Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Citation - WoS: 7Citation - Scopus: 8Invariants of Multi-Linkoids(Springer Basel Ag, 2023) Gabrovsek, Bostjan; Güğümcü, NeslihanIn this paper, we extend the definition of a knotoid to multilinkoids that consist of a finite number of knot and knotoid components. We study invariants of multi-linkoids, such as the Kauffman bracket polynomial, ordered bracket polynomial, the Kauffman skein module, and the T-invariant in relation with generalized T-graphs.Article Citation - WoS: 1Citation - Scopus: 1Initial Stages of Gravity-Driven Flow of Two Fluids of Equal Depth(American Institute of Physics, 2023) Korobkin, Alexander; Yılmaz, OğuzShort-time behavior of gravity-driven free surface flow of two fluids of equal depth and different densities is studied. Initially, the fluids are at rest and separated with a vertical rigid plate of negligible thickness. Then, the plate disappears suddenly and a gravity-driven flow of the fluids starts. The flow in an early stage is described by the potential theory. The initial flow in the leading order is described by a linear problem, which is solved by the Fourier series method. The motions of the interface between the fluids and their free surfaces are investigated. The singular behaviors of the velocity field at the bottom point, where the interface meets the rigid bottom, and the top point, where the interface meets both free surfaces, are analyzed in detail. The flow velocity is shown to be log-singular at the bottom point. The leading-order inner asymptotic solution is constructed in a small vicinity of this point. It is shown that the flow close to the bottom point is self-similar. The motion of the interface is independent of any parameters, including the density ratio, of the problem in specially stretched variables. In the limiting case of negligible density of one of the fluids, the results of the classical dam break problem are recovered. The Lagrangian representation is employed to capture the behavior of the interface and the free surfaces at the top, where the fluid interface meets the free surfaces. The shapes of the free surfaces and the interface in the leading order computed by using the Lagrangian variables show a jump discontinuity of the free surface near the top point where the free surfaces and the interface meet. Inner region formulation is derived near the top point.Article Citation - WoS: 9Citation - Scopus: 8Taylor Wavelets Collocation Technique for Solving Fractional Nonlinear Singular Pdes(Springer, 2022) Aghazadeh, Nasser; Mohammadi, Amir; Tanoğlu, GamzeA novel technique has been introduced to solve the Emden-Fowler equations. It has been derived from the Taylor wavelets collocation method. The proposed scheme has been successfully implemented in order to solve the singular equations. The singular problem converts to a system of algebraic equations that can be solved numerically. Moreover, the technique is very effective to remove the strong singularity point at x = 0. The numerical experiments have been checked out with the exact and approximate solutions that have been achieved by others including the Adomian decomposition method (Wazwaz in Appl Math Comput 166:638-651, 2005), Modified Homotopy Perturbation Method (Singh et al. J Math Chem 54(4):918-931, 2016). Also, the error analysis of the technique has been considered.Article A Reliable and Fast Mesh-Free Solver for the Telegraph Equation(Springer, 2022) İmamoğlu Karabaş, Neslişah; Korkut, Sıla Övgü; Gürarslan, Gürhan; Tanoğlu, GamzeIn the presented study, the hyperbolic telegraph equation is taken as the focus point. To solve such an equation, an accurate, reliable, and efficient method has been proposed. The developed method is mainly based on the combination of a kind of mesh-free method and an adaptive method. Multiquadric radial basis function mesh-free method is considered on spatial domain and the adaptive fifth-order Runge–Kutta method is used on time domain. The validity and the performance of the proposed method have been checked on several test problems. The approximate solutions are compared with the exact solution, it is shown that the proposed method has more preferable to the other methods in the literature.Article Citation - Scopus: 3Stabilization of Higher Order Schrödinger Equations on a Finite Interval: Part Ii(American Institute of Mathematical Sciences, 2022) Özsarı, Türker; Yılmaz, Kemal CemBackstepping based controller and observer models were designed for higher order linear and nonlinear Schrödinger equations on a finite interval in [3] where the controller was assumed to be acting from the left endpoint of the medium. In this companion paper, we further the analysis by considering boundary controller(s) acting at the right endpoint of the domain. It turns out that the problem is more challenging in this scenario as the associated boundary value problem for the backstepping kernel becomes overdetermined and lacks a smooth solution. The latter is essential to switch back and forth between the original plant and the so called target system. To overcome this difficulty we rely on the strategy of using an imperfect kernel, namely one of the boundary conditions in kernel PDE model is disregarded. The drawback is that one loses rapid stabilization in comparison with the left endpoint con-trollability. Nevertheless, the exponential decay of the L2-norm with a certain rate still holds. The observer design is associated with new challenges from the point of view of wellposedness and one has to prove smoothing properties for an associated initial boundary value problem with inhomogeneous boundary data. This problem is solved by using Laplace transform in time. However, the Bromwich integral that inverts the transformed solution is associated with certain analyticity issues which are treated through a subtle analysis. Numerical algorithms and simulations verifying the theoretical results are given.Article Citation - WoS: 8Citation - Scopus: 8Stabilization of Higher Order Schrödinger Equations on a Finite Interval: Part I(American Institute of Mathematical Sciences, 2021) Batal, Ahmet; Özsarı, Türker; Yılmaz, Kemal CemWe study the backstepping stabilization of higher order linear and nonlinear Schrödinger equations on a finite interval, where the boundary feedback acts from the left Dirichlet boundary condition. The plant is stabilized with a prescribed rate of decay. The construction of the backstepping kernel is based on a challenging successive approximation analysis. This contrasts with the case of second order pdes. Second, we consider the case where the full state of the system cannot be measured at all times but some partial information such as measurements of a boundary trace are available. For this problem, we simultaneously construct an observer and the associated backstepping controller which is capable of stabilizing the original plant. Wellposedness and regularity results are provided for all pde models. Although the linear part of the model is similar to the KdV equation, the power type nonlinearity brings additional difficulties. We give two examples of boundary conditions and partial measurements. We also present numerical algorithms and simulations verifying our theoretical results to the fullest extent. Our numerical approach is novel in the sense that we solve the target systems first and obtain the solution to the feedback system by using the bounded invertibility of the backstepping transformation. © 2021, American Institute of Mathematical Sciences. All rights reserved.Article Citation - WoS: 2Citation - Scopus: 2An Inverse Parameter Problem With Generalized Impedance Boundary Condition for Two-Dimensional Linear Viscoelasticity(Society for Industrial and Applied Mathematics Publications, 2021) Ivanyshyn Yaman, Olha; Le Louer, FrederiqueWe analyze an inverse boundary value problem in two-dimensional viscoelastic media with a generalized impedance boundary condition on the inclusion via boundary integral equation methods. The model problem is derived from a recent asymptotic analysis of a thin elastic coating as the thickness tends to zero [F. Caubet, D. Kateb, and F. Le Louer, J. Elasticity, 136 (2019), pp. 17-53]. The boundary condition involves a new second order surface symmetric operator with mixed regularity properties on tangential and normal components. The well-posedness of the direct problem is established for a wide range of constant viscoelastic parameters and impedance functions. Extending previous research in the Helmholtz case, the unique identification of the impedance parameters from measured data produced by the scattering of three independent incident plane waves is established. The theoretical results are illustrated by numerical experiments generated by an inverse algorithm that simultaneously recovers the impedance parameters and the density solution to the equivalent boundary integral equation reformulation of the direct problem.Article Citation - WoS: 1Citation - Scopus: 1Fully Three-Dimensional Analysis of a Photonic Crystal Assisted Silicon on Insulator Waveguide Bend(World Scientific Publishing, 2018) Eti, Neslihan; Çetin, Zebih; Sözüer, Hüseyin SamiA detailed numerical study of low-loss silicon on insulator (SOI) waveguide bend is presented using the fully three-dimensional (3D) finite-difference time-domain (FDTD) method. The geometrical parameters are optimized to minimize the bending loss over a range of frequencies. Transmission results for the conventional single bend and photonic crystal assisted SOI waveguide bend are compared. Calculations are performed for the transmission values of TE-like modes where the electric field is strongly transverse to the direction of propagation. The best obtained transmission is over 95% for TE-like modes.Article On Smoothers for Multigrid of the Second Kind(John Wiley and Sons Inc., 2019) Aksoylu, Burak; Kaya, AdemWe study smoothers for the multigrid method of the second kind arising from Fredholm integral equations. Our model problems use nonlocal governing operators that enforce local boundary conditions. For discretization, we utilize the Nystrom method with the trapezoidal rule. We find the eigenvalues of matrices associated to periodic, antiperiodic, and Dirichlet problems in terms of the nonlocality parameter and mesh size. Knowing explicitly the spectrum of the matrices enables us to analyze the behavior of smoothers. Although spectral analyses exist for finding effective smoothers for 1D elliptic model problems, to the best of our knowledge, a guiding spectral analysis is not available for smoothers of a multigrid of the second kind. We fill this gap in the literature. The Picard iteration has been the default smoother for a multigrid of the second kind. Jacobi-like methods have not been considered as viable options. We propose two strategies. The first one focuses on the most oscillatory mode and aims to damp it effectively. For this choice, we show that weighted-Jacobi relaxation is equivalent to the Picard iteration. The second strategy focuses on the set of oscillatory modes and aims to damp them as quickly as possible, simultaneously. Although the Picard iteration is an effective smoother for model nonlocal problems under consideration, we show that it is possible to find better than ones using the second strategy. We also shed some light on internal mechanism of the Picard iteration and provide an example where the Picard iteration cannot be used as a smoother.Article Citation - WoS: 7Citation - Scopus: 7Complex Ginzburg–landau Equations With Dynamic Boundary Conditions(Elsevier Ltd., 2018) Corrêa, Wellington José; Özsarı, TürkerThe initial-dynamic boundary value problem (idbvp) for the complex Ginzburg–Landau equation (CGLE) on bounded domains of RN is studied by converting the given mathematical model into a Wentzell initial–boundary value problem (ibvp). First, the corresponding linear homogeneous idbvp is considered. Secondly, the forced linear idbvp with both interior and boundary forcings is studied. Then, the nonlinear idbvp with Lipschitz nonlinearity in the interior and monotone nonlinearity on the boundary is analyzed. The local well-posedness of the idbvp for the CGLE with power type nonlinearities is obtained via a contraction mapping argument. Global well-posedness for strong solutions is shown. Global existence and uniqueness of weak solutions are proven. Smoothing effect of the corresponding evolution operator is proved. This helps to get better well-posedness results than the known results on idbvp for nonlinear Schrödinger equations (NLS). An interesting result of this paper is proving that solutions of NLS subject to dynamic boundary conditions can be obtained as inviscid limits of the solutions of the CGLE subject to same type of boundary conditions. Finally, long time behavior of solutions is characterized and exponential decay rates are obtained at the energy level by using control theoretic tools.