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    Approximation of the KdVB equation by the quintic B-spline differential quadrature method
    (Academic Publication Council, 2015) Bashan, Ali; Karakoc, Seydi Battal Gazi; Geyikli, Turabi
    In this paper, the Korteweg-de Vries-Burgers' (KdVB) equation is solved numerically by a new differential quadrature method based on quintic B-spline functions. The weighting coefficients are obtained by semi-explicit algorithm including an algebraic system with five-band coefficient matrix. The L-2 and L-infinity error norms and lowest three invariants I-1, I-2 and I-3 have computed to compare with some earlier studies. Stability analysis of the method is also given. The obtained numerical results show that the present method performs better than the most of the methods available in the literature.
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    Numerical approximation to a solution of the modified regularized long wave equation using quintic B-splines
    (Springeropen, 2013) Karakoc, Seydi Battal Gazi; Yagmurlu, Nuri Murat; Ucar, Yusuf
    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 and 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 Solution of the Modified Equal Width Wave Equation
    (Hindawi Ltd, 2012) Karakoc, Seydi Battal Gazi; Geyikli, Turabi
    Numerical solution of the modified equal width wave equation is obtained by using lumped Galerkin method based on cubic B-spline finite element method. Solitary wave motion and interaction of two solitary waves are studied using the proposed method. Accuracy of the proposed method is discussed by computing the numerical conserved laws L-2 and L-infinity error norms. The numerical results are found in good agreement with exact solution. A linear stability analysis of the scheme is also investigated.
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    Numerical solutions of the MRLW equation by cubic B-spline Galerkin finite element method
    (Academic Publication Council, 2015) Karakoc, Seydi Battal Gazi; Ucar, Yusuf; Yagmurlu, Nurimurat
    In this paper, a numerical solution of the modified regularized long wave (MRLW) equation has been obtained by a numerical technique based on a lumped Galerkin method using cubic B-spline finite elements. Solitary wave motion, interaction of two and three solitary waves have been studied to validate the proposed method. The three invariants (I-1, I-2, I-3) of the motion have been calculated to determine the conservation properties of the scheme. Error norms L-2 and L-infinity have been used to measure the differences between the exact and numerical solutions. Also, a linear stability analysis of the scheme is proposed.
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    Petrov-Galerkin finite element method for solving the MRLW equation
    (Springer Heidelberg, 2013) Karakoc, Seydi Battal Gazi; Geyikli, Turabi
    In this article, a Petrov-Galerkin method, in which the element shape functions are cubic and weight functions are quadratic B-splines, is introduced to solve the modified regularized long wave (MRLW) equation. The solitary wave motion, interaction of two and three solitary waves, and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. Accuracy and efficiency of the method are demonstrated by computing the numerical conserved laws and L-2, L-infinity error norms. The computed results show that the present scheme is a successful numerical technique for solving the MRLW equation. A linear stability analysis based on the Fourier method is also investigated.
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    Two Different Methods for Numerical Solution of the Modified Burgers' Equation
    (Hindawi Ltd, 2014) Karakoc, Seydi Battal Gazi; Bashan, Ali; Geyikli, Turabi
    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-infinity 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.