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

In the present study, numerical solutions of the fractional diffusion and fractional diffusion-wave equations where fractional derivatives are considered in the Caputo sense have been obtained by a Galerkin finite element method using quadratic B-spline base functions. For the fractional diffusion equation, the L1 discretizaton formula is applied, whereas the L2 discretizaton formula is applied for the fractional diffusion-wave equation. The error norms L 2 and L ∞ are computed to test the accuracy of the proposed method. It is shown that the present scheme is unconditionally stable by applying a stability analysis to the approximation obtained by the proposed scheme.

Abstract:The aim of the present paper is to obtain the approximate analytical solutions of time-fractional damped burger and CahnAllen equations by means of the homotopy analysis method (HAM). In the HAM solution, there exists an auxiliary parameterh which provides a convenient way to adjust and check the convergence region of the solution series. In the model problems, an appropriate choice of the auxiliary parameter has been examined for increasing values of time.

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A collocation finite element method for solving fractional diffusion equation for force-free case is considered. In this paper, we develop an approximation method based on collocation finite elements by cubic B-spline functions to solve fractional diffusion equation for force-free case formulated with Riemann-Liouville operator. Some numerical examples of interest are provided to show the accuracy of the method. A comparison between exact analytical solution and a numerical one has been made.