A quartic subdomain finite element method for the modified kdv equation

In this article, we have obtained numerical solutions of the modified Korteweg-de Vries (MKdV) equation by a numerical technique attributed on subdomain finite element method using quartic B-splines. The proposed numerical algorithm is controlled by applying three test problems including single solitary wave, interaction of two and three solitary waves. To inspect the performance of the newly applied method, the error norms, L2 and L∞, as well as the four lowest invariants, I1, I2, I3 and I4 have been computed. Linear stability analysis of the algorithm is also examined

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

Two different methods for numerical solution of the modified burgers’ equation

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 computing \u1d43f����2 and \u1d43f����∞ 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

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 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

Modified differential transformation method for solving classes of non-linear differential equations

In this research article, a numerical scheme namely modified differential transformation method (MDTM) is employed successfully to obtain accurate approximate solutions for classes of nonlinear differential equations. This scheme based on differential transform method (DTM), Laplace transform and Pad´e approximants. Validity and efficiency of MDTM are tested upon several examples and comparisons.are made to demonstrate that. The results lead to conclude that the MDTM is effective, explicit and easy to use.Publisher's Versio

New exact solutionsand numerical approximations of the generalized kdv equation

This paper is devoted to create new exact and numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with ansatz method and Galerkin finite element method based on cubic B-splines over finite elements. Propagation of single solitary wave is investigated to show the efficiency and applicability of the proposed methods. The performance of the numerical algorithm is proved by computing L2 and L∞ error norms. Also, three invariants I1, I2, and I3 have been calculated to determine the conservation properties of the presented algorithm. The obtained numerical solutions are compared with some earlier studies for similar parameters. This comparison clearly shows that the obtained results are better than some earlier results and they are found to be in good agreement with exact solutions. Additionally, a linear stability analysis based on Von Neumann’s theory is surveyed and indicated that our method is unconditionally stable

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 solution of the MEW equaiton using sextic B splines

In this article, a numerical solution of the modified equal width wave (MEW) equation, based on subdomain method using sextic B-spline is used to simulate the motion of single solitary wave and interaction of two solitary waves. The three invariants of the motion are calculated to determine the conservation properties of the system. L2 and L∞ error norms are used to measure differences between the analytical and numerical solutions. The obtained results are compared with some published numerical solutions. A linear stability analysis of the scheme is also investigate