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In this paper, an exact method named Riccati–Bernoulli sub-ODE method and a numerical method named Subdomain finite element method are proposed for solving the nonlinear generalized regularized lon...

The generalized regularized long wave (GRLW) equation has been developed to model a variety of physical phenomena such as ion-acoustic and magnetohydrodynamic waves in plasma, nonlinear transverse waves in shallow water and phonon packets in nonlinear crystals. This paper aims to develop and analyze a powerful numerical scheme for the nonlinear generalized regularized long wave (GRLW) equation by Petrov--Galerkin method in which the element shape functions are cubic and weight functions are quadratic B-splines. The suggested method is performed to three test problems involving propagation of the single solitary wave, interaction of two solitary waves and evolution of solitons with the Maxwellian initial condition. The variational formulation and semi-discrete Galerkin scheme of the equation are firstly constituted. We estimate accuracy of such a spatial approximation. Then Fourier stability analysis of the linearized scheme shows that it is unconditionally stable. To verify practicality and robustness of the new scheme error norms $L_{2}$, $L_{\infty }$ and three invariants $I_{1},I_{2}$ and $I_{3}$ are calculated. The obtained numerical results are compared with other published results and shown to be precise and effective., 28 pages

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.

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 L 2 norms and for the other problem computing three invariant quantities I 1 , I 2 and I 3 . 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.

This work deals with the constitute of numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with Petrov-Galerkin finite element approach utilising a cubic B-spline function as the trial function and a quadratic function as the test function. Accurateness and effectiveness of the submitted methods are shown by employing propagation of single solitary wave. The L 2 , L ∞ error norms and I 1 , I 2 and I 3 invariants are used to validate the applicability and durability of our numerical algorithm. Implementing the Von-Neumann theory, it is manifested that the suggested method is marginally stable. Furthermore, supernonlinear traveling wave solution of the GKdV equation is presented using phase plots. It is seen that the GKdV equation supports superperiodic traveling wave solution only and it is significantly affected by velocity and nonlinear parameters. Also, considering a superficial periodic forcing multistability of traveling waves of perturbed GKdV equation is presented. It is found that the perturbed GKdV equation supports coexisting chaotic and various quasiperiodic features with same parametric values at different initial conditions.

In this article, a collocation fi nite element method based on septic B-splines as a tool has been carried out to obtain the numerical solutions of the nonlinear generalized Rosenau-RLW equation. One of the advantages of this method is that when the bases are chosen at a high degree, better numerical solutions are obtained. Effectiveness of the method is demonstrated by solving the equation with various initial and boundary conditions. Also, in order to detect the performance of the method we have computed L2 and L1 error norms and two lowest invariants IM and IE: The obtained numerical results have been compared with some of those in the literature for similar parameters. This comparison clearly shows that the obtained results are better than and found in good conformity with the some earlier results. Stability analysis denotes that our algorithm, based on a Crank Nicolson approximation in time, is unconditionally stable.

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 B-splines. 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 L 2 and L ∞. A linear stability analysis based on a Fourier method shows that the numerical scheme is unconditionally stable.

In this article, numerical solutions of the modified Korteweg-de Vries (MKdV) equation have been obtained by a numerical technique attributed on collocation method using quintic B-spline finite elements. The suggested numerical scheme is controlled by applying three test problems involving single solitary wave, interaction of two and three solitary waves. To check the performance of the newly applied method, the error norms, L2 and L∞, as well as the three lowest invariants, I1, I2 and I3, have been calculated. The acquired numerical results are compared with some of those available in the literature. Linear stability analysis of the algorithm is also examined.

In this article, modified Korteweg-de Vries (mKdV) equation is solved numerically by using lumped Petrov-Galerkin approach, where weight functions are quadratic and the element shape functions are cubic B-splines. The proposed numerical scheme is tested by applying four test problems including single solitary wave, interaction of two and three solitary waves, and evolution of solitons with the Gaussian initial condition. In order to show the performance of the algorithm, the error norms, L-2, L-infinity, and a couple of conserved quantities are computed. For the linear stability analysis of numerical algorithm, Fourier method is also investigated. As a result, the computed results show that the presented numerical scheme is a successful numerical technique for solving the mKdV equation. Therefore, the presented method is preferable to some recent numerical methods. (c) 2017 Sharif University of Technology. All rights reserved.

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.

Bu çalışmada, modifiye edilmiş Korteweg-de Vries (MKdV) denkleminin sayısal çözümleri septik B-spline kollokasyon sonlu eleman yöntemi kullanılarak elde edilmiştir. Önerilen sayısal algoritmanın doğruluğu, tek soliton dalga, iki ve üç soliton dalganın girişimi gibi üç test probleminin uygulanması ile kontrol edilmiştir. Zamana bağlı Crank Nicolson yaklaşımına dayanan sayısal algoritmamız şartsız olarak kararlıdır. Yeni uygulanan yöntemin performansını kontrol etmek için L2, Lsonsuz , hata normları ile I1,I2 , I3 ve I4 değişmezlerinin değerleri hesaplanmıştır. Elde edilen sayısal sonuçlar literatürde bulunan diğer sonuçlarla karşılaştırılmıştır., In this paper, we have obtained numerical solutions of the modified Korteweg-de Vries (MKdV) equation by using septic B-spline collocation finite element method. The suggested numerical algorithm is controlled by applying three test problems including; single soliton wave, interaction of two and three soliton waves. Our numerical algorithm, attributed to a Crank Nicolson approximation in time, is unconditionally stable. To control the performance of the newly applied method, the error norms L2 ,Lsonsuz and invariants I1 ,I2 ,I3, and I4 have been calculated. The obtained numerical results are compared with some of those available in the literature.

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 normsL2andL∞and the invariantsI1,I2, andI3. 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.

A numerical solution of the modified Burgers’ equation (MBE) is obtained by using quartic B-spline subdomain finite element method (SFEM) over which the nonlinear term is locally linearized and using quartic B-spline differential quadrature (QBDQM) method. The accuracy and efficiency of the methods are discussed by computingL2andL∞error norms. Comparisons are made with those of some earlier papers. The obtained numerical results show that the methods are effective numerical schemes to solve the MBE. A linear stability analysis, based on the von Neumann scheme, shows the SFEM is unconditionally stable. A rate of convergence analysis is also given for the DQM.

Abstract 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 L 2, 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 investigated.

Numerical solutions of linear and nonlinear integrodifferential-difference equations are presented using homotopy analysis method. The aim of the paper is to present an efficient numerical procedure for solving linear and nonlinear integrodifferential-difference equations. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system.

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 L 2 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 results. MSC: 97N40, 65N30, 65D07, 76B25, 74S05.

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 ${L}_{2}$ and ${L}_{\infty }$ error norms. The numerical results are found in good agreement with exact solution. A linear stability analysis of the scheme is also investigated.