Diğer Yayınlar / Other Publications

Permanent URI for this collectionhttps://ada.atilim.edu.tr/handle/123456789/27

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An Unconventional Splitting for Korteweg de Vries–Burgers Equation
(EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS, 2015) Aydın, Ayhan; Mathematics
Numerical 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.
Conservative finite difference schemes for the chiral nonlinear Schrödinger equation
(Boundary Value Problems, 2015) Ismaıl, Mohammed S.; Al-basyounı, Khalil S.; Aydın, Ayhan; Mathematics
In 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.
LIE-POISSON INTEGRATORS FOR A RIGID SATELLITE ON A CIRCULAR ORBIT
(2011) Aydın, Ayhan; Mathematics
In 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.
Lobatto IIIA–IIIB discretization of the strongly coupled nonlinear Schrödinger equation
(Journal of Computational and Applied Mathematics, 2009) Aydın, Ayhan; Karasözen, Bülent; Mathematics
In 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.
Multi-symplectic integration of coupled non-linear Schrödinger system with soliton solutions
(International Journal of Computer Mathematics, 2009) Aydın, Ayhan; Karasözen, Bülent; Mathematics
Systems 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.
Operator Splitting of the KdV-Burgers Type Equation with Fast and Slow Dynamics
(2015) Aydın, Ayhan; Karasözen, Bülent; Mathematics
The 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.
SEMI-EXPLICIT MULTI-SYMPLECTIC INTEGRATION OF NONLINEAR SCHRODINGER EQUATION
(2015) Aydın, Ayhan; Mathematics
In 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.
Symplectic and multi-symplectic methods for coupled nonlinear Schrödinger equations with periodic solutions
(Computer Physics Communications, 2007) Aydın, Ayhan; Karasözen, Bülent; Mathematics
We 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.

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