Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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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.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: 32Citation - Scopus: 37Symbolic Computation and Construction of New Exact Traveling Wave Solutions To Fitzhugh-Nagumo and Klein-Gordon Equations(Walter de Gruyter GmbH, 2009) Öziş, Turgut; Aslan, İsmailWith the aid of the symbolic computation system Mathematica, many exact solutions for the Fitzhugh-Nagumo equation and the Klein-Gordon equation with a quadratic nonlinearity are constructed by an auxiliary equation method, the so-called (G'/G)-expansion method, where the new and more general forms of solutions are also obtained. Periodic and solitary traveling wave solutions capable of moving in both directions are observed.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: 9Citation - Scopus: 9Exactly Solvable Quantum Sturm-Liouville Problems(American Institute of Physics, 2009) Atılgan Büyükaşık, Şirin; Pashaev, Oktay; Tığrak Ulaş, EsraThe harmonic oscillator with time-dependent parameters covers a broad spectrum of physical problems from quantum transport, quantum optics, and quantum information to cosmology. Several methods have been developed to quantize this fundamental system, such as the path integral method, the Lewis-Riesenfeld time invariant method, the evolution operator or dynamical symmetry method, etc. In all these methods, solution of the quantum problem is given in terms of the classical one. However, only few exactly solvable problems of the last one, such as the damped oscillator or the Caldirola-Kanai model, have been treated. The goal of the present paper is to introduce a wide class of exactly solvable quantum models in terms of the Sturm-Liouville problem for classical orthogonal polynomials. This allows us to solve exactly the corresponding quantum parametric oscillators with specific damping and frequency dependence, which can be considered as quantum Sturm-Liouville problems.Article Citation - WoS: 14Citation - Scopus: 14Cofinitely Weak Supplemented Lattices(Indian National Science Academy, 2009) Alizade, Rafail; Toksoy, Sultan EylemIn this paper it is shown that an E-complemented complete modular lattice L with small radical is weakly supplemented if and only if it is semilocal. L is a cofinitely weak supplemented lattice if and only if every maximal element of L has a weak supplement in L. If )α/0 is a cofinitely weak supplemented (weakly supplemented) sublattice and 1/α has no maximal element (1/α is weakly supplemented and a has a weak supplement in L), then L is cofinitely weak supplemented (weakly supplemented).Article Citation - WoS: 1Citation - Scopus: 1Casimir energies for some single cavities(TUBITAK, 2006) Ahmedov, Hacı; Duru, İsmail HakkıCasimir energies for some single cavities. Casimir energies are discussed for some cavities.Article Citation - WoS: 26Citation - Scopus: 26On a Recent Generalization of Semiperfect Rings(Cambridge University Press, 2008) Büyükaşık, Engin; Christian, LompIn a recent paper by Wang and Ding, it was stated that any ring which is generalized supplemented as a left module over itself is semiperfect. The purpose of this note is to show that Wang and Ding's claim is not true and that the class of generalized supplemented rings lies properly between the classes of semilocal and semiperfect rings. Moreover, we propose a corrected version of the theorem by introducing a wider notion of 'local' for submodules. © 2008 Australian Mathematical Society.Conference Object Citation - WoS: 24Citation - Scopus: 24Solitons of the Resonant Nonlinear Schrödinger Equation With Nontrivial Boundary Conditions: Hirota Bilinear Method(Pleiades Publishing, 2007) Lee, Jyh Hao; Pashaev, OktayWe use the Hirota bilinear approach to consider physically relevant soliton solutions of the resonant nonlinear Schrödinger equation with nontrivial boundary conditions, recently proposed for describing uniaxial waves in a cold collisionless plasma. By the Madelung representation, the model transforms into the reaction-diffusion analogue of the nonlinear Schrödinger equation, for which we study the bilinear representation, the soliton solutions, and their mutual interactions.