Yusuf Ucar

A Galerkin Finite Element Method to Solve Fractional Diffusion and Fractional Diffusion-Wave Equations

Abstract 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.

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

A Quadratic B-Spline Galerkin Approach for Solving a Coupled KdV Equation

Abstract In this paper, a quadratic B-spline Galerkin finite element approach is applied to one-dimensional coupled KdV equation in order to obtain its numerical solutions. The performance of the method is examined on three test problems. Computed results are compared with the exact results and also other numerical results given in the literature. A Fourier stability analysis of the approach is also done.

Numerical Solutions of the Modified Burgers’ Equation by Finite Difference Methods

Abstract In this study, a numerical solution of the modified Burgers’ equation is obtained by the finite difference methods. For the solution process, two linearization techniques have been applied to get over the non-linear term existing in the equation. Then, some comparisons have been made between the obtained results and those available in the literature. Furthermore, the error norms L2 and L∞ are computed and found to be sufficiently small and compatible with others in the literature. The stability analysis of the linearized finite difference equations obtained by two different linearization techniques has been separately conducted via Fourier stability analysis method.

A new approach on numerical solutions of the Improved Boussinesq type equation using quadratic B-spline Galerkin finite element method

In the present manuscript, some numerical solutions of an Improved Boussinesq type equation are obtained by means of quadratic B-spline Galerkin finite element method. Then, error norms L2 and L∞ have been calculated to test the accuracy of the current method. In the manuscript, solitary wave movement and interaction of solitary-antisolitary waves are considered as test problems.

Numerical solution of the complex modified Korteweg-de Vries equation by DQM

In this paper, a method based on the differential quadrature method with quintic B- spline has been applied to simulate the solitary wave solution of the complex modified Korteweg- de Vries equation (CMKdV). Three test problems, namely single solitary wave, interaction of two solitary waves and interaction of three solitary waves have been investigated. The efficiency and accuracy of the method have been measured by calculating maximum error norm L∞ for single solitary waves having analytical solutions. Also, the three lowest conserved quantities and obtained numerical results have been compared with some of the published numerical results.

A New Perspective on The Numerical Solution for Fractional Klein Gordon Equation

In the present manuscript, a new numerical scheme is presented for solving the time fractional nonlinear Klein-Gordon equation. The approximate solutions of the fractional equation are based on cubic B-spline collocation finite element method and L2 algorithm. The fractional derivative in the given equation is handled in terms of Caputo sense. Using the methods, fractional differential equation is converted into algebraic equation system that are appropriate for computer coding. Then, two model problems are considered and their error norms are calculated to demonstrate the reliability and efficiency of the proposed method. The newly calculated error norms show that numerical results are in a good agreement with the exact solutions.