Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection
Permanent URI for this collectionhttps://hdl.handle.net/11147/7148
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Article Citation - Scopus: 2A Conserved Linearization Approach for Solving Nonlinear Oscillation Problems(Natural Sciences Publishing, 2018) Korkut, Sıla Övgü; Gücüyenen Kaymak, Nurcan; Tanoğlu, GamzeNonlinear oscillation problems are extensively used in engineering and applied sciences. Due to non-availability of the analytic solutions, numerical approaches have been used for these equations. In this study, a numerical method which is based on Newton-Raphson linearization and Fréchet derivative is suggested. The convergence analysis is also studied locally. The present method is tested on three examples: damped oscillator, Van-der Pol equation and Schrödinger equation. It is shown that the obtained solutions via the present method are more accurate than those of the well-known second order Runge-Kutta method. When examining the present method, preservation of characteristic properties of these equations is also considered. The obtained results show that the present method is applicable with respect to the efficiency and the physical compatibility.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: 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 Cmmse-Convergence Analysis for Operator Splitting Methods With Application To Burgers-Huxley Equation(Natural Sciences Publishing, 2015) Çiçek, Yeşim; Tanoğlu, GamzeWe provide an error analysis of the operator splitting method of the Lie-Trotter type applied to the Burgers-Huxley equation ut + αuux - εuxx = β(1 - u)(u - γ)u. We show that the Lie-Trotter splitting method converges with the expected rate in Hs(R), where Hs(R) is the Sobolev space and s is an arbitrary nonnegative integer. We split the equation into linear and nonlinear parts and apply numerical methods for these subproblems. We present errors and confirm the theoretical results with the numerical example.Article Citation - WoS: 5Citation - Scopus: 7Iterative Operator Splitting Method for Capillary Formation Model in Tumor Angiogenesis Problem: Analysis and Application(John Wiley and Sons Inc., 2011) Gücüyenen, Nurcan; Tanoğlu, GamzeIterative operator splitting method is used to solve numerically the mathematical model for capillary formation in tumor angiogenesis problem. The method is based on first splitting the complex problem into simpler sub-problems. Then each sub-equation is combined with iterative schemes. The algorithms are obtained by applying the proposed method to the given model problem. The explicit local error bounds are derived to show consistency. We also explained the stability by constructing the stability functions. The obtained numerical results show that iterative splitting method provides high accuracy and efficiency with respect to other classical methods in the literature.