Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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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.Article Citation - WoS: 77Citation - Scopus: 110Analytic Study on Two Nonlinear Evolution Equations by Using the (g'/g)-expansion Method(Elsevier Ltd., 2009) Aslan, İsmail; Öziş, TurgutThe validity and reliability of the so-called (G′/G)-expansion method is tested by applying it to two nonlinear evolutionary equations. Solutions in more general forms are obtained. When the parameters are taken as special values, it is observed that the previously known solutions can be recovered. New rational function solutions are also presented. Being concise and less restrictive, the method can be applied to many nonlinear partial differential equations.Article Citation - WoS: 50Citation - Scopus: 56On the Validity and Reliability of the (g'/g)-expansion Method by Using Higher-Order Nonlinear Equations(Elsevier Ltd., 2009) Aslan, İsmail; Öziş, TurgutIn this study, we demonstrate the validity and reliability of the so-called (G′/G)-expansion method via symbolic computation. For illustrative examples, we choose the sixth-order Boussinesq equation and the ninth-order Korteweg-de-Vries equation. As a result, the power of the employed method is confirmed.Article Citation - WoS: 41Citation - Scopus: 56Exact and Explicit Solutions To Some Nonlinear Evolution Equations by Utilizing the (g'/g)-expansion Method(Elsevier Ltd., 2009) Aslan, İsmailIn this paper, we demonstrate the effectiveness of the so-called (G′/G)-expansion method by examining some nonlinear evolution equations with physical interest. Our work is motivated by the fact that the (G′/G)-expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.Article Citation - WoS: 4Citation - Scopus: 6Dissipative Hierarchies and Resonance Solitons for Kp-Ii and Mkp-Ii(Elsevier Ltd., 2007) Francisco, Meltem L. Y.; Lee, Jyh Hao; Pashaev, OktayWe show that dissipative solitons (dissipatons) of the second and the third members of SL(2,R) AKNS hierarchy give rise to the real solitons of KP-II, while for SL(2,R) Kaup-Newell hierarchy they give solitons of MKP-II. By the Hirota bilinear form for both flows, we find new bilinear system for these equations, and one- and two-soliton solutions. For special values of parameters our solutions show resonance behaviour with creation of four virtual solitons. We first time created four virtual soliton resonance solution for KP-II and established relations of it with degenerate four-soliton solution in the Hirota-Satsuma bilinear form for KP-II. Our approach allows one to interpret the resonance soliton as a composite object of two dissipative solitons in 1 + 1 dimensions.Article Citation - WoS: 30Citation - Scopus: 36Discrete Exact Solutions To Some Nonlinear Differential-Difference Equations Via the (g'/g)-expansion Method(Elsevier Ltd., 2009) Aslan, İsmailWe extended the (G′/G)-expansion method to two well-known nonlinear differential-difference equations, the discrete nonlinear Schrödinger equation and the Toda lattice equation, for constructing traveling wave solutions. Discrete soliton and periodic wave solutions with more arbitrary parameters, as well as discrete rational wave solutions, are revealed. It seems that the utilized method can provide highly accurate discrete exact solutions to NDDEs arising in applied mathematical and physical sciences.Article Citation - WoS: 37Citation - Scopus: 40Finite Element Method Solution of Electrically Driven Magnetohydrodynamic Flow(Elsevier Ltd., 2006) Neslitürk, Ali İhsan; Tezer, MünevverThe magnetohydrodynamic (MHD) flow in a rectangular duct is investigated for the case when the flow is driven by the current produced by electrodes, placed one in each of the walls of the duct where the applied magnetic field is perpendicular. The flow is steady, laminar and the fluid is incompressible, viscous and electrically conducting. A stabilized finite element with the residual-free bubble (RFB) functions is used for solving the governing equations. The finite element method employing the RFB functions is capable of resolving high gradients near the layer regions without refining the mesh. Thus, it is possible to obtain solutions consistent with the physical configuration of the problem even for high values of the Hartmann number. Before employing the bubble functions in the global problem, we have to find them inside each element by means of a local problem. This is achieved by approximating the bubble functions by a nonstandard finite element method based on the local problem. Equivelocity and current lines are drawn to show the well-known behaviours of the MHD flow. Those are the boundary layer formation close to the insulated walls for increasing values of the Hartmann number and the layers emanating from the endpoints of the electrodes. The changes in direction and intensity with respect to the values of wall inductance are also depicted in terms of level curves for both the velocity and the induced magnetic field.Article Citation - WoS: 6Citation - Scopus: 5Asymptotic Behavior of Linear Impulsive Integro-Differential Equations(Elsevier Ltd., 2008) Akhmet, Marat; Tleubergenova, M. A.; Yılmaz, OğuzAsymptotic equilibria of linear integro-differential equations and asymptotic relations between solutions of linear homogeneous impulsive differential equations and those of linear integro-differential equations are established. A new Gronwall-Bellman type lemma for integro-differential inequalities is proved. An example is given to demonstrate the validity of one of the results. © 2008 Elsevier Ltd. All rights reserved.Article Integrable Vortex Dynamics in Anisotropic Planar Spin Liquid Model(Elsevier Ltd., 2008) Gürkan, Zeynep Nilhan; Pashaev, OktayThe problem of magnetic vortex dynamics in an anisotropic spin liquid model is considered. For incompressible flow the model admits reduction to saturating Bogomolny inequality analytic projections of spin variables, subject the linear holomorphic Schrödinger equation. It allows us to construct N vortex configurations in terms of the complex Hermite polynomials. Using complex Galilean boost transformations, the interaction of the vortices and the vortex chain lattices (vortex crystals) is studied. By the complexified Cole-Hopf transformation, integrable N vortex dynamics is described by the holomorphic Burgers equation. Mapping of the point vortex problem to N-particle problem, the complexified Calogero-Moser system, showing its integrability and the Hamiltonian structure, is given. © 2006 Elsevier Ltd. All rights reserved.Article Citation - WoS: 2Citation - Scopus: 2Applying Mathematica and Webmathematica To Graph Coloring(Elsevier Ltd., 2007) Ufuktepe, Ünal; Bacak, GökşenThis paper analyzes some graph issues by using the symbolic program Mathematica and its version for the Web, webMathematica. In particular, we consider the problem of graph coloring: the assignment of colors to the vertices/edges of the graph such that adjacent vertices/edges are colored differently. In addition, we address the problem of obtaining the tenacity of binomial trees with Mathematica. Finally, we describe briefly an example of the application of our software to a scheduling problem.