Exact solitary solution and a three-level linearly implicit conservative finite difference method for the generalized Rosenau–Kawahara-RLW equation with generalized Novikov type perturbation

He, Dongdong

doi:10.1007/s11071-016-2700-x

Cited by 33 publications

(16 citation statements)

References 77 publications

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“…Readers can refer to previous studies. [25][26][27][28][29][30] As for the KdV equation, its interest is that it comes up naturally in all kinds of models. Among them are internal gravity waves in a stratified fluid (highly relevant for geophysics), waves in an astrophysical plasma, electrical transmission line behavior, and even blood pressure waves-soliton solutions to the KdV equation, which explain why our pulse, coming from a localized pressure wave in our arteries, is detectable all over our body and persists despite changes in local conditions and artery geometry in the circulation system.…”

Section: Introductionmentioning

confidence: 99%

A new conservative fourth‐order accurate difference scheme for solving a model of nonlinear dispersive equations

Ghiloufi

Rouatbi

Omrani

2018

Math Methods in App Sciences

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This article is devoted to the study of a nonlinear conservative fourth‐order difference scheme for a model of nonlinear dispersive equations that is governed by the RLW‐KdV equation. The existence of the approximate solution and the convergence of the difference scheme are proved, by using the energy method. In addition, the convergent order in maximum norm is 2 in temporal direction and 4 in spatial direction. The unconditional stability as well as uniqueness of the difference scheme is also derived. An application on the RLW and MRLW equations is discussed numerically in details. Furthermore, interaction of solitary waves with different amplitudes are shown. The 3 invariants of the motion are evaluated to determine the conservation proprieties of the system. The temporal evaluation of a Maxwellian initial pulse is then studied. Some numerical examples are given to validate the theoretical results.

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Section: Introductionmentioning

confidence: 99%

A new conservative fourth‐order accurate difference scheme for solving a model of nonlinear dispersive equations

Ghiloufi

Rouatbi

Omrani

2018

Math Methods in App Sciences

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“…Crank-Nicolson Scheme and Richardson's Extrapolation. We use a three-level Crank-Nicolson scheme (see [18][19][20][21]) of second-order accuracy to solve the nonlinear and inhomogeneous partial differential equation given by (19) and use Richardson's extrapolation technique for further improving accuracy. Numerically, one can only solve (19) over a finite domain…”

Section: Methodsmentioning

confidence: 99%

Numerical Solutions to Optimal Portfolio Selection and Consumption Strategies under Stochastic Volatility

Zhang²

2020

Complexity

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Based on the method of dynamic programming, this paper uses analysis methods governed by the nonlinear and inhomogeneous partial differential equation to study modern portfolio management problems with stochastic volatility, incomplete markets, limited investment scope, and constant relative risk aversion (CRRA). In this paper, a three-level Crank–Nicolson finite difference scheme is used to determine numerical solutions under this general setting. One of the main contributions of this paper is to apply this three-level technology to solve the portfolio selection problem. In addition, we have used a technique to deal with the nonlinear term, which is another novelty in performing the Crank–Nicolson algorithm. The Crank–Nicolson algorithm has also been extended to third-order accuracy by performing Richardson’s extrapolation. The accuracy of the proposed algorithm is much higher than the traditional finite difference method. Lastly, experiments are conducted to show the performance of the proposed algorithm.

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“…In [33], two conservative high-order accurate finite difference schemes for the periodic initial value generalized Rosenau-Kawahara-RLW equation were introduced and extensively studied. For more related nonlinear wave equations, readers can refer to [34][35][36][37][38][39][40][41][42][43].…”

Section: Introductionmentioning

confidence: 99%

“…In [29], Wongsaijai and Poochinapan used the sine-cosine method to find the exact solution of the Rosenau-RLW-KdV equation. He and Pan [32] also used the sine-cosine method to obtain the solitary solution for the generalized Rosenau-Kawahara-RLW equation, and the solution for the Rosenau-Kawahara-RLW equation with, notably, the generalized Novikov type perturbation was solely derived by He [38]. The solution of (2 + 1) dimensional of nonlinear wave equation using modified exponential function method and Ansatz function technique with symbolic computation was proposed in [61].…”

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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Exact solitary solution and a three-level linearly implicit conservative finite difference method for the generalized Rosenau–Kawahara-RLW equation with generalized Novikov type perturbation

Cited by 33 publications

References 77 publications

A new conservative fourth‐order accurate difference scheme for solving a model of nonlinear dispersive equations

A new conservative fourth‐order accurate difference scheme for solving a model of nonlinear dispersive equations

Numerical Solutions to Optimal Portfolio Selection and Consumption Strategies under Stochastic Volatility

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

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