Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article The Application of a Finite Difference Method To a Dynamical Interface Problem(Acad. Publications, 2003) Tanoğlu, Gamze; Ağıroğlu, İzzet OnurA multiple-order-parameter model for Cu-Au system on a face cubic centered lattice was recently developed in the presence of anisotropy. In that model, three order parameters (non-conserved) and one concentration order parameter (conserved), which has been taken as a constant, were considered. Later on, the model has been extended, so that, concentration has been taken as a variable. It has been seen that two models were in a good agreement near critical temperature since the non-conserved order parameter behaves like a constant near critical temperature in both models.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 Hirota Method for Solving Reaction-Diffusion Equations With Generalized Nonlinearity(World Academic Press, 2006) Tanoğlu, GamzeThe Hirota Method is applied to find an exact solitary wave solution to evolution equation with generalized nonlinearity. By introducing the power form of Hirota ansatz the bilinear representation for this equation is derived and the traveling wave solution is constructed by Hirota perturbation. We show that velocity of this solution is naturally fixed by truncating the Hirota’s perturbation expansion. So in our approach, this truncate on works similarly to the way Ablowitz and Zeppetella obtained an exact travelling wave solution of Fisher’s equation by finding the special wave speed for which the resulting ODE is of the Painleve type. In the special case the model admits N shock soliton solution and the reduction to Burgers’ equation.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: 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.Article Citation - WoS: 16Citation - Scopus: 18Solitary Wave Solution of Nonlinear Multi-Dimensional Wave Equation by Bilinear Transformation Method(Elsevier Ltd., 2007) Tanoğlu, GamzeThe Hirota method is applied to construct exact analytical solitary wave solutions of the system of multi-dimensional nonlinear wave equation for n-component vector with modified background. The nonlinear part is the third-order polynomial, determined by three distinct constant vectors. These solutions have not previously been obtained by any analytic technique. The bilinear representation is derived by extracting one of the vector roots (unstable in general). This allows to reduce the cubic nonlinearity to a quadratic one. The transition between other two stable roots gives us a vector shock solitary wave solution. In our approach, the velocity of solitary wave is fixed by truncating the Hirota perturbation expansion and it is found in terms of all three roots. Simulations of solutions for the one component and one-dimensional case are also illustrated.Article Citation - WoS: 31Citation - Scopus: 34Vector Shock Soliton and the Hirota Bilinear Method(Elsevier Ltd., 2005) Pashaev, Oktay; Tanoğlu, GamzeThe Hirota bilinear method is applied to find an exact shock soliton solution of the system reaction-diffusion equations for n-component vector order parameter, with the reaction part in form of the third order polynomial, determined by three distinct constant vectors. The bilinear representation is derived by extracting one of the vector roots (unstable in general), which allows us reduce the cubic nonlinearity to a quadratic one. The vector shock soliton solution, implementing transition between other two roots, as a fixed points of the potential from continuum set of the values, is constructed in a simple way. In our approach, the velocity of soliton is fixed by truncating the Hirota perturbation expansion and it is found in terms of all three roots. Shock solitons for extensions of the model, by including the second order time derivative term and the nonlinear transport term are derived. Numerical solutions illustrating generation of solitary wave from initial step function, depending of the polynomial roots are given.