Browsing by Author "AYDIN, Ayhan"
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Item An Unconventional Splitting for Korteweg de Vries–Burgers Equation(EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS, 2015-08-08) AYDIN, AyhanNumerical solutions of the Korteweg de Vries–Burgers (KdVB) equation based on splitting is studied. We put a real parameter into a KdVB equation and split the equation into two parts. The real parameter that is inserted into the KdVB equation enables us to play with the splitted parts. The real parameter enables to write the each splitted equation as close to the Korteweg de Vries (KdV) equation as we wish and as far from the Burgers equation as we wish or vice a versa. Then we solve the splitted parts numerically and compose the solutions to obtained the integrator for the KdVB equation. Finally we present some numerical experiments for the solution of the KdV, Burger’s and KdVB equations. The numerical experiments shows that the new splitting gives feasible and valid results.Item Conservative finite difference schemes for the chiral nonlinear Schrödinger equation(Boundary Value Problems, 2015-08-08) ISMAIL, Mohammed S.; AL-BASYOUNI, Khalil S.; AYDIN, AyhanIn this paper, we derive three finite difference schemes for the chiral nonlinear Schrödinger equation (CNLS). The CNLS equation has two kinds of progressive wave solutions: bright and dark soliton. The proposed methods are implicit, unconditionally stable and of second order in space and time directions. The exact solutions and the conserved quantities are used to assess the efficiency of these methods. Numerical simulations of single bright and dark solitons are given. The interactions of two bright solitons are also displayed.Item Inverse spectral problem for finite Jacobi matrices with zero diagonal(Inverse Problems in Science and Engineering, 2015-08-08) AYDIN, Ayhan; GUSEINOV, Gusien Sh.In this study, the necessary and sufficient conditions for solvability of an inverse spectral problem about eigenvalues and normalizing numbers for finite-order real Jacobi matrices with zero diagonal elements are established.An explicit procedure of reconstruction of the matrix from the spectral data consisting of the eigenvalues and normalizing numbers is given. Numerical examples and error analysis are provided to demonstrate the solution technique of the inverse problem. The results obtained are used to justify the solving procedure of the finite Langmuir lattice by the method of inverse spectral problem.Item LIE-POISSON INTEGRATORS FOR A RIGID SATELLITE ON A CIRCULAR ORBIT(2011-08-08) AYDIN, AyhanIn the last two decades, many structure preserving numerical methods like Poisson integrators have been investigated in numerical studies. Since the structure matrices are different in many Poisson systems, no integrator is known yet to preserve the Poisson structure of any Poisson system. In the present paper, we propose Lie– Poisson integrators for Lie–Poisson systems whose structure matrix is different from the ones studied before. In particular, explicit Lie-Poisson integrators for the equations of rotational motion of a rigid body (the satellite) on a circular orbit around a fixed gravitational center have been constructed based on the splitting. The splitted parts have been composed by a first, a second and a third order compositions. It has been shown that the proposed schemes preserve the quadratic invariants of the equation. Numerical results reveal the preservation of the energy and agree with the theoretical treatment that the invariants lie on the sphere in long–term with different orders of accuracy.Item LINEARLY IMPLICIT SCHEMES FOR THE ROSENAU-KORTEWED-DE VRIES REGULARIZED LONG WAVE EQUATIONS(2015-06-25) AL- OMAİRİ, Salim Mahmood Yaseen; AYDIN, AyhanIn this thesis, we consider the numerical solution of the Rosenau– Korteweg de Vries– Reularized Long Wave (Rosenau KdV–RLW) equation which couples the general Rosenau–KdV and Rosenau–RLW equations. Two conserved quantities of the Rose nau KdV–RLW equation has been proven, namely the mass and the energy. The aim is to develop numerical methods to preserve these conserved quantities exactly or preserve with a small error. Two linearly implicit schemes for the initial–boundary value problem of the Rosenau KdV–RLW equation are proposed. One method is con servative and the other method is nonconservative. It is proved that the conservative method preserves the energy of the equation exactly. It is also shown that the scheme is second order accurate and unconditionally stable. The second scheme is nonconser vative. It is first order accurate and conditionally stable. Numerical results shown that both scheme well simulates the solitary wave of the equation in long time. Moreover, numerical results verify the exact energy conservation of the conservative scheme.Item Lobatto IIIA–IIIB discretization of the strongly coupled nonlinear Schrödinger equation(Journal of Computational and Applied Mathematics, 2009-12-24) AYDIN, Ayhan; KARASÖZEN, BülentIn this paper, we construct a second order semi-explicit multi-symplectic integrator for the strongly coupled nonlinear Schrödinger equation based on the two-stage Lobatto IIIA–IIIB partitioned Runge–Kutta method. Numerical results for different solitary wave solutions including elastic and inelastic collisions, fusion of two solitons and with periodic solutions confirm the excellent long time behavior of the multi-symplectic integrator by preserving global energy, momentum and mass.Item Multi-symplectic integration of coupled non-linear Schrödinger system with soliton solutions(International Journal of Computer Mathematics, 2009-04-23) AYDIN, Ayhan; KARASÖZEN, BülentSystems of coupled non-linear Schrödinger equations with soliton solutions are integrated using the six-point scheme which is equivalent to the multi-symplectic Preissman scheme. The numerical dispersion relations are studied for the linearized equation. Numerical results for elastic and inelastic soliton collisions are presented. Numerical experiments confirm the excellent conservation of energy, momentum and norm in long-term computations and their relations to the qualitative behaviour of the soliton solutions.Item Operator Splitting of the KdV-Burgers Type Equation with Fast and Slow Dynamics(2015-09-17) AYDIN, Ayhan; KARASÖZEN, BülentThe Korteweg de Vries-Burgers (KdV-Burgers) type equation arising from the discretiza tion of the viscous Burgers equation with fast dispersion and slow diffusion is solved using operator splitting. The dispersive and diffusive parts are discretized in space by second order conservative finite differences. The resulting system of ordinary differential equations are composed using the time reversible Strang splitting. The numerical results reveal that the periodicity of the solutions and the invariants of the KdV-Burgers equation are well preserved.Item SEMI-EXPLICIT MULTI-SYMPLECTIC INTEGRATION OF NONLINEAR SCHRODINGER EQUATION(2015-08-11) AYDIN, AyhanIn this paper we apply Lobatto IIIA-IIIB type multi-symplectic discretization in space and time to the nonlinear Schrödinger equation. The resulting scheme is semi-explicit in time and therefore more efficient than implicit multisymplectic schemes. Numerical results confirm excellent long time conservation of the local and global conserved quantities like the energy, momentum and norm.Item Symplectic and multi-symplectic methods for coupled nonlinear Schrödinger equations with periodic solutions(Computer Physics Communications, 2007-05-18) AYDIN, Ayhan; KARASÖZEN, BülentWe consider for the integration of coupled nonlinear Schrödinger equations with periodic plane wave solutions a splitting method from the class of symplectic integrators and the multi-symplectic six-point scheme which is equivalent to the Preissman scheme. The numerical experiments show that both methods preserve very well the mass, energy and momentum in long-time evolution. The local errors in the energy are computed according to the discretizations in time and space for both methods. Due to its local nature, the multi-symplectic six-point scheme preserves the local invariants more accurately than the symplectic splitting method, but the global errors for conservation laws are almost the same.