Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Conference Object On the Relativistic Supersymmetric Quantum Mechanics(Springer Verlag, 2002) Mir-Kasimov, Rufat M.; Kasım, Rıfat MirThe present paper is devoted to the one-dimensional relativistic supersymmetric quantum mechanics (RSUSYQM). A short formulation of RSUSYQM is given. We show that RSUSYQM is a q-deformed non-relativistic SUSYQM. Two simple examples are given.Conference Object Derivative and Integration on Time Scale With Mathematica(Imperial College Press, 2003) Yantır, AhmetMathematical modelling of time dependent systems is always interesting for applied mathematicians. First continuous and then discrete mathematical models were built in the mathematical development from ancient to modem times. With the discovery of time scale, the problem of irregular systems was solved in the 1990s. In this paper we explain the derivative and integral of functions of time scales and the solution of some basic calculus problems using Mathematica.Conference Object Measure on Time Scales With Mathematica(Springer Verlag, 2006) Ufuktepe, Ünal; Yantır, AhmetIn this paper we study the Lebesgue Delta-measure on time scales. We refer to [3, 4] for the main notions and facts from the general measure and Lebesgue Delta integral theory. The objective of this paper is to show how the main concepts of Mathematica can be applied to fundamentals of Lebesgue Delta- and Lebesgue Delta- measure on an arbitrary time scale and also on a discrete time scale whose rule is given by the reader. As the time scale theory is investigated in two parts, by means of alpha and rho operators, we named the measures on time scales by the set function DMeasure and NMeasure respectively for arbitrary time scales.Conference Object Citation - Scopus: 1Hipokampüsün El ve Atlas Tabanlı Otomatik Bölütlenmesinin Hacimsel Olarak Karşılaştırılması(Institute of Electrical and Electronics Engineers Inc., 2009) Kutucu, Hakan; Eker, Çağdaş; Kitiş, Ömer; Gönül, Ali SaffetHigh-resolution Magnetic resonance imaging (MRI) is helpful in diagnosing diseases such as schizophrenia, alzheimer, dementia etc. Brain segmentation is an important preprocess in medical imaging applications. In this study we compare atlas based segmentation and manual segmentation of hippocampus for volumetric measures. A statistically difference was obtained between automatic and manual measurement. We conclude that contemporary techniques are not adequate to obtain sensitive data in some barin structures such as hippocampus core.Article Citation - WoS: 4Relativistic Burgers and Nonlinear Schrödinger Equations(Pleiades Publishing, 2009) Pashaev, OktayWe construct relativistic complex Burgers-Schrodinger and nonlinear Schrodinger equations. In the nonrelativistic limit, they reduce to the standard Burgers and nonlinear Schrodinger equations and are integrable through all orders of relativistic corrections.Conference Object The Hirota Method for Reaction-Diffusion Equations With Three Distinct Roots(American Institute of Physics, 2004) Tanoğlu, Gamze; Pashaev, OktayThe Hirota Method, with modified background is applied to construct exact analytical solutions of nonlinear reaction-diffusion equations of two types. The first equation has only nonlinear reaction part, while the second one has in addition the nonlinear transport term. For both cases, the reaction part has the form of the third order polynomial with three distinct roots. We found analytic one-soliton solutions and the relationships between three simple roots and the wave speed of the soliton. For the first case, if one of the roots is the mean value of other two roots, the soliton is static.We show that the restriction on three distinct roots to obtain moving soliton is removed in the second case by, adding nonlinear transport term to the first equation.Article Hirota Method for Solving Reaction-Diffusion Equations With Generalized Nonlinearity(World Academic Press, 2006) Tanoğlu, GamzeThe Hirota Method is applied to find an exact solitary wave solution to evolution equation with generalized nonlinearity. By introducing the power form of Hirota ansatz the bilinear representation for this equation is derived and the traveling wave solution is constructed by Hirota perturbation. We show that velocity of this solution is naturally fixed by truncating the Hirota’s perturbation expansion. So in our approach, this truncate on works similarly to the way Ablowitz and Zeppetella obtained an exact travelling wave solution of Fisher’s equation by finding the special wave speed for which the resulting ODE is of the Painleve type. In the special case the model admits N shock soliton solution and the reduction to Burgers’ equation.Article Citation - WoS: 3Citation - Scopus: 3A1-L10 Phase Boundaries and Anisotropy Via Multiple-Order Theory for an Fcc Alloy(European Mathematical Society Publishing House, 2003) Tanoğlu, Gamze; Braun, Richard J.; Cahn, John W.; McFadden, Geoffrey B.The dependence of thermodynamic properties of planar interphase boundaries (IPBs) and antiphase boundaries (APBs) in a binary alloy on an fcc lattice is studied as a function of their orientation. Using a recently developed diffuse interface model based on three non-conserved order parameters and the concentration, and a free energy density that gives a realistic phase diagram with one disordered phase (A1) and two ordered phases (L12 and L10) such as occur in the Cu-Au system, we are able to find IPBs and APBs between any pair of phases and domains, and for all orientations. The model includes bulk and gradient terms in a free energy functional, and assumes that there is no mismatch in the lattice parameters for the disordered and ordered phases.We catalog the appropriate boundary conditions for all IPBs and APBs. We then focus on the IPB between the disordered A1 phase and the L10 ordered phase. For this IPB we compute the numerical solution of the boundary value problem to find its interfacial energy, γ as a function of orientation, temperature, and chemical potential (or composition). We determine the equilibrium shape for a precipitate of one phase within the other using the Cahn-Hoffman "-vector" formalism. We find that the profile of the interface is determined only by one conserved and one non-conserved order parameter, which leads to a surface energy which, as a function of orientation, is "transversely isotropic" with respect to the tetragonal axis of the L10 phase. We verify the model's consistency with the Gibbs adsorption equation.Article Citation - WoS: 81Citation - Scopus: 79The Resonant Nonlinear Schrödinger Equation in Cold Plasma Physics. Application of Bäcklund-Darboux Transformations and Superposition Principles(Cambridge University Press, 2007) Lee, Jiunhung; Pashaev, Oktay; Rogers, Colin; Schief, W. K.A system of nonlinear equations governing the transmission of uni-axial waves in a cold collisionless plasma subject to a transverse magnetic field is reduced to the recently proposed resonant nonlinear Schrödinger (RNLS) equation. This integrable variant of the standard nonlinear Schrödinger equation admits novel nonlinear superposition principles associated with Bäcklund-Darboux transformations. These are used here, in particular, to construct analytic descriptions of the interaction of solitonic magnetoacoustic waves propagating through the plasma.Article Citation - WoS: 11Citation - Scopus: 10A Stabilizing Subgrid for Convection-Diffusion Problem(World Scientific Publishing Co. Pte Ltd, 2006) Neslitürk, Ali İhsanA stabilizing subgrid which consists of a single additional node in each triangular element is analyzed by solving the convection-diffusion problem, especially in the case of small diffusion. The choice of the location of the subgrid node is based on minimizing the residual of a local problem inside each element. We study convergence properties of the method under consideration and its connection with previously suggested stabilizing subgrids. We prove that the standard Galerkin finite element solution on augmented grid produces a discrete solution that satisfy the same a priori error estimates that are typically obtained with SUPG and RFB methods. Some numerical experiments that confirm the theoretical findings are also presented.