Sine-Gordon expansion method for exact solutions to conformable time fractional equations in RLW-class

The bright soliton solutions and singular solutions are constructed for the space-time fractional EW and the space-time fractional modified EW (MEW) equations. Both equations are reduced to ordinary differential equations by the use of fractional complex transform (FCT) and properties of modified Riemann–Liouville derivative. Then, various ansatz method are implemented to construct the solutions for both equations.

Exact solutions of space-time fractional EW and modified EW equations

In this paper, we investigate the physical behavior of the basic elements that related to phase decomposition in ternary alloys of (Fe–Cr–Mo) and (Fe–Cr–Cu) according to analytical and approximate simulation. We study the dynamic of the separation phase for the ternary alloys of iron. The dynamical process of this separation has been described in a mathematical model called the Cahn–Hilliard equation. The minor element behavior in the process has been described by the Cahn–Hilliard equation. It describes the process of phase separation for two components of a binary fluid in ternary alloys of (Fe–Cr–Mo) and (Fe–Cr–Cu). We implement a modified auxiliary equation method and the cubic B-spline scheme on this mathematical model to show the dynamical process of phase separation and the concentration of one of two components in a system. We try obtaining the solitary and approximate solutions of this model to show the relation between the components in this phase. We discuss our solutions in view of a Stefan, Thomas-Windle, and Navier–Stokes models. Whereas, these models describe the motion of viscous fluid substance.

Analytical and numerical simulations for the kinetics of phase separation in iron (Fe-Cr-X (X=Mo, Cu)) based on ternary alloys

Numerical difficulties arise in the solution of Burgers' equation for the case of a large Reynolds number. The aim of this paper is to extend the finite element method proposed by Caldwell et al. to the general case of n elements. The method is tested out on Burgers' equation for two different initial conditions and proves to be much more accurate than finite difference methods for a large Reynolds number.

Solution of Burgers' equation with a large Reynolds number

Interactions between exotic multi-valued solitons of the (2+1)-dimensional Korteweg-de Vries equation describing shallow water wave

In this paper, the Korteweg-de Vries-Burgers’ (KdVB) equation is solved numerically by anew differential quadrature method based on quintic B-spline functions. The weightingcoefficients are obtained by semi-explicit algorithm including an algebraic system with fivebandcoefficient matrix. The L2 and L∞ error norms and lowest three invariants 1 2 I , I and3 I have computed to compare with some earlier studies. Stability analysis of the method isalso given. The obtained numerical results show that the present method performs better thanthe most of the methods available in the literature.

https://journalskuwait.org/kjs/index.php/KJS/article/download/198/75

Approximation of the KdVB equation by the quintic B-spline differential quadrature method

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

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Numerical approximation to a solution of the modified regularized long wave equation using quintic B-splines

In this article, we study solitary-wave solutions of the nonlinear Benjamin–Bona–Mahony–Burgers(BBM–Burgers) equation based on a lumped Galerkin technique using cubic B- spline finite elements for the spatial approximation. The existence and uniqueness of solutions of the Galerkin version of the solutions have been established. An accuracy analysis of the Galerkin finite element scheme for the spatial approximation has been well studied. The proposed scheme is carried out for four test problems including dispersion of single solitary wave, interaction of two, three solitary waves and development of an undular bore. Then we propose a full discrete scheme for the resulting IVP. Von Neumann theory is used to establish stability analysis of the full discrete numerical algorithm. To display applicability and durableness of the new scheme, error norms L 2 , L ∞ and three invariants I 1 , I 2 and I 3 are computed and the acquired results are demonstrated both numerically and graphically. The obtained results specify that our new scheme ensures an apparent and an operative mathematical instrument for solving nonlinear evolution equation.

Galerkin finite element solution for Benjamin–Bona–Mahony–Burgers equation with cubic B-splines

Numerical solutions of the modified equal width wave equation are obtained by using collocation method with septic B-spline finite elements with three different linearization techniques. The motion of a single solitary wave, interaction of two solitary waves and birth of solitons are studied using the proposed method. Accuracy of the method is discussed by computing the numerical conserved laws error norms L2 and L∞. The numerical results show that the present method is a remarkably successful numerical technique for solving the MEW equation. A linear stability analysis shows that this numerical scheme, based on a Crank Nicolson approximation in time, is unconditionally stable.

/pdf/septic-b-spline-collocation-method-for-the-numerical-1zokjyjusb.pdf

Septic B-Spline Collocation Method for the Numerical Solution of the Modified Equal Width Wave 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 L 2 and L ∞ 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.

/pdf/two-different-methods-for-numerical-solution-of-the-modified-3jmn3tugfb.pdf

Seydi Battal Gazi Karakoç

Papers

Approximation of the KdVB equation by the quintic B-spline differential quadrature method

Numerical approximation to a solution of the modified regularized long wave equation using quintic B-splines

Galerkin finite element solution for Benjamin–Bona–Mahony–Burgers equation with cubic B-splines

Septic B-Spline Collocation Method for the Numerical Solution of the Modified Equal Width Wave Equation

Two different methods for numerical solution of the modified Burgers' equation.