A septic B-spline collocation method for solving the generalized equal width wave equation

In this work, a septic B-spline collocation method is implemented to find the numerical solution of the generalized equal width (GEW) wave equation by using two different linearization techniques. Test problems including single soliton, interaction of solitons and Maxwellian initial condition are solved to verify the proposed method by calculating the error norms L2 and L∞ and the invariants I1, I2 and I3. Applying the Von-Neumann stability analysis, the proposed method is shown to be unconditionally stable. As a result, the obtained results are found in good agreement with the some recent results

Application of the collocation method with b-splines to the gew equation

In this paper, the generalized equal width (GEW) wave equation is solved numerically by using a quintic B-spline collocation algorithm with two different linearization techniques. Also, a linear stability analysis of the numerical scheme based on the von Neumann method is investigated. The numerical algorithm is applied to three test problems consisting of a single solitary wave, the interaction of two solitary waves, and a Maxwellian initial condition. In order to determine the performance of the numerical method, we compute the error in the L2- and L∞ norms and in the invariants I1, I2, and I3 of the GEW equation. These calculations are compared with earlier studies. Afterwards, the motion of solitary waves according to different parameters is designe

Subdomain finite element method with quartic B-splines for the modified equal width wave equation

In this paper, a numerical solution of the modified equal width wave (MEW) equation, has been obtained by a numerical technique based on Subdomain finite element method with quartic Bsplines. Test problems including the motion of a single solitary wave and interaction of two solitary waves are studied to validate the suggested method. Accuracy and efficiency of the proposed method are discussed by computing the numerical conserved laws and error norms L2 and L∞. A linear stability analysis based on a Fourier method shows that the numerical scheme is unconditionally stable

Septic B spline collocation method for the numerical solution of the modified equal width wave equation

Numerical solutions of the modified equal width wave equation are obtained by using collocation method with septic B-spline finite elements with three different linearization techniques. The motion of a single solitary wave, interaction of two solitary waves and birth of solitons are studied using the proposed method. Accuracy of the method is discussed by computing the numerical conserved laws error norms L2 and L∞. The numerical results show that the present method is a remarkably successful numerical technique for solving the MEW equation. A linear stability analysis shows that this numerical scheme, based on a Crank Nicolson approximation in time, is unconditionally stable

Petrov galerkin method with cubic B splines for solving the MEW equation

In the present paper, we introduce a numerical solution algorithm based on a Petrov-Galerkin method in which the element shape functions are cubic B-splines and the weight functions quadratic B-splines . The motion of a single solitarywave and interaction of two solitarywaves are studied. Accuracy and efficiency of the proposed method are discussed by computing the numerical conserved laws and L2 , L¥ error norms. The obtained results show that the present method is a remarkably successful numerical technique for solving the modified equal width wave(MEW) equation. A linear stability analysis of the scheme shows that it is unconditionally stable

A Powerful Robust Cubic Hermite Collocation Method for the Numerical Calculations and Simulations of the Equal Width Wave Equation

In this article, non-linear Equal Width-Wave (EW) equation will be numerically solved . For this aim, the non-linear term in the equation is firstly linearized by Rubin-Graves type approach. After that, to reduce the equation into a solvable discretized linear algebraic equation system which is the essential part of this study, the Crank-Nicolson type approximation and cubic Hermite collocation method are respectively applied to obtain the integration in the temporal and spatial domain directions. To be able to illustrate the validity and accuracy of the proposed method, six test model problems that is single solitary wave, the interaction of two solitary waves, the interaction of three solitary waves, the Maxwellian initial condition, undular bore and finally soliton collision will be taken into consideration and solved. Since only the single solitary wave has an analytical solution among these solitary waves, the error norms Linf and L2 are computed and compared to a few of the previous works available in the literature. Furthermore, the widely used three invariants I1, I2 and I3 of the proposed problems during the simulations are computed and presented. Beside those, the relative changes in those invariants are presented. Also, a comparison of the error norms Linf and L2 and these invariants obviously shows that the proposed scheme produces better and compatible results than most of the previous works using the same parameters. Finally, von Neumann analysis has shown that the present scheme is unconditionally stable.Comment: 25 pages, 9 tables, 6 figure

Numerical solution of the modified equal width wave equation

Numerical solution of the modified equal width wave equation is obtained by using lumped Galerkin method based on cubic B-spline finite element method. Solitary wave motion and interaction of two solitary waves are studied using the proposed method. Accuracy of the proposed method is discussed by computing the numerical conserved laws L2 and L∞ error norms. The numerical results are found in good agreement with exact solution. A linear stability analysis of the scheme is also investigated

Numerical solutions of the generalized equal width wave equation using the Petrov–Galerkin method

In this article, we consider a generalized equal width wave (GEW) equation which is a significant nonlinear wave equation as it can be used to model many problems occurring in applied sciences. Here we study a Petrov–Galerkin method for the model problem, in which element shape functions are quadratic and weight functions are linear B-splines. We investigate the existence and uniqueness of solutions of the weak form of the equation. Then, we establish the theoretical bound of the error in the semi-discrete spatial scheme as well as of a full discrete scheme at t = t n. Furthermore, a powerful Fourier analysis has been applied to show that the proposed scheme is unconditionally stable. Finally, propagation of solitary waves and evolution of solitons are analyzed to demonstrate the efficiency and applicability of the proposed scheme. The three invariants (I1, I2 and I3) of motion have been commented to verify the conservation features of the proposed algorithms. Our proposed numerical scheme has been compared with other published schemes and demonstrated to be valid, effective and it outperforms the others

Numerical investigations of shallow water waves via generalized equal width (GEW) equation

In this article, a mathematical model representing solution of the nonlinear generalized equal width (GEW) equation has been considered. Here we aim to investigate solutions of GEW equation using a numerical scheme by using sextic B-spline Subdomain finite element method. At first Galerkin finite element method is proposed and a priori bound has been established. Then a semi-discrete and a Crank-Nicolson Galerkin finite element approximation have been studied respectively. In addition to that a powerful Fourier series analysis has been performed and indicated that our method is unconditionally stable. Finally, proficiency and practicality of the method have been demonstrated by illustrating it on two important problems of the GEW equation including propagation of single solitons and collision of double solitary waves. The performance of the numerical algorithm has been demonstrated for the motion of single soliton by computing L∞ and L2 norms and for the other problem computing three invariant quantities I1, I2 and I3. The presented numerical algorithm has been compared with other established schemes and it is observed that the presented scheme is shown to be effectual and valid