Petrov Galerkin finite element method for solving the MRLW equation

In this article, a Petrov-Galerkin method, in which the element shape functions are cubic and weight functions are quadratic B-splines, is introduced to solve the modified regularized long wave (MRLW) equation. The solitary wave motion, interaction of two and three solitary waves, and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. Accuracy and efficiency of the method are demonstrated by computing the numerical conserved laws and L2, L∞ error norms. The computed results show that the present scheme is a successful numerical technique for solving the MRLW equation. A linear stability analysis based on the Fourier method is also investigate

A numerical solution of the modified regularized long wave (MRLW) equation using quartic B-splines

In this paper, a numerical solution of the modified regularized long wave (MRLW) equation is obtained by subdomain finite element method using quartic B-spline functions. Solitary wave motion, interaction of two and three solitary waves and the development of the Maxwellian initial condition into solitary waves are studied using the proposed method. Accuracy and efficiency of the proposed method are tested by calculating the numerical conserved laws and error norms L₂ and L∞. The obtained results show that the method is an effective numerical scheme to solve the MRLW equation. In addition, a linear stability analysis of the scheme is found to be unconditionally stable.Publisher's Versio

Numerical solutions of the MRLW equation by cubic B-spline Galerkin finite element method

In this paper, a numerical solution of the modified regularized long wave (MRLW) equation has been obtained by a numerical technique based on a lumped Galerkin method using cubic B-spline finite elements. Solitary wave motion, interaction of two and three solitary waves have been studied to validate the proposed method. The three invariants ( 1 2 3 I ,I ,I ) of the motion have been calculated to determine the conservation properties of the scheme. Error norms L2 and L∞ have been used to measure the differences between the exact and numerical solutions. Also, a linear stability analysis of the scheme is proposed

Numerical approximation to a solution of the modified regularized long wave equation using quintic B splines

In this work, a numerical solution of the modified regularized long wave (MRLW) equation is obtained by the method based on collocation of quintic B-splines over the finite elements. A linear stability analysis shows that the numerical scheme based on Von Neumann approximation theory is unconditionally stable. Test problems including the solitary wave motion, the interaction of two and three solitary waves and the Maxwellian initial condition are solved to validate the proposed method by calculating error norms L2 and L∞ that are found to be marginally accurate and efficient. The three invariants of the motion have been calculated to determine the conservation properties of the scheme. The obtained results are compared with other earlier result

An efficient approach to numerical study of the MRLW equation with b spline collocation method

A septic B-spline collocation method is implemented to find the numerical solution of the modified regularized long wave (MRLW) equation. Three test problems including the single soliton and interaction of two and three solitons are studied to validate the proposed method by calculating the error norms \u1d43f��2 and \u1d43f��∞ and the invariants \u1d43c��1, \u1d43c��2, and \u1d43c��3. Also, we have studied the Maxwellian initial condition pulse.The numerical results obtained by the method show that the present method is accurate and efficient. Results are compared with some earlier results given in the literature. A linear stability analysis of the method is also investigated

An efficient numerical method for the modified regularized long wave equation using Fourier spectral method

AbstractThe modified regularized long wave (MRLW) equation is numerically solved using Fourier spectral collection method. The MRLW equation is discretized in space variable by the Fourier spectral method and Leap-Frog method for time dependence. To validate the efficiency, accuracy and simplicity of the used method, four cases study are solved. The single soliton wave motion, interaction of two solitary waves, interaction of three solitary waves and a Maxwellian initial condition pulse are studied. The L2 and L∞ error norms are computed for the motion of single solitary waves. To determine the conservation properties of the MRLW equation three invariants of motion are evaluated for all test problems

Solitary-wave solutions of the GRLW equation using septic B-spline collocation method

In this work, solitary-wave solutions of the generalized regularized long wave (GRLW) equation are obtained by using septic B-spline collocation method with two different lin- earization techniques. To demonstrate the accuracy and efficiency of the numerical scheme, three test problems are studied by calculating the error norms L 2 and L ∞ and the invari- ants I 1 , I 2 and I 3 . A linear stability analysis based on the von Neumann method of the numerical scheme is also investigated. Consequently, our findings indicate that our numer- ical scheme is preferable to some recent numerical schemes

Numerical approximation of the generalized regularized long wave equation using Petrov–Galerkin finite element method

The generalized regularized long wave (GRLW) equation has been developed to model a variety of physical phenomena such as ion-acoustic and magnetohydro dynamic waves in plasma,nonlinear transverse waves in shallow water and phonon packets in nonlinear crystals. This paper aims to develop andanalyze a powerful numerical scheme for the nonlinear GRLWequation by Petrov–Galerkin method in which the elementshape functions are cubic and weight functions are quadratic B-splines. The proposed method is implemented to three ref-erence problems involving propagation of the single solitarywave, interaction of two solitary waves and evolution of solitons with the Maxwellian initial condition. The variational for-mulation and semi-discrete Galerkin scheme of the equation are firstly constituted. We estimate rate of convergence of such an approximation. Using Fourier stability analysis of thelinearized scheme we show that the scheme is uncondition-ally stable. To verify practicality and robustness of the new scheme error norms L2, L∞ and three invariants I1, I2,and I3 are calculated. The computed numerical results are compared with other published results and confirmed to be precise and effective