On the identification of nonlinear terms in the generalized Camassa-Holm equation involving dual-power law nonlinearities

Nanta, Supawan; Yimnet, Suriyon; Poochinapan, Kanyuta; Wongsaijai, Ben

doi:10.1016/j.apnum.2020.10.006

Cited by 16 publications

(3 citation statements)

References 56 publications

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“…Remark 16 In Schemes II and III the difference approximation of the nonlinear term uu x is widely used to ensure the energy conservation, stability, and convergence of the numerical solutions. For further information, the reader should consult references [43][44][45][46][47] for various types of equations with the same treatments of the nonlinear term uu x .…”

Section: Numerical Experimentsmentioning

confidence: 99%

Novel advances in high-order numerical algorithm for evaluation of the shallow water wave equations

Poochinapan

Wongsaijai

2023

Adv Cont Discr Mod

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In this paper, we propose a high-order nonlinear algorithm based on a finite difference method modification to the regularized long wave equation and the Benjamin–Bona–Mahony–Burgers equation subject to the homogeneous boundary. The consequence system of nonlinear equations typically trades with high computation burden. This dilemma can be overcome by establishing a fast numerical algorithm procedure without a reduction of numerical accuracy. The proposed algorithm forms a linear system with constant coefficient matrix at each time step and produces numerical solutions, which remarkably gains many computational advantages. In terms of analysis, a priori estimation for the numerical solution is derived to obtain the convergence and stability analysis. Additionally, the algorithm is globally mass preserving to avoid nonphysical behavior. Two benchmarks, including a single solitary wave to both equations, are given to validate the applicability and accuracy of the proposed method. Numerical results are obtained and compared to other approaches available in the literature. From the comparisons it is clear that the proposed approach produces accurate and precise results at low computational cost. Besides, the proposed scheme is applied to study the effect of the viscous term on a single solitary wave. It is shown that the viscous term results in the amplitude and width of the initial condition but not in its velocities in the case of a single solitary wave. As a consequence, theoretical and numerical findings provide a new area to investigate and expand the high-order algorithm for the family of wave equations.

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Section: Numerical Experimentsmentioning

confidence: 99%

Novel advances in high-order numerical algorithm for evaluation of the shallow water wave equations

Poochinapan

Wongsaijai

2023

Adv Cont Discr Mod

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“…Due to the method's preservation of physical properties, it had a similar or greater impact on numerical simulations [27,37,38,[41][42][43][44][45][46][47]. Therefore, in this example, we intended to provide and monitor the numerical simulations obtained in Examples 1 and 2 to validate the energy-preserving property discussed in Theorem 4.…”

Section: Conservative Approximationsmentioning

confidence: 99%

Analytical and numerical studies on dynamic of complexiton solution for complex nonlinear dispersive model

Phumichot,

Kuntiya,

Poochinapan

et al. 2023

Math Methods in App Sciences

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In this paper, we apply the basic one‐way propagator in order to introduce complex regularized long wave (RLW) equation. The complex RLW can be transformed into a system of nonlinear equations. By adopting a consequence nonlinear system, we can derive an energy conservation law of a complex regularized long wave equation. We then investigate how the Ansatz method may be applied to find a class of solitary wave solutions. Simultaneously, a numerical scheme for solving the model is implemented using a finite difference method based on the energy‐preserving Crank–Nicolson/Adams–Bashforth technique. It is worth mentioning that the obtained system is nonlinear. However, by using the present algorithm, we are able to linearize this system and solve it because of the implicit nature of the system of equations. An a priori estimate of the numerical solutions is derived to obtain a convergence and stability analysis; this yields second‐order accuracy in both time and space. Additionally, some numerical experiments verify computational efficiency. The results indicate that this method is an excellent way to preserve energy conservation, providing second‐order accuracy both in time and space with a maximum norm. In addition, we use the proposed scheme to study the effects of dispersive parameters when proceeding with an initial complex Gaussian condition.

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“…In 2010, Lai [50] established the existence and uniqueness of a local solution of the CH equation in Sobolev space H s ðℝÞ, and the well-posedness was established by Li and Olver [49]. Very recently, Nanta et al [51] obtained the numerical study of the generalized Camassa-Holm equation involving dual-power law nonlinearities. Other studies of CH-related equation are also reported by various publications [52][53][54][55][56][57].…”

Section: Introductionmentioning

confidence: 99%

On Solitary Wave Solutions for the Camassa-Holm and the Rosenau-RLW-Kawahara Equations with the Dual-Power Law Nonlinearities

Sukantamala

Nanta

2021

Abstract and Applied Analysis

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The nonlinear wave equation is a significant concern to describe wave behavior and structures. Various mathematical models related to the wave phenomenon have been introduced and extensively being studied due to the complexity of wave behaviors. In the present work, a mathematical model to obtain the solution of the nonlinear wave by coupling the classical Camassa-Holm equation and the Rosenau-RLW-Kawahara equation with the dual term of nonlinearities is proposed. The solution properties are analytically derived. The new model still satisfies the fundamental energy conservative property as the original models. We then apply the energy method to prove the well-posedness of the model under the solitary wave hypothesis. Some categories of exact solitary wave solutions of the model are described by using the Ansatz method. In addition, we found that the dual term of nonlinearity is essential to obtain the class of analytic solution. Besides, we provide some graphical representations to illustrate the behavior of the traveling wave solutions.

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On the identification of nonlinear terms in the generalized Camassa-Holm equation involving dual-power law nonlinearities

Cited by 16 publications

References 56 publications

Novel advances in high-order numerical algorithm for evaluation of the shallow water wave equations

Novel advances in high-order numerical algorithm for evaluation of the shallow water wave equations

Analytical and numerical studies on dynamic of complexiton solution for complex nonlinear dispersive model

On Solitary Wave Solutions for the Camassa-Holm and the Rosenau-RLW-Kawahara Equations with the Dual-Power Law Nonlinearities

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