions, thus indicating that the blowup time decreases as the smoothness of the initial conditions decreases. The blowup time has also been found to decrease as the relaxation time, degree of nonlinearity, linear drift coefficient and amplitude of the initial conditions are increased, and as the width of the initial condition is decreased, but it increases as the viscosity coefficient is increased. No blowup has been observed for relaxation times smaller than one-hundredth, viscosity coefficients larger than ten-thousandths, quadratic and cubic nonlinearities, and initial Gaussian, triangular and rectangular conditions of unity amplitude. Originality/value The blowup of a one-dimensional, bidirectional equation that is a model for the propagation of waves in shallow water, longitudinal displacement in homogeneous viscoelastic bars, nerve conduction, nonlinear acoustics and heat transfer in very small devices and/or at very high transfer rates has been determined numerically as a function of the linear and nonlinear drift coefficients, power of the nonlinear drift, viscosity coefficient, viscous attenuation, and amplitude, smoothness and width of the initial conditions for nonzero relaxation times.

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Numerical approximation with the splitting algorithm to a solution of the modified regularized long wave equation

KARTA

2023

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

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In this article, a Lie-Totter splitting algorithm, which is highly reliable, flexible and convenient, is proposed along with the collocation finite element method to approximate solutions of the modified regular long wave equation. For this article, quintic B-spline approximation functions are used in the implementation of collocation methods. Four numerical examples including a single solitary wave, the interaction of two- three solitary waves, and a Maxwellian initial condition are presented to test the closeness of the solutions obtained by the proposed algorithm to the exact solutions. The solutions produced are compared with those in some studies with the same parameters that exist in the literature. The fact that the present algorithm produces results as intended is a proof of how useful, accurate and reliable it is. It can be stated that this fact will be very useful the application of the presented technique for other partial differential equations, with the thought that it may lead the reader to obtain superior results from this study.

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

Cited by 3 publications

References 32 publications

Embedded Solitons With X(2) Nonlinear Susceptibility

Embedded Solitons With X(2) Nonlinear Susceptibility

Effect of initial conditions on a one-dimensional model of small-amplitude wave propagation in shallow water. II: Blowup for nonsmooth conditions

Numerical approximation with the splitting algorithm to a solution of the modified regularized long wave equation

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