Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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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: 30Citation - Scopus: 33The Initial Stage of Dam-Break Flow(Springer Verlag, 2009) Korobkin, Alexandre; Yılmaz, OğuzThe liquid flow and the free surface shape during the initial stage of dam breaking are investigated. The method of matched asymptotic expansions is used to derive the leading-order uniform solution of the classical dam-break problem. The asymptotic analysis is performed with respect to a small parameter which characterizes the short duration of the stage under consideration. The second-order outer solution is obtained in the main flow region. This solution is not valid in a small vicinity of the intersection point between the initially vertical free surface and the horizontal rigid bottom. The dimension of this vicinity is estimated with the help of a local analysis of the outer solution close to the intersection point. Stretched local coordinates are used in this vicinity to resolve the flow singularity and to derive the leading-order inner solution, which describes the formation of the jet flow along the bottom. It is shown that the inner solution is self-similar and the corresponding boundary-value problem can be reduced to the well-known Cauchy-Poisson problem for water waves generated by a given pressure distribution along the free surface. An analysis of the inner solution reveals the complex shape of the jet head, which would be difficult to simulate numerically. The asymptotic solution obtained is expected to be helpful in the analysis of developed gravity-driven flows.Article Citation - WoS: 4Citation - Scopus: 4Power-Series Solution for the Two-Dimensional Inviscid Flow With a Vortex and Multiple Cylinders(Springer Verlag, 2009) Pashaev, Oktay; Yılmaz, OğuzThe problem of a point vortex and N fixed cylinders in a two-dimensional inviscid fluid is studied and an analytical-numerical solution in the form of an infinite power series for the velocity field is obtained using complex analysis. The velocity distribution for the case of two cylinders is compared with the existing results of the problem of a vortex in an annular region which is conformally mapped onto the exterior of two cylinders. Limiting cases of N cylinders and the vortex, being far away from each other are studied. In these cases, "the dipole approximation" or "the point-island approximation" is derived, and its region of validity is established by numerical tests. The velocity distribution for a geometry of four cylinders placed at the vertices of a square and a vortex is presented. The problem of vortex motion with N cylinders addressed in the paper attracted attention recently owing to its importance in many applications. However, existing solutions using Abelian function theory are sophisticated and the theory is not one of the standard techniques used by applied mathematicians and engineers. Moreover, in the N ≥ 3 cylinder problem, the infinite product involved in the presentation of the Schottky-Klein prime function must also be truncated. So, the approach used in the paper is simple and an alternative to existing methods. This is the main motivation for this study.Article Citation - WoS: 17Citation - Scopus: 19Lebesgue-Stieltjes Measure on Time Scales(TUBITAK, 2009) Deniz, Aslı; Ufuktepe, ÜnalThe theory of time scales was introduced by Stefan Hilger in his Ph. D. thesis in 1988, supervised by Bernd Auldbach, in order to unify continuous and discrete analysis [5]. Measure theory on time scales was first constructed by Guseinov [4], then further studies were made by Guseinov-Bohner [1], Cabada-Vivero [2] and Rzezuchowski [6]. In this article, we adapt the concept of Lebesgue-Stieltjes measure to time scales. We define Lebesgue-Stieltjes Δ and ▶-measures and by using these measures, we define an integral adapted to time scales, specifically Lebesgue-Stieltjes Δ-integral. We also establish the relation between Lebesgue-Stieltjes measure and Lebesgue-Stieltjes Δ-measure, consequently between Lebesgue-Stieltjes integral and Lebesgue-Stieltjes Δ-integral.Article Boundary Shape and Casimir Energy(IOP Publishing Ltd., 2009) Ahmedov, Hacı; Duru, İsmail HakkıCasimir energy changes are investigated for geometries obtained by small but arbitrary deformations of a given geometry for which the vacuum energy is already known for the massless scalar field. As a specific case, deformation of a spherical shell is studied. From the deformation of the sphere we show that the Casimir energy is a decreasing function of the surface-to-volume ratio. The decreasing rate is higher for less smooth deformations.Article Citation - WoS: 15Citation - Scopus: 17A New Dynamical Model of Brainstorming: Linear, Nonlinear, Continuous (simultaneous) and Impulsive (sequential) Cases(Academic Press Inc., 2009) Coşkun, Hamit; Yılmaz, OğuzIn this paper, we extended the linear dynamical model of [Brown, V., Paulus, P. B. (1996). A simple dynamic model of social factors in group brainstorming. Small Group Research, 27, 91-114] on two accounts. First, we modelled the sequential type brainstorming using impulsive differential equations by treating each category as an impulse and tested its validity in the two experiments that investigated and demonstrated the beneficial effects of sequential priming and memory in individual brainstorming. Finally, we considered the nonlinear case of brainstorming in writing or brainwriting where dyads exchanged their ideas in a written format and that eliminated negative factors occurring in oral brainstorming (e.g., evaluation apprehension, free-riding, production blocking) and enhanced the upward performance matching, and conducted the second experiment in order to test its validity in this paradigm with the effects of sequential priming and memory. Comparisons showed good agreement between results of experiments and those of the mathematical model.Article Citation - WoS: 15Citation - Scopus: 16Vortex Images and Q-Elementary Functions(IOP Publishing Ltd., 2008) Pashaev, Oktay; Yılmaz, OğuzIn the present paper, the problem of vortex images in the annular domain between two coaxial cylinders is solved by the q-elementary functions. We show that all images are determined completely as poles of the q-logarithmic function, and are located at sites of the q-lattice, where a dimensionless parameter q = r 2 2/r 2 1 is given by the square ratio of the cylinder radii. The resulting solution for the complex potential is represented in terms of the Jackson q-exponential function. Our approach in this paper provides an efficient path to rediscover known solutions for the vortex-cylinder pair problem and yields new solutions as well. By composing pairs of q-exponents to the first Jacobi theta function and conformal mapping to a rectangular domain we show that our solution coincides with the known one, obtained before by elliptic functions. The Schottky-Klein prime function for the annular domain is factorized explicitly in terms of q-exponents. The Hamiltonian, the Kirchhoff-Routh and the Green functions are constructed. As a new application of our approach, the uniformly rotating exact N-vortex polygon solutions with the rotation frequency expressed in terms of q-logarithms at Nth roots of unity are found. In particular, we show that a single vortex orbits the cylinders with constant angular velocity, given as the q-harmonic series. Vortex images in two particular geometries with only one cylinder as the q → ∞ limit are studied in detail.Article Citation - WoS: 6Citation - Scopus: 6Integrable Hierarchies and Information Measures(IOP Publishing Ltd., 2008) Parwani, Rajesh R.; Pashaev, OktayIn this paper we investigate integrable models from the perspective of information theory, exhibiting various connections. We begin by showing that compressible hydrodynamics for a one-dimensional isentropic fluid, with an appropriately motivated information theoretic extension, is described by a general nonlinear Schrödinger (NLS) equation. Depending on the choice of the enthalpy function, one obtains the cubic NLS or other modified NLS equations that have applications in various fields. Next, by considering the integrable hierarchy associated with the NLS model, we propose higher order information measures which include the Fisher measure as their first member. The lowest members of the hierarchy are shown to be included in the expansion of a regularized Kullback-Leibler measure while, on the other hand, a suitable combination of the NLS hierarchy leads to a Wootters type measure related to a NLS equation with a relativistic dispersion relation. Finally, through our approach, we are led to construct integrable relativistic NLS equations.Article Second Order Diffraction of Water Waves by a Bottom Mounted Vertical Circular Cylinder and Some Related Numerical Problems(The American Society of Mechanical Engineers(ASME), 2007) Yılmaz, OğuzA Hankel transformation is used to obtain the second order diffraction solution of vertical cylinder of circular cross section. The improper integral over the free surface is tackled carefully. The singularity at the free surface is overcome effectively using a third order nonlinear transformation. Numerical results for free surface elevations compare well with the published data.Article Citation - WoS: 13Citation - Scopus: 15Analysis of a Corner Layer Problem in Anisotropic Interfaces(Southwest Missouri State University, 2006) Alikakos, N. D.; Bates, P. W.; Cahn, J. W.; Fife, P. C.; Fusco, G.; Tanoğlu, GamzeWe investigate a model of anisotropic diffuse interfaces in ordered FCC crystals introduced recently by Braun et al and Tanoglu et al [3, 18, 19], focusing on parametric conditions which give extreme anisotropy. For a reduced model, we prove existence and stability of plane wave solutions connecting the disordered FCC state with the ordered Cu3Au state described by solutions to a system of three equations. These plane wave solutions correspond to planar interfaces. Different orientations of the planes in relation to the crystal axes give rise to different surface energies. Guided by previous work based on numerics and formal asymptotics, we reduce this problem in the six dimensional phase space of the system to a two dimensional phase space by taking advantage of the symmetries of the crystal and restricting attention to solutions with corresponding symmetries. For this reduced problem a standing wave solution is constructed that corresponds to a transition that, in the extreme anisotropy limit, is continuous but not differentiable. We also investigate the stability of the constructed solution by studying the eigenvalue problem for the linearized equation. We find that although the transition is stable, there is a growing number 0(1/ε), of critical eigenvalues, where 1/ε ≫ 1 is a measure of the anisotropy. Specifically we obtain a discrete spectrum with eigenvalues λn = ε2/3 μn with μn ∼ Cn2/3, as n → +∞. The scaling characteristics of the critical spectrum suggest a previously unknown microstructural instability.