A three-level average implicit finite difference scheme to solve equation obtained by coupling the Rosenau–KdV equation and the Rosenau–RLW equation

Wongsaijai, Ben; Poochinapan, Kanyuta

doi:10.1016/j.amc.2014.07.075

Cited by 53 publications

(89 citation statements)

References 21 publications

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“…And in this paper, we assume that the wave peak is initially located at x = 0, and x l , x r , which are large numbers, are used to assure that the solitary wave peak is always located inside the domain [x l , x r ] during the time interval [0, T ]. Similar setup is used in [34,35].…”

Section: Introductionmentioning

confidence: 92%

“…For numerical investigation, [35] proposed a threelevel implicit conservative finite difference method for the above Rosenau-KdV-RLW equation. For theoretical studies, [36][37][38][39] studied the solitary wave, shock waves, conservation laws as well as the asymptotic behavior for the Rosenau-KdV-RLW equation with power law nonlinearity.…”

Section: Introductionmentioning

confidence: 99%

“…For numerical investigations, [34] proposed a second-order conservative finite difference method for the Rosenau-KdV equation. By coupling the above Rosenau-RLW equation and Rosenau-KdV equation, one can obtain the following Rosenau-KdV-RLW equation [35][36][37][38][39],…”

Section: Introductionmentioning

confidence: 99%

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New solitary solutions and a conservative numerical method for the Rosenau–Kawahara equation with power law nonlinearity

2015

Nonlinear Dyn

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Recently, Biswas et al. (Phys Wave Phenom 19:24-29, 2011) derived exact bright and dark solitary solutions for the Rosenau-Kawahara equation with power law nonlinearity, and Hu et al. (Adv Math Phys 11, Article ID 217393, 2014) proposed two conservative finite difference schemes for the usual RosenauKawahara equation. In this paper, we obtain another set of exact solitary solutions for the Rosenau-Kawahara equation with power law nonlinearity. More importantly, a conservative finite difference method is presented. The fundamental energy conservation law is preserved by the current numerical scheme. And the existence and uniqueness of the numerical solution are proved. The convergence and stability of the numerical solution are also shown. The method is second-order convergent both in time and in space. Finally, numerical results confirm well with the theoretical results and show that the current method can be well used to study the solitary wave at long time.

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Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

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New solitary solutions and a conservative numerical method for the Rosenau–Kawahara equation with power law nonlinearity

2015

Nonlinear Dyn

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“…Some finite difference schemes for the solution of Rosenau-KdV equation and the generalized Rosenau-KdV equation can be seen in [7,24]. The conservation laws of the Rosenau-KdV-RLW equation are computed with power law nonlinearity by the aid of multiplier approach in Lie symmetry analysis [13,23]. A numerical approach with a new formulation for a nonlinear wave proposed by coupling the Rosenau-KdV equation and the Rosenau-RLW equation is presented.…”

Section: Introductionmentioning

confidence: 99%

Numerical Study of Rosenau-KDV Equation Using Finite Element Method Based on Collocation Approach

Dhawan

Karakoç

et al. 2017

Mathematical Modelling and Analysis

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Abstract. In the present paper, a numerical method is proposed for the numerical solution of Rosenau-KdV equation with appropriate initial and boundary conditions by using collocation method with septic B-spline functions on the uniform mesh points. The method is shown to be unconditionally stable using von-Neumann technique. To check accuracy of the error norms L2 and L∞ are computed. Interaction of two and three solitary waves are used to discuss the effect of the behavior of the solitary waves during the interaction. Furthermore, evolution of solitons is illustrated by undular bore initial condition. These results show that the technique introduced here is suitable to investigate behaviors of shallow water waves.

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“…Those classical finite difference schemes often use the formula ( ) = ( /(1 + ))[ −1 + ( ) ] to construct second-order convergent linear finite difference scheme [11-15, 18, 20, 22]. In [23], the authors used the formula ( 2 ) = 2 +(1− )( 2 ) and proposed a linear finite difference scheme, but the proof of the prior estimate in ∞ -norm is not perfect.…”

Section: Introductionmentioning

confidence: 99%

A New Linear Difference Scheme for Generalized Rosenau-Kawahara Equation

Chen

Xiang

Chen

et al. 2018

Mathematical Problems in Engineering

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We introduce in this paper a new technique, a semiexplicit linearized Crank-Nicolson finite difference method, for solving the generalized Rosenau-Kawahara equation. We first prove the second-order convergence in ∞ -norm of the difference scheme by an induction argument and the discrete energy method, and then we obtain the prior estimate in ∞ -norm of the numerical solutions. Moreover, the existence, uniqueness, and satiability of the numerical solution are also shown. Finally, numerical examples show that the new scheme is more efficient in terms of not only accuracy but also CPU time in implementation.

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A three-level average implicit finite difference scheme to solve equation obtained by coupling the Rosenau–KdV equation and the Rosenau–RLW equation

Cited by 53 publications

References 21 publications

New solitary solutions and a conservative numerical method for the Rosenau–Kawahara equation with power law nonlinearity

New solitary solutions and a conservative numerical method for the Rosenau–Kawahara equation with power law nonlinearity

Numerical Study of Rosenau-KDV Equation Using Finite Element Method Based on Collocation Approach

A New Linear Difference Scheme for Generalized Rosenau-Kawahara Equation

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