Solitary Wave Solutions of the Generalized Rosenau-KdV-RLW Equation

This paper investigates the solitary wave solutions of the generalized Rosenau–Korteweg-de Vries-regularized-long wave equation. This model is obtained by coupling the Rosenau–Korteweg-de Vries and Rosenau-regularized-long wave equations. The solution of the equation is approximated by a local meshless technique called radial basis function (RBF) and the finite-difference (FD) method. The association of the two techniques leads to a meshless algorithm that does not requires the linearization of the nonlinear terms. First, the partial differential equation is transformed into a system of ordinary differential equations (ODEs) using radial kernels. Then, the ODE system is solved by means of an ODE solver of higher-order. It is shown that the proposed method is stable. In order to illustrate the validity and the efficiency of the technique, five problems are tested and the results compared with those provided by other schemes.info:eu-repo/semantics/publishedVersio

A detailed numerical study on generalized ROSENAU-KDV equation with finite element method

In this study, we have got numerical solutions of the generalized RosenauKdV equation by using collocation finite element method in which septic B-splines are used as approximate functions. Effectivity and proficiency of the method are shown by solving the equation with different initial and boundary conditions. Also, to do this L and L 2 error norms and two lowest invariants MI and EI have been computed. A linear stability analysis indicates that our algorithm, based on a Crank Nicolson approximation in time, is unconditionally stable. An error analysis of the new algorithm has been made. The obtained numerical solutions are compared with some earlier studies. This comparison clearly indicates that the obtained results are better than the earlier results

A numerical study using finite element method for generalized RosenauKawahara-RLW equation

In this paper, we are going to obtain the soliton solution of the generalized RosenauKawahara-RLW equation that describes the dynamics of shallow water waves in oceans and rivers. We confirm that our new algorithm is energy-preserved and unconditionally stable. In order to determine the performance of our numerical algorithm, we have computed the error norms L2 and L∞. Convergence of full discrete scheme is firstly studied. Numerical experiments are implemented to validate the energy conservation and effectiveness for longtime simulation. The obtained numerical results have been compared with a study in the literature for similar parameters. This comparison clearly shows that our results are much better than the other results

Numerical simulation for treatment of dispersive shallow water waves with Rosenau-KdV equation

In this paper, numerical solutions for the Rosenau-Korteweg-de Vries equation are studied by using the subdomain method based on the sextic B-spline basis functions. Numerical results for five test problems including the motion of single solitary wave, interaction of two and three well-separated solitary waves of different amplitudes, evolution of solitons with Gaussian and undular bore initial conditions are obtained. Stability and a priori error estimate of the scheme are discussed. A comparison of the values of the obtained invariants and error norms for single solitary wave with earlier results is also made. The results show that the present method is efficient and reliable

Peakompactons: Peaked compact nonlinear waves

This paper is meant as an accessible introduction to/tutorial on the analytical construction and numerical simulation of a class of nonstandard solitary waves termed peakompactons. These peaked compactly supported waves arise as solutions to nonlinear evolution equations from a hierarchy of nonlinearly dispersive Korteweg–de Vries-type models. Peakompactons, like the now-well-known compactons and unlike the soliton solutions of the Korteweg–de Vries equation, have finite support, i.e., they are of finite wavelength. However, unlike compactons, peakompactons are also peaked, i.e., a higher spatial derivative suffers a jump discontinuity at the wave’s crest. Here, we construct such solutions exactly by reducing the governing partial differential equation to a nonlinear ordinary differential equation and employing a phase-plane analysis. A simple, but reliable, finite-difference scheme is also designed and tested for the simulation of collisions of peakompactons. In addition to the peakompacton class of solu..

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

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

Genelleştirilmiş Rosenau-KdV ve genelleştirilmiş Rosenau-RLW denklemlerinin kollokasyon yöntemi ile nümerik çözümleri

Bu tezde genelleştirilmiş Rosenau-KdV ve genelleştirilmiş Rosenau-RLW denklemlerinin sayısal çözümleri yedinci (septic) dereceden B-spline fonksiyonlar kullanılarak Kollokasyon yöntemi ile elde edilmiştir. Bu tez dört bölümden oluşmaktadır. Tezin birinci bölümünde Sonlu Elemanlar yöntemi, Kollokasyon yöntemi ve B-spline fonksiyonlar hakkında bilgiler sunulmuştur. Tezin ikinci bölümünde geneleştirilmiş Rosenau-KdV denklemi tanıtıldı ve yedinci dereceden B-spline fonksiyonlar kullanılarak Kollokasyon yöntemi ile nümerik çözümleri elde edilmiştir. Tezin üçüncü bölümünde genelleştirilmiş Rosenau-RLW denklemi verilerek yedinci dereceden B-spline fonksiyonlar kullanılarak Kollokasyon yöntemi ile nümerik çözümleri elde edilmiştir. Tezin son bölümünde ise elde ettiğimiz nümerik değerlere ilişkin sonuç ve öneriler sunulmuştur