The Korteweg–de Vries–Kawahara equation in a bounded domain and some numerical results

Ceballos, Juan Carlos; Sepúlveda, Mauricio; Villagrán, Octavio Paulo Vera

doi:10.1016/j.amc.2007.01.107

Cited by 39 publications

(41 citation statements)

References 19 publications

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“…The order of convergence decreases by increasing collocation points for a fixed time step Δt = 0.001. The forward movement of the solitary wave at different time levels in comparison with the exact solution (26) is shown in Figure 6, same as in [52]. The two solitary waves propagate towards right as the time progresses.…”

Section: Numerical Applicationmentioning

confidence: 68%

Meshless Method of Lines for Numerical Solution of Kawahara Type Equations

Bibi¹,

Tirmizi²,

Haq³

2011

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In this work, an algorithm based on method of lines coupled with radial basis functions namely meshless method of lines (MMOL) is presented for the numerical solution of Kawahara, modified Kawahara and KdV Kawahara equations. The motion of a single solitary wave, interaction of two and three solitons and the phenomena of wave generation is discussed. The results are compared with the exact solution and with the results in the relevant literature to show the efficiency of the method.

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Section: Numerical Applicationmentioning

confidence: 68%

Meshless Method of Lines for Numerical Solution of Kawahara Type Equations

Bibi¹,

Tirmizi²,

Haq³

2011

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“…This forward differences approximation is done in order to obtain a positive definite matrix I + δt A, and was used in a first time for a simple KdV equation in a bounded domain in [11] (see, also an application of the same idea for a fifth order dispersive equation in [10,32], and for a coupled system of KdV types equations in [4]). The approximation of the damping function a = a(x) is given by…”

Section: Description Of the Numerical Schemementioning

confidence: 99%

“…The numerical schemes proposed here, are based on unconditional stable finite difference method used and described for the KdV equation in Colin and Gisclon [11], for the KdV-Kawahara equation in Ceballos et al [10,32], and for a system of KdV equation in Bisognin et al [4]. However, in the case of GKdV-4 equation, we had to make a special treatment of the nonlinearity u 4 u x , rewriting it in a particular and a bit sophisticated way, taking into account the invariance of the mass scaling in the L 2 -norm.…”

Section: Introductionmentioning

confidence: 99%

Uniform stabilization of numerical schemes for the critical generalized Korteweg-de Vries equation with damping

2010

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This work is devoted to the analysis of a fully-implicit numerical scheme for the critical generalized Korteweg-de Vries equation (GKdV with p = 4) in a bounded domain with a localized damping term. The damping is supported in a subset of the domain, so that the solutions of the continuous model issuing from small data are globally defined and exponentially decreasing in the energy space. Based in this asymptotic behavior of the solution, we introduce a finite difference scheme, which despite being one of the first order, has the good property to converge in L 4 -strong. Combining this strong convergence with discrete multipliers and a contradiction argument, we show that the smallness of the initial condition leads to the uniform (with This work was supported by grant 49.0987/2005-2 of the Cooperation CNPq/CONICYT (Brasil-Chile). M.S. has been also supported by Fondecyt Project #1070694, FONDAP and BASAL projects CMM, Universidad de Chile, and CI 2 MA, Universidad de Concepción. O.V.V. has been supported by Postdoctoral fellowship of LNCC (National Laboratory for Scientific Computation), and he is grateful for the dependences of the LNCC while he realized postdoctorate.

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“…Since applicability of the analytical methods to these equations is limited for a wider class of initial and boundary conditions, it is of considerable numerical interest for the understanding of the nonlinear behavior of the Kawahara-type equations for variety of the initial and boundary conditions. A finite difference scheme was proposed to find the numerical solution of the KdV-Kawahara equation in [12]. A dual-Petrov Galerkin method for the Kawahara type equations is developed in [6].…”

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confidence: 99%