Theoretical and computational structures on solitary wave solutions of Benjamin Bona Mahony-Burgers equation

This paper aims to obtain exact and numerical solutions of the nonlinear Benjamin Bona Mahony-Burgers (BBM-Burgers) equation. Here, we propose the modi ed Kudryashov method for getting the exact traveling wave solutions of BBM-Burgers equation and a septic B-spline collocation nite element method for numerical investigations. The numerical method is validated by studying solitary wave motion. Linear stability analysis of the numerical scheme is done with Fourier method based on von-Neumann theory. To show suitability and robustness of the new numerical algorithm, error norms L2, L1 and three invariants I1; I2 and I3 are calculated and obtained results are given both numerically and graphically. The obtained results state that our exact and numerical schemes ensure evident and they are penetrative mathematical instruments for solving nonlinear evolution equation

Galerkin finite element solution for benjamin-bona-mahony-burgers equation with cubic b-splines

In this article, we study solitary-wave solutions of the nonlinear Benjamin–Bona–Mahony– Burgers(BBM–Burgers) equation based on a lumped Galerkin technique using cubic Bspline finite elements for the spatial approximation. The existence and uniqueness of solutions of the Galerkin version of the solutions have been established. An accuracy analysis of the Galerkin finite element scheme for the spatial approximation has been well studied. The proposed scheme is carried out for four test problems including dispersion of single solitary wave, interaction of two, three solitary waves and development of an undular bore. Then we propose a full discrete scheme for the resulting IVP. Von Neumann theory is used to establish stability analysis of the full discrete numerical algorithm. To display applicability and durableness of the new scheme, error norms L2, L∞ and three invariants I1, I2 and I3 are computed and the acquired results are demonstrated both numerically and graphically. The obtained results specify that our new scheme ensures an apparent and an operative mathematical instrument for solving nonlinear evolution equation

The meshless methods for numerical solution of the nonlinear Klein-Gordon equation

In this paper, we develop the numerical solution of nonlinear Klein-Gordon equation (NKGE) using the meshless methods. The finite difference scheme and the radial basis functions (RBFs) collocation methods are used to discretize time derivative and spatial derivatives, respectively. Numerical results are given to confirm the accuracy and efficiency of the presented schemes.Publisher's Versio

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

A new approach for numerical solution of modified korteweg-de vries equation

In this paper, a lumped Galerkin method is applied with cubic B-spline interpolation functions to find the numerical solution of the modified Korteweg-de Vries (mKdV) equation. Test problems including motion of single solitary wave, interaction of two solitons, interaction of three solitons, and evolution of solitons are solved to verify the proposed method by calculating the error norms L2 and L1 and the conserved quantities mass, momentum and energy. Applying the von-Neumann stability analysis, the proposed method is shown to be unconditionally stable. Consequently, the obtained results are found to be harmony with the some recent result

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