Aydın, Ayhan

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https://hdl.handle.net/20.500.14411/6774
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Aydın,A.
Aydin, Ayhan
A.,Aydın
A., Ayhan
Aydın, Ayhan
Aydin,Ayhan
Ayhan Aydın
A., Aydın
Aydin A.
A.,Aydin
Ayhan, Aydin
Aydın A.
AYDIN A.
A.,Ayhan
A., Aydin
Aydin,A.
Ayhan, Aydın
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Profesör Doktor
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ayhan.aydin@atilim.edu.tr
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Mathematics
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0000-0002-0837-9364
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WoS Researcher ID
AAL-5690-2020

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Documents

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171

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Turkish Journal of Mathematics2
Journal of Mathematical Physics2
Applied Mathematics and Computation1
Boundary Value Problems1
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Now showing 1 - 10 of 19
  • Thumbnail Image
    Article
    Citation - WoS: 24
    Citation - Scopus: 24
    Symplectic and multisymplectic Lobatto methods for the "good" Boussinesq equation
    (Amer inst Physics, 2008) Aydin, A.; Karasoezen, B.
    In this paper, we construct second order symplectic and multisymplectic integrators for the "good" Boussineq equation using the two-stage Lobatto IIIA-IIIB partitioned Runge-Kutta method, which yield an explicit scheme and is equivalent to the classical central difference approximation to the second order spatial derivative. Numerical dispersion properties and the stability of both integrators are investigated. Numerical results for different solitary wave solutions confirm the excellent long time behavior of symplectic and multisymplectic integrators by preservink local and global energy and momentum. (C) 2008 American Institute of Physics.
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    Article
    Citation - WoS: 4
    Citation - Scopus: 5
    An Unconventional Finite Difference Scheme for Modified Korteweg-De Vries Equation
    (Hindawi Ltd, 2017) Koroglu, Canan; Aydin, Ayhan
    A numerical solution of the modified Korteweg-de Vries (MKdV) equation is presented by using a nonstandard finite difference (NSFD) scheme with theta method which includes the implicit Euler and a Crank-Nicolson type discretization. Local truncation error of the NSFD scheme and linear stability analysis are discussed. To test the accuracy and efficiency of the method, some numerical examples are given. The numerical results of NSFD scheme are compared with the exact solution and a standard finite difference scheme. The numerical results illustrate that the NSFD scheme is a robust numerical tool for the numerical integration of the MKdV equation.
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    Article
    Citation - WoS: 23
    Citation - Scopus: 25
    Multisymplectic Integration of n-coupled Nonlinear Schrodinger Equation With Destabilized Periodic Wave Solutions
    (Pergamon-elsevier Science Ltd, 2009) Aydin, Ayhan
    N-coupled nonlinear Schrodinger equation (N-CNLS) is shown to be in multisymplectic form. 3-CNLS equation is studied for analytical and numerical purposes. A new six-point scheme which is equivalent to the multisymplectic Preissman scheme is derived for 3-CNLS equation. A new periodic wave solution is obtained and its stability analysis is discussed. 3-CNLS equation is integrated for destabilized periodic solutions both for integrable and non-integrable cases by multisymplectic six-point scheme. Different kinds of evolutions are observed for different parameters and coefficients of the system. Numerical results show that, the multisymplectic six-point scheme has excellent local and global conservation properties in long-time computation. (C) 2008 Elsevier Ltd. All rights reserved.
  • Thumbnail Image
    Article
    Citation - WoS: 9
    Citation - Scopus: 10
    A nonstandard numerical method for the modified KdV equation
    (indian Acad Sciences, 2017) Aydin, Ayhan; Koroglu, Canan
    A linearly implicit nonstandard finite difference method is presented for the numerical solution of modified Korteweg-de Vries equation. Local truncation error of the scheme is discussed. Numerical examples are presented to test the efficiency and accuracy of the scheme.
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    Article
    Citation - WoS: 17
    Citation - Scopus: 16
    Multisymplectic Box Schemes for the Complex Modified Korteweg-De Vries Equation
    (Amer inst Physics, 2010) Aydin, A.; Karasozen, B.
    In this paper, two multisymplectic integrators, an eight-point Preissman box scheme and a narrow box scheme, are considered for numerical integration of the complex modified Korteweg-de Vries equation. Energy and momentum preservation of both schemes and their dispersive properties are investigated. The performance of both methods is demonstrated through numerical tests on several solitary wave solutions. (C) 2010 American Institute of Physics. [doi:10.1063/1.3456068]
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    Article
    Citation - WoS: 3
    Citation - Scopus: 3
    Inverse Spectral Problem for Finite Jacobi Matrices With Zero Diagonal
    (Taylor & Francis Ltd, 2015) Aydin, Ayhan; Guseinov, Gusein 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. Anexplicit 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.
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    Article
    Symplectic and multi-symplectic methods for coupled nonlinear Schrödinger equations with periodic solutions
    (Computer Physics Communications, 2007) Aydın, Ayhan; Karasözen, Bülent
    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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    Article
    Lobatto Iiia–iiib Discretization of the Strongly Coupled Nonlinear Schrödinger Equation
    (Journal of Computational and Applied Mathematics, 2009) Aydın, Ayhan; Karasözen, Bülent
    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.
  • Thumbnail Image
    Publication
    Semi-Explicit Multi-Symplectic Integration of Nonlinear Schrodinger Equation
    (2015) Aydın, Ayhan
    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.
  • Thumbnail Image
    Book Part
    Multisymplectic Integrators for Coupled Nonlinear Partial Differential Equations
    (Nova Science Publishers, Inc., 2012) Karas̈ozen,B.; Aydın, Ayhan; Aydin,A.; Aydın, Ayhan; Mathematics; Mathematics
    The numerical solution of nonlinear partial differential equations (PDEs) using symplectic geometric integrators has been the subject of many studies in recent years. Many nonlinear partial differential equations can be formulated as an infinite dimensional Hamiltonian system. After semi-discretization in the space variable, a system of Hamiltonian ordinary differential equations (ODEs) is obtained, for which various symplectic integrators can be applied. Numerical results show that symplectic schemes have superior performance, especially in long time simulations. The concept of multisymplectic PDEs and multisymplectic schemes can be viewed as the generalization of symplectic schemes. In the last decade, many multisymplectic methods have been proposed and applied to nonlinear PDEs, like to nonlinear wave equation, nonlinear Schr̈odinger equation, Korteweg de Vries equation, Dirac equation, Maxwell equation and sine-Gordon equation. In this review article, recent results of multisymplectic integration on the coupled nonlinear PDEs, the coupled nonlinear Schr̈odinger equation, the modified complex Korteweg de Vries equation and the Zakharov system will be given. The numerical results are discussed with respect to the stability of the schemes, accuracy of the solutions, conservation of the energy and momentum, preservation of dispersion relations. © 2012 Nova Science Publishers, Inc. All rights reserved.