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Article Citation - WoS: 4Citation - Scopus: 5An Unconventional Finite Difference Scheme for Modified Korteweg-De Vries Equation(Hindawi Ltd, 2017) Koroglu, Canan; Aydin, AyhanA 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.Article Citation - WoS: 3Citation - Scopus: 5Global Energy Preserving Model Reduction for Multi-Symplectic Pdes(Elsevier Science inc, 2023) Uzunca, Murat; Karasozen, Bulent; Aydin, AyhanMany Hamiltonian systems can be recast in multi-symplectic form. We develop a reduced -order model (ROM) for multi-symplectic Hamiltonian partial differential equations (PDEs) that preserves the global energy. The full-order solutions are obtained by finite difference discretization in space and the global energy preserving average vector field (AVF) method. The ROM is constructed in the same way as the full-order model (FOM) applying proper orthogonal decomposition (POD) with the Galerkin projection. The reduced-order system has the same structure as the FOM, and preserves the discrete reduced global energy. Ap-plying the discrete empirical interpolation method (DEIM), the reduced-order solutions are computed efficiently in the online stage. A priori error bound is derived for the DEIM ap-proximation to the nonlinear Hamiltonian. The accuracy and computational efficiency of the ROMs are demonstrated for the Korteweg de Vries (KdV) equation, Zakharov-Kuznetzov (ZK) equation, and nonlinear Schrodinger (NLS) equation in multi-symplectic form. Preser-vation of the reduced energies shows that the reduced-order solutions ensure the long-term stability of the solutions.(c) 2022 Elsevier Inc. All rights reserved.Article Citation - WoS: 3Citation - Scopus: 3Inverse 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.Article New Accurate Conservative Finite Difference Schemes for 1-D and 2-D Schrödinger-Boussinesq Equations(Sivas Cumhuriyet University, 2024) Aydin, Ayhan; Mohammed, TahaIn this paper, first-order and second-order accurate structure-preserving finite difference schemes are proposed for solving the Schrödinger- Boussinesq equations. The conservation of the discrete energy and mass of the present schemes are analytically proved. Numerical experiments are given to support the theoretical results. Numerical examples show the efficiency of the proposed scheme and the correction of the theoretical proofsArticle Citation - WoS: 12Citation - Scopus: 12Conservative Finite Difference Schemes for the Chiral Nonlinear Schrodinger Equation(Springer international Publishing Ag, 2015) Ismail, Mohammad S.; Al-Basyouni, Khalil S.; Aydin, AyhanIn this paper, we derive three finite difference schemes for the chiral nonlinear Schrodinger 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.
