WoS İndeksli Yayınlar Koleksiyonu / WoS Indexed Publications Collection
Permanent URI for this collectionhttps://hdl.handle.net/11147/7150
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Article Citation - WoS: 4Citation - Scopus: 4A New Numerical Algorithm Based on Quintic B-Spline and Adaptive Time Integrator for Cou- Pled Burger's Equation(Tabriz University, 2023) Çiçek, Yeşim; Gücüyenen Kaymak, Nurcan; Bahar, Ersin; Gürarslan, Gürhan; Tanoğlu, GamzeIn this article, the coupled Burger's equation which is one of the known systems of the nonlinear parabolic partial differential equations is studied. The method presented here is based on a combination of the quintic B-spline and a high order time integration scheme known as adaptive Runge-Kutta method. First of all, the application of the new algorithm on the coupled Burger's equation is presented. Then, the convergence of the algorithm is studied in a theorem. Finally, to test the efficiency of the new method, coupled Burger's equations in literature are studied. We observed that the presented method has better accuracy and efficiency compared to the other methods in the literature. © 2023 University of Tabriz. All Rights Reserved.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 An Operator Splitting Approximation Combined With the Supg Method for Transport Equations With Nonlinear Reaction Term(Tech Science Press, 2012) Baysal, Onur; Tanoğlu, GamzeIn this work, an operator splitting method is proposed in order to obtain a stable numerical solution for transport equation with non-linear reaction term. We split the transport equation into a reaction part and an advection diffusion part. The former one which becomes a nonlinear ordinary differential equation can be approximated by the simple higher order integrator or solved exactly. The later one is approximated by the Streamline-Upwind Petrov-Galerkin (SUPG) method combined with the generalized Euler time integration (q-method). Numerical results that illustrate the good performance of this method are reported.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: 5Citation - Scopus: 5Convergence Analysis and Numerical Solution of the Benjamin-Bona Equation by Lie-Trotter Splitting(TUBITAK, 2018) Zürnacı, Fatma; Gücüyenen Kaymak, Nurcan; Seydaoğlu, Muaz; Tanoğlu, GamzeIn this paper, an operator splitting method is used to analyze nonlinear Benjamin-Bona-Mahony-type equations. We split the equation into an unbounded linear part and a bounded nonlinear part and then Lie-Trotter splitting is applied to the equation. The local error bounds are obtained by using the approach based on the differential theory of operators in a Banach space and the quadrature error estimates via Lie commutator bounds. The global error estimate is obtained via Lady Windermere's fan argument. Finally, to confirm the expected convergence order, numerical examples are studied.Article Citation - WoS: 2Citation - Scopus: 4An Efficient Iterative Algorithm for Solving Non-Linear Oscillation Problems(Faculty of Sciences and Mathematics, University of Nis, 2017) Korkut Uysal, Sıla Övgü; Tanoğlu, GamzeA new iterative method is presented for numerical solution of nonlinear evolutionary problems. The convergence properties of the proposed method are analysed in abstract framework by using the concepts of consistency, stability and order. Both the ϕ-functions and semigroup properties are used to overcome the presence of unboundedness of the operator. In order to confirm the theoretical results, the method is applied to three benchmark problems from the literature. The numerical results are compared with traditional splitting methods and confirm that the proposed method is more accurate as well as more efficient than the traditional splitting methods.Article Citation - WoS: 16Citation - Scopus: 16Strang Splitting Method for Burgers-Huxley Equation(Elsevier Ltd., 2016) Çiçek, Yeşim; Tanoğlu, GamzeWe derive an analytical approach to the Strang splitting method for the Burgers-Huxley equation (BHE) ut+αuux-ε uXX=β(1-u)(u-γ)u. We proved that Srtang splitting method has a second order convergence in Hs(R), where Hs(R) is the Sobolev space and s is an arbitrary nonnegative integer. We numerically solve the BHE by Strang splitting method and compare the results with the reference solution.Article Citation - WoS: 2Citation - Scopus: 2The Convergence of a New Symmetric Iterative Splitting Method for Non-Autonomous Systems(Taylor and Francis Ltd., 2012) Tanoğlu, Gamze; Korkut, SılaThe iterative splitting methods have been extensively applied to solve complicated systems of differential equations. In this process, we split the complex problem into several sub-problems, each of which can be solved sequentially. In this paper, we construct a new symmetric iterative splitting scheme based on the Magnus expansion for solving non-autonomous problems. We also study its convergence properties by using the concepts of stability, consistency, and order. Several numerical examples are illustrated to confirm the theoretical results by comparing frequently used methods. © 2012 Copyright Taylor and Francis Group, LLC.Article Citation - WoS: 11Citation - Scopus: 14Higher Order Operator Splitting Methods Via Zassenhaus Product Formula: Theory and Applications(Elsevier Ltd., 2011) Geiser, Jürgen; Tanoğlu, Gamze; Gücüyenen, NuranIn this paper, we contribute higher order operator splitting methods improved by Zassenhaus product. We apply the contribution to classical and iterative splitting methods. The underlying analysis to obtain higher order operator splitting methods is presented. While applying the methods to partial differential equations, the benefits of balancing time and spatial scales are discussed to accelerate the methods. The verification of the improved splitting methods are done with numerical examples. An individual handling of each operator with adapted standard higher order time-integrators is discussed. Finally, we conclude the higher order operator splitting methods.Article Citation - WoS: 4Citation - Scopus: 7On the Numerical Solution of Korteweg-De Vries Equation by the Iterative Splitting Method(Elsevier Ltd., 2011) Gücüyenen, Nurcan; Tanoğlu, GamzeIn this paper, we apply the method of iterative operator splitting on the Korteweg-de Vries (KdV) equation. The method is based on first, splitting the complex problem into simpler sub-problems. Then each sub-equation is combined with iterative schemes and solved with suitable integrators. Von Neumann analysis is performed to achieve stability criteria for the proposed method applied to the KdV equation. The numerical results obtained by iterative splitting method for various initial conditions are compared with the exact solutions. It is seen that they are in a good agreement with each other. © 2011 Elsevier Inc. All rights reserved.