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    A New Highly Accurate Numerical Scheme for Benjamin-Bona-Mahony-Burgers Equation Describing Small Amplitude Long Wave Propagation
    (Springer Basel Ag, 2023) Kutluay, Selcuk; Ozer, Sibel; Yagmurlu, Nuri Murat
    In this article, a new highly accurate numerical scheme is proposed and used for solving the initial-boundary value problem of the Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation. The BBM-Burgers equation is fully discretized by the Crank-Nicolson type method using the first-order forward finite difference approximation for the derivative in time and the standard second-order central difference approximations for all spatial derivatives. The nonlinear term appearing in the implicit scheme is firstly linearized in terms of a new dependent variable by utilizing the well known Taylor series expansion and then the resulting tri-diagonal linear algebraic equation system is solved by a direct solver method. To test the accuracy and efficiency of the scheme, three experimental test problems are taken into consideration of which the two have analytical solutions and the other one has not an analytical one. The computed results are compared with those of some studies in the literature for the same values of parameters. It is shown that the obtained results from the present method, which is stable and easy-to-use, get closer and closer to the exact solutions when the step sizes refine. This fact is also an other evidence of the accuracy and reliability of the method. Moreover, a low level data storage requirement and easy-to-implement algorithm of the present method can be considered among its notable advantages over other numerical methods. In addition, the unconditionally stability of the present scheme is shown by the von Neumann method.
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    Numerical solution by quintic B-spline collocation finite element method of generalized Rosenau-Kawahara equation
    (Springer Heidelberg, 2022) Ozer, Sibel
    In this study, numerical solution of generalized Rosenau-Kawahara equation with quintic B-spline collocation finite element method has been obtained. First, the generalized Rosenau-Kawahara equation is converted into a coupled differential equation system by the change of variable for the derivative with respect to space variable. Then, the numerical integrations of the resulting system according to time and space were obtained using the Crank-Nicolson-type formulation and quintic B-spline functions, respectively. The obtained numerical scheme has been applied to four model problems. It is seen that the results obtained from the presented scheme are compatible with the analytical solution, the error norms are smaller than those given in the literature, and conservation constants remain virtually unchanged.
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    Numerical solution of the Rosenau-KdV-RLW equation by operator splitting techniques based on B-spline collocation method
    (Wiley, 2019) Ozer, Sibel
    In the present study, the operator splitting techniques based on the quintic B-spline collocation finite element method are presented for calculating the numerical solutions of the Rosenau-KdV-RLW equation. Two test problems having exact solutions have been considered. To demonstrate the efficiency and accuracy of the present methods, the error norms L-2 and L-infinity with the discrete mass Q and energy E conservative properties have been calculated. The results obtained by the method have been compared with the exact solution of each problem and other numerical results in the literature, and also found to be in good agreement with each other. A Fourier stability analysis of each presented method is also investigated.
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    Numerical solutions of nonhomogeneous Rosenau type equations by quintic B-spline collocation method
    (Wiley, 2022) Ozer, Sibel; Yagmurlu, Nuri Murat
    In this study, a numerical scheme based on a collocation finite element method using quintic B-spline functions for getting approximate solutions of nonhomogeneous Rosenau type equations prescribed by initial and boundary conditions is proposed. The numerical scheme is tested on four model problems with known exact solutions. To show how accurate results the proposed scheme produces, the error norms defined by L-2 and L-infinity are calculated. Additionally, the stability analysis of the scheme is done by means of the von Neuman method.
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    Two efficient numerical methods for solving Rosenau-KdV-RLW equation
    (Academic Publication Council, 2021) Ozer, Sibel
    In this study, two efficient numerical schemes based on B-spline finite element method (FEM) and time-splitting methods for solving Rosenau-KdV-RLW equation are presented. In the first method, the equation is solved by cubic B-spline Galerkin FEM. For the second method, after splitting Rosenau-KdV-RLW equation in time, it is solved by Strang timesplitting technique using cubic B-spline Galerkin FEM. The differential equation system in the methods is solved by the fourth-order Runge-Kutta method. The stability analysis of the methods is performed. Both methods are applied to an example. The obtained numerical results are compared with some methods available in the literature via the error norms L-2 and L-infinity, convergence rates, and mass and energy conservation constants. The present results are found to be consistent with the compared ones.