A Galerkin finite element method to solve fractional diffusion and fractional diffusion-wave equations

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

A Numerical Solution to Fractional Diffusion Equation for Force-Free Case

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

THE LEFT (RIGHT) ROUGH APPROXIMATIONS IN A GROUP

WOS: 000464544500007Rough set theory proposes a new mathematical approach to model vagueness. In this paper, we introduce a new definition for rough approximations with respect to the subgroups of a group, which is called the left (right) lower and upper approximations. In fact, we prove that this definition is a generalization of definition in [5]. Then we give some properties of these rough approximations

Numerical solution of time fractional Burgers equation

In this article, the time fractional order Burgers equation has been solved by quadratic B-spline Galerkin method. This method has been applied to three model problems. The obtained numerical solutions and error norms L2 and L∞ have been presented in tables. Absolute error graphics as well as those of exact and numerical solutions have been given

Collocation solutions for the time fractional telegraph equation using cubic B-spline finite elements

In this study, we investigate numerical solutions of the fractional telegraph equation with the aid of cubic B-spline collocation method. The fractional derivatives have been considered in the Caputo forms. The L1and L2 formulae are used to discretize the Caputo fractional derivative with respect to time. Some examples have been given for determining the accuracy of the regarded method. Obtained numerical results are compared with exact solutions arising in the literature and the error norms L2 and L∞ have been computed. In addition, graphical representations of numerical results are given. The obtained results show that the considered method is effective and applicable for obtaining the numerical results of nonlinear fractional partial differential equations (FPDEs)

New wave solutions of an integrable dispersive wave equation with a fractional time derivative arising in ocean engineering models

The fractional Camassa-Holm equation is generally used as a powerful tool in computer simulations of water waves in shallow water, coastal and harbor models. In this paper, new wave solutions of this equation are obtained by using a new extended direct algebraic method. Thirty-six completely new solutions are obtained and are graphically represented. These solutions may motivate future research on the topic. © 2020 University of Kuwait. All rights reserved

Approximate Analytical Solution to Time-Fractional Damped Burger and Cahn-Allen Equations

The aim of the present paper is to obtain the approximate analytical solutions of time-fractional damped burger and Cahn- Allen equations by means of the homotopy analysis method (HAM). In the HAM solution, there exists an auxiliary parameter ¯h 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

Analytical solutions of Cahn-Hillard phase-field model for spinodal decomposition of a binary system

Spinodal decomposition is a very important and challenging issue not for only materials science but for also many other fields in science. Phase-field models, which have become very popular in recent years, are very promising for the evaluation of phase transformations such as spinodal decomposition. In this study, the Cahn-Hillard equation which is one of the most trending phase-field models is analytically solved by the double exp-function and the modified extended exp-function methods. Spatio-temporal energy and concentration distributions were evaluated by applying two of the ten obtained analytical solutions to the free energy equation. The results obtained were found to be successful in predicting spinodal decomposition as is expected from the Cahn-Hillard model