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 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

Solving fractional diffusion and fractional diffusion-wave equations by Petrov-Galerkin finite element method

In the last few years, it has become highly evident that fractional calculus has been widely used in several areas of science. Because of this fact, their numerical solutions also have become urgently important. In this manuscript, numerical solutions of both the fractional diffusion and fractional diffusion-wave equations have been obtained by a Petrov-Galerkin finite element method using quadratic B-spline base functions as trial functions and linear B-spline base functions as the test functions. In those equations, fractional derivatives are used in terms of the Caputo sense. While the L1 discretizaton formula has been applied to fractional diffusion equation, the L2 discretizaton formula has been applied to the fractional diffusion-wave equation. Finally, the error norms L2 and L∞ have been calculated for testing the accuracy of the proposed scheme.Publisher's Versio

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

Different linearization techniques for the numerical solution of the MEW equation

The modi ed equal width wave (MEW) equation is solved numeri- cally by giving two di¤erent linearization techniques based on collocation nite element method in which cubic B-splines are used as approximate functions. To support our work three test problems; namely, the motion of a single solitary wave, interaction of two solitary waves and the birth of solitons are studied. Results are compared with other published numerical solutions available in the literature. Accuracy of the proposed method is discussed by computing the nu- merical conserved laws L2 and L1 error norms. A linear stability analysis of the approximation obtained by the scheme shows that the method is unconditionally stable