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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Turkish Journal of Mathematics2
Journal of Mathematical Physics2
Applied Mathematics and Computation1
Boundary Value Problems1
Chaos, Solitons & Fractals1
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  • 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
    New Accurate Conservative Finite Difference Schemes for 1-D and 2-D Schrödinger-Boussinesq Equations
    (Sivas Cumhuriyet University, 2024) Aydin, Ayhan; Mohammed, Taha
    In 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 proofs
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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.
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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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    Master Thesis
    Üçlü Lineer Olmayan Schrödinger Denklemi için Yapı Koruyan Sayısal Yöntemler
    (2016) Ertuğ, Sevim; Aydın, Ayhan
    Birleşik N denklemli lineer olmayan Schrödinger (N--CNLS) denklemi fizik, optik, kuantum mekaniği ve akışkanlar dinamiği gibi birçok alanda sıklıkla kullanılan önemli matematiksel modellerden biridir. Son yıllarda lineer olmayan Schrödinger (NLS) denklemi ve ikili lineer olmayan Schrödinger (2-CNLS) denklemi için yapılmış çok sayıda çalışma varken, üçlü lineer olmayan Schrödinger (3-CNLS) denklem sistemi için yapılan sayısal çalışma sayısı oldukça azdır. Bu denklem sistemlerinin kütle korunumu ve enerji korunumu gibi bazı fiziksel (ya da geometrik) korunum özellikleri vardir. Standard sayısal yöntemler bu tür korunumları korumamakta ve korunum sayısal çözümde bozulmaktadır. Son yıllarda bu tip özellikleri koruyan sayısal yöntemler geliştirme çalışmalarına ilgi araştırmacılar arasında hızla artmaktadır. Bu tezin amacı, üçlü lineer olmayan Schrödinger (3-CNLS) denkleminin bir veya birden fazla fiziksel (ya da geometrik) özelliğini koruyan sayısal yöntemler geliştirmektir. 3-CNLS denkleminin enerji ve kütle olmak üzere iki korunum özelliği elde edilmiştir. Daha sonra, periyodik ve homojen sınır şartları gibi uygun sınır şartları altında, bu korunumların ayrık hallerini koruyan üç tane sayısal yöntem geliştirilmiştir. İlk olarak, Ortalama Vektör Alanı (AVF) olarak bilinen bir yöntem kullanılarak, enerji koruyan sayısal yöntem tasarlanmıştır. Daha sonra denklemin kütlesini koruyan iki adımlı (ya da üç basamaklı) bir sayısal yöntem tasarlanmıştır. Son olarak, denklemin hem kütle hem de enerjisini koruyan bir adımlı (ya da iki basamaklı) sayısal yöntem tasarlanmıştır. Tasarlanan sayısal yöntemlerin doğrusal kararlılık, doğruluk ve yakınsaklık analizleri yapılmıştır. Enerji ve kütle koruyan sayısal yöntemlerin dağılım özellikleri incelenmiştir. Sayısal yöntemlerin etkinliğini ve yapı koruma özelliklerini doğrulamak için bir çok sayısal uygulamalar yapılmıştır. Sayısal sonuçlar uzun zaman aralığında her üç sayısal yöntemin de denklemin periyodik, bir soliton ve çarpışan soliton çözümlerinin de çok iyi sonuçlar verdiğini göstermektedir.
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    Master Thesis
    Doğrusal Olmayan Black-scholes Denklemi için Üstel Sonlu Fark Yöntemi
    (2017) Omar, Fathıa; Aksoy, Ümit; Aydın, Ayhan
    Bu tezde, likit olmayan bir piyasada ortaya çıkan doğrusal olmayan Black-Scholes denklemi için üstel sonlu fark yöntemi çalışılmıştır. 1. Bölüm opsiyon fiyatlandırması problemi terminolojisi, temel tanımlar ve literatür taramasına ayrılmıştır. 2. Bölümde Black-Scholes modeli ve Black-Scholes denklemi için sonlu fark yöntemleri gözden geçirilmiştir. 3. Bölümde doğrusal olmayan Black-Scholes denklemi için açık sonlu fark yöntemi, monotonluk, kararlılık ve tutarlılık sonuçları ile birlikte çalışılmıştır. 4. Bölümde doğrusal ve doğrusal olmayan Black-Scholes denklemleri için üstel sonlu fark yöntemi uygulanmıştır. Ayrıca, yöntemin tutarlılığı ve yakınsaklığı araştırılmıştır. Teorik sonuçları doğrulamak için sayısal örnekler verilmiştir. Sayısal sonuçlar, üstel sonlu fark yönteminin açık sonlu fark yönteminden daha iyi performans sergilediğini göstermiştir. 5. Bölüm sonuç kısmına ayrılmıştır.
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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: 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
    A New Conservative Numerical Method for Strongly Coupled Nonlinear Schrödinger Equations
    (Springer Heidelberg, 2025) Ors, Ridvan Fatih; Koroglu, Canan; Aydin, Ayhan
    In this paper, a numerical method based on the conservative finite difference scheme is constructed to numerically solve the strongly coupled nonlinear Schr & ouml;dinger (SCNLS) equation. Conservative properties such as energy and mass of the SCNLS equation have been proven. In particular a fourth-order central difference scheme is used to discretize the the spatial derivative and a second-order Crank-Nicolson type discretization is used to discretize the temporal derivative. It has been shown that the proposed scheme preserves the discrete mass and energy. The existence of discrete solution is also investigated. Several numerical results are given to demonstrate the preservation properties of the new method. Also, the effect of the linear coupling parameters on the evolution of solitary waves is investigated.
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    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.