Mathematics / Matematik

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  • Article
    Citation - WoS: 7
    Citation - Scopus: 7
    Complex Ginzburg–landau Equations With Dynamic Boundary Conditions
    (Elsevier Ltd., 2018) Corrêa, Wellington José; Özsarı, Türker
    The initial-dynamic boundary value problem (idbvp) for the complex Ginzburg–Landau equation (CGLE) on bounded domains of RN is studied by converting the given mathematical model into a Wentzell initial–boundary value problem (ibvp). First, the corresponding linear homogeneous idbvp is considered. Secondly, the forced linear idbvp with both interior and boundary forcings is studied. Then, the nonlinear idbvp with Lipschitz nonlinearity in the interior and monotone nonlinearity on the boundary is analyzed. The local well-posedness of the idbvp for the CGLE with power type nonlinearities is obtained via a contraction mapping argument. Global well-posedness for strong solutions is shown. Global existence and uniqueness of weak solutions are proven. Smoothing effect of the corresponding evolution operator is proved. This helps to get better well-posedness results than the known results on idbvp for nonlinear Schrödinger equations (NLS). An interesting result of this paper is proving that solutions of NLS subject to dynamic boundary conditions can be obtained as inviscid limits of the solutions of the CGLE subject to same type of boundary conditions. Finally, long time behavior of solutions is characterized and exponential decay rates are obtained at the energy level by using control theoretic tools.
  • Article
    Citation - WoS: 7
    Citation - Scopus: 7
    Nonlinear Integral Equations for Bernoulli's Free Boundary Value Problem in Three Dimensions
    (Elsevier Ltd., 2017) Ivanyshyn Yaman, Olha; Kress, Rainer
    In this paper we present a numerical solution method for the Bernoulli free boundary value problem for the Laplace equation in three dimensions. We extend a nonlinear integral equation approach for the free boundary reconstruction (Kress, 2016) from the two-dimensional to the three-dimensional case. The idea of the method consists in reformulating Bernoulli's problem as a system of boundary integral equations which are nonlinear with respect to the unknown shape of the free boundary and linear with respect to the boundary values. The system is linearized simultaneously with respect to both unknowns, i.e., it is solved by Newton iterations. In each iteration step the linearized system is solved numerically by a spectrally accurate method. After expressing the Fréchet derivatives as a linear combination of single- and double-layer potentials we obtain a local convergence result on the Newton iterations and illustrate the feasibility of the method by numerical examples.
  • Article
    Citation - WoS: 9
    Citation - Scopus: 9
    Multi-Wave and Rational Solutions for Nonlinear Evolution Equations
    (Walter de Gruyter GmbH, 2010) Aslan, İsmail
    Nonlinear evolution equations always admit multi-soliton and rational solutions. The Burgers equation is used as an example, and the exp-function method is used to eluciadte the solution procedure.
  • Article
    Citation - WoS: 9
    Citation - Scopus: 8
    Similarity Solutions To Burgers' Equation in Terms of Special Functions of Mathematical Physics
    (Jagellonian University, 2017) Öziş, Turgut; Aslan, İsmail
    In this paper, the Lie group method is used to investigate some closed form solutions of famous Burgers' equation. A detailed and complete symmetry analysis is performed. By similarity transformations, the equation is reduced to ordinary differential equations whose general solutions are written in terms of the error function, Kummer's confluent hypergeometric function Φ(a; b; x) and Bessel functions Jp, showing the strong connection between the best mathematical modelling equations and the special functions of mathematical physics.
  • Article
    Citation - WoS: 11
    Citation - Scopus: 11
    Variations on a Theme of Q-Oscillator
    (IOP Publishing Ltd., 2015) Pashaev, Oktay
    We present several ideas in the direction of physical interpretation of q- and f-oscillators as nonlinear oscillators. First we show that an arbitrary one-dimensional integrable system in action-angle variables can be naturally represented as a classical and quantum f-oscillator. As an example, the semi-relativistic oscillator as a descriptive of the Landau levels for relativistic electron in magnetic field is solved as an f-oscillator. By using dispersion relation for q-oscillator we solve the linear q-Schrödinger equation and corresponding nonlinear complex q-Burgers equation. The same dispersion allows us to construct integrable q-NLS model as a deformation of cubic NLS in terms of recursion operator of NLS hierarchy. A peculiar property of the model is to be completely integrable at any order of expansion in deformation parameter around q = 1. As another variation on the theme, we consider hydrodynamic flow in bounded domain. For the flow bounded by two concentric circles we formulate the two circle theorem and construct the solution as the q-periodic flow by non-symmetric q-calculus. Then we generalize this theorem to the flow in the wedge domain bounded by two arcs. This two circular-wedge theorem determines images of the flow by extension of q-calculus to two bases: the real one, corresponding to circular arcs and the complex one, with q as a primitive root of unity. As an application, the vortex motion in annular domain as a nonlinear oscillator in the form of classical and quantum f-oscillator is studied. Extending idea of q-oscillator to two bases with the golden ratio, we describe Fibonacci numbers as a special type of q-numbers with matrix Binet formula. We derive the corresponding golden quantum oscillator, nonlinear coherent states and Fock-Bargman representation. Its spectrum satisfies the triple relations, while the energy levels' relative difference approaches asymptotically to the golden ratio and has no classical limit.
  • Article
    Citation - WoS: 1
    Citation - Scopus: 1
    Q-Shock soliton evolution
    (Elsevier Ltd., 2012) Pashaev, Oktay; Nalcı, Şengül
    By generating function based on Jackson's q-exponential function and the standard exponential function, we introduce a new q-analogue of Hermite and Kampe-de Feriet polynomials. In contrast to q-Hermite polynomials with triple recurrence relations similar to [1], our polynomials satisfy multiple term recurrence relations, which are derived by the q-logarithmic function. It allows us to introduce the q-Heat equation with standard time evolution and the q-deformed space derivative. We find solution of this equation in terms of q-Kampe-de Feriet polynomials with arbitrary number of moving zeros, and solved the initial value problem in operator form. By q-analog of the Cole-Hopf transformation we obtain a new q-deformed Burgers type nonlinear equation with cubic nonlinearity. Regular everywhere, single and multiple q-shock soliton solutions and their time evolution are studied. A novel, self-similarity property of the q-shock solitons is found. Their evolution shows regular character free of any singularities. The results are extended to the linear time dependent q-Schrödinger equation and its nonlinear q-Madelung fluid type representation. © 2012 Elsevier Ltd. All rights reserved.
  • Article
    Citation - WoS: 10
    Citation - Scopus: 11
    Some Exact and Explicit Solutions for Nonlinear Schrödinger Equations
    (Polish Academy of Sciences, 2013) Aslan, İsmail
    Nonlinear models occur in many areas of applied physical sciences. This paper presents the first integral method to carry out the integration of Schrödinger-type equations in terms of traveling wave solutions. Through the established first integrals, exact traveling wave solutions are obtained under some parameter conditions.
  • Article
    Citation - WoS: 17
    Citation - Scopus: 25
    The First Integral Method for Constructing Exact and Explicit Solutions To Nonlinear Evolution Equations
    (John Wiley and Sons Inc., 2012) Aslan, İsmail
    Problems that are modeled by nonlinear evolution equations occur in many areas of applied sciences. In the present study, we deal with the negative order KdV equation and the generalized Zakharov system and derive some further results using the so-called first integral method. By means of the established first integrals, some exact traveling wave solutions are obtained in a concise manner.
  • Article
    Citation - WoS: 6
    Citation - Scopus: 6
    Construction of Exact Solutions for Fractional-Type Difference-Differential Equations Via Symbolic Computation
    (Elsevier Ltd., 2013) Aslan, İsmail
    This paper deals with fractional-type difference-differential equations by means of the extended simplest equation method. First, an equation related to the discrete KdV equation is considered. Second, a system related to the well-known self-dual network equations through a real discrete Miura transformation is analyzed. As a consequence, three types of exact solutions (with the aid of symbolic computation) emerged; hyperbolic, trigonometric and rational which have not been reported before. Our results could be used as a starting point for numerical procedures as well.
  • Article
    Citation - WoS: 3
    Citation - Scopus: 2
    Application of the Exp-Function Method To the (2+1)-Dimensional Boiti-Leon Equation Using Symbolic Computation
    (Taylor and Francis Ltd., 2011) Aslan, İsmail
    Locate full-text(opens in a new window)|Full Text(opens in a new window)|View at Publisher| Export | Download | Add to List | More... International Journal of Computer Mathematics Volume 88, Issue 4, March 2011, Pages 747-761 Application of the Exp-function method to the (2+1)-dimensional Boiti-Leon-Pempinelli equation using symbolic computation (Article) Aslan, I. Department of Mathematics, Izmir Institute of Technology, Urla, Izmir 35430, Turkey View references (47) Abstract This paper deals with the so-called Exp-function method for studying a particular nonlinear partial differential equation (PDE): the (2+1)-dimensional Boiti-Leon-Pempinelli equation. The method is constructive and can be carried out in a computer with the aid of a computer algebra system. The obtained generalized solitary wave solutions contain more arbitrary parameters compared with the earlier works, and thus, they are wider. This means that our method is effective and powerful for constructing exact and explicit analytic solutions to nonlinear PDEs.