Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Citation - WoS: 9Citation - Scopus: 8A Numerical Method Based on Legendre Wavelet and Quasilinearization Technique for Fractional Lane-Emden Type Equations(Springer, 2024) İdiz, F.; Tanoǧlu, G.; Aghazadeh, N.In this research, we study the numerical solution of fractional Lane-Emden type equations, which emerge mainly in astrophysics applications. We propose a numerical approach making use of Legendre wavelets and the quasilinearization technique. The nonlinear term in fractional Lane-Emden type equations is iteratively linearized using the quasilinearization technique. The linearized equations are then solved using the Legendre wavelet collocation method. The proposed method is quite effective to overcome the singularity in fractional Lane-Emden type equations. Convergence and error analysis of the proposed method are given. We solve some test problems to compare the effectiveness of the proposed method with some other numerical methods in the literature. © 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.Article Citation - WoS: 11Citation - Scopus: 8Local Well-Posedness of the Higher-Order Nonlinear Schrödinger Equation on the Half-Line: Single-Boundary Condition Case(Wiley, 2024) Alkın, A.; Mantzavinos, D.; Özsarı, T.We establish local well-posedness in the sense of Hadamard for a certain third-order nonlinear Schrödinger equation with a multiterm linear part and a general power nonlinearity, known as higher-order nonlinear Schrödinger equation, formulated on the half-line (Formula presented.). We consider the scenario of associated coefficients such that only one boundary condition is required and hence assume a general nonhomogeneous boundary datum of Dirichlet type at (Formula presented.). Our functional framework centers around fractional Sobolev spaces (Formula presented.) with respect to the spatial variable. We treat both high regularity ((Formula presented.)) and low regularity ((Formula presented.)) solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, this is no longer the case and, instead, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial-boundary value problems, as it involves proving boundary-type Strichartz estimates that are not common in the study of Cauchy (initial value) problems. The linear analysis, which forms the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method (also known as the unified transform) for the associated forced linear problem. In this connection, we note that the higher-order Schrödinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivatives in the linear part of the equation. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; and (iii) complicated oscillatory kernels in the weak solution formula for the linear initial-boundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data. The present work provides a first, complete treatment via the Fokas method of a nonhomogeneous initial-boundary value problem for a partial differential equation associated with a multiterm linear differential operator. © 2023 Wiley Periodicals LLC.Article Parity of an Odd Dominating Set(2022) Batal, AhmetFor a simple graph $G$ with vertex set $V(G)={v_1,...,v_n}$, we define the closed neighborhood set of a vertex $u$ as $N[u]={v in V(G) ; | ; v ; text{is adjacent to} ; u ; text{or} ; v=u }$ and the closed neighborhood matrix $N(G)$ as the matrix whose $i$th column is the characteristic vector of $N[v_i]$. We say a set $S$ is odd dominating if $N[u]cap S$ is odd for all $uin V(G)$. We prove that the parity of the cardinality of an odd dominating set of $G$ is equal to the parity of the rank of $G$, where rank of $G$ is defined as the dimension of the column space of $N(G)$. Using this result we prove several corollaries in one of which we obtain a general formula for the nullity of the join of graphs.Article Relativistic Dissipatons in Integrable Nonlinear Majorana Type Spinor Model(Yunnan University, 2022) Pashaev, Oktay; Lee, J. H.By method of moving frame, the relativistic integrable nonlinear model for real, Majorana type spinor fields in 1+1 dimensions is introduced and gauge equivalence of this model with Papanicolau spin model on one sheet hyperboloid is established. In terms of the so called double numbers, the model is represented also as hyperbolic complex relativistic model, in the form similar to the massive Thirring model. By using Hirota's bilinear method, the one dissipaton solution of this model is constructed. We calculate first integrals of motion for this dissipaton and show that it represents a relativistic particle with highly nonlinear mass. Analyzing resonance conditions for scattering of two relativistic dissipatons, we find a solution describing resonant property of the dissipatons.Conference Object Citation - WoS: 1Citation - Scopus: 4Uniform Asymptotic Stability by Indefinite Lyapunov Functions(IEEE, 2022) Sahan, Gokhan; Ozdemir, DeryaIn this work, we consider Uniform Asymptotic Stability (UAS) of nonlinear time-varying systems. We utilize an indefinite signed polynomial of Lyapunov Function (LF) for the upper bound of the derivative of LF. This special bound is especially useful for perturbation problems. Compared to the ones in the literature we improve the upper bound of the LF and its related properties. Since UAS is the first step for input to state stability (ISS) and integral ISS, it should be thought that these improvements will give rise to new advances in real-world applications as well.Article Two Numerical Solutions for Solving a Mathematical Model of the Avascular Tumor Growth(Dokuz Eylül Üniversitesi, 2021) Korkut, Sıla Övgü; İmamoğlu Karabaş, Neslişah; Başbınar, YaseminObjective: Cancer which is one of the most challenging health problems overall the world is composed of various processes: tumorigenesis, angiogenesis, and metastasis. Attempting to understand the truth behind this complicated disease is one of the common objectives of many experts and researchers from different fields. To provide deeper insights any prognostic and/or diagnostic scientific contribution to this topic is so crucial. In this study, the avascular tumor growth model which is the earliest stage of tumor growth is taken into account from a mathematical point of view. The main aim is to solve the mathematical model of avascular tumor growth numerically. Methods: This study has focused on the numerical solution of the continuum mathematical model of the avascular tumor growth described by Sharrett and Chaplin. Unlike the existing recent literature, the study has focused on the methods for the temporal domain. To obtain the numerical schemes the central difference method has been used in the spatial coordinates. This discretization technique has reduced the main partial differential equation into an ordinary differential equation which will be solved successively by two alternative techniques: the 4th order Runge-Kutta method (RK4) and the three-stage strongly-stability preserving Runge-Kutta method (SSP-RK3). Results: The model has been solved by the proposed methods. The numerical results are discussed in both mathematical and biological angles. The biological compatibility of the methods is depicted in various figures. Besides biological outputs, the accuracies of the methods have been listed from a mathematical point of view. Furthermore, the rate of convergence of the proposed methods has also been discussed computationally. Conclusion: All recorded results are evidence that the proposed schemes are applicable for solving such models. Moreover, all exhibited figures have proved the biological compatibility of the methods. It is observed that the quiescent cells which are one of the most mysterious cells in clinics tend to become proliferative for the selected parameters.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.Article Relaxation of Conditions of Lyapunov Functions(2021) Şahan, GökhanIn this study, stability conditions are given for nonlinear time varying systems using the classical Lyapunov 2nd Method and its arguments. A novel approach is utilized and so that uniform stability can also be proved by using an unclassical Lyapunov Function. In contrast with the studies in the literature, Lyapunov Function is allowed to be negative definite and increasing through the system. To construct a classical Lyapunov Function, we use a reverse time approach methodology for the intervals where the unclassical one is increasing. So we prove the stability using a new Lyapunov Function construction methodology. The main result shows that the existence of such a function guarantees the stability of the origin. Some numerical examples are also given to demonstrate the efficiency of the method we use.Article Some Properties of Rickart Modules(Ankara Üniversitesi, 2012) Üngör, Burcu; Kafkas, Gizem; Halıcıoğlu, Sait; Harmancı, AbdullahR birimli bir halka, M saº g R-mod¸l ve M nin endomorÖzma halkas¨ S = EndR(M) olsun. Her f 2 S iÁin rM(f) = eM olacak biÁimde e2 = e 2 S varsa (denk olarakKerf,Mmod¸l¸n¸nbirdirekttoplanan¨ise)MyeRickartmod¸lad¨verilmi?stir[8]. BuÁal¨?smadaRickartmod¸llerinˆzellikleriincelenmeyedevamedilmi?stir. M birRickart mod¸l olmak ¸zere, M nin S-kat¨ (s¨ras¨yla S-indirgenmi?s, S-simetrik, S-yar¨ deºgi?smeli, S-Armendariz)mod¸l olmas¨ iÁin gerek ve yeter ?sart¨n S nin kat¨ (s¨ras¨yla indirgenmi?s, simetrik, yar¨ deºgi?smeli, Armendariz) halka olduºgu gˆsterilmi?stir. M[x], S[x] halkas¨na gˆre Rickart mod¸l iken M nin de Rickart mod¸l oldugu,tersinin M nin S-Armendariz olmas¨ durumunda doºgru olduºgu ispatlanm¨?st¨r. Ayrıca bir M mod¸l¸n¸n Rickart ol- mas¨iÁingerekveyeter?sart¨nhersaºgmod¸l¸nM-temelprojektifolduºgueldeedilmi?stir.Conference Object Can Cpt Be Violated Through Extended Time Reversal?(World Scientific Publishing, 2001) Erdem, Recai; Ufuktepe, ÜnalWe consider the implications of the extension of time reversal through Wigner types and group extensions. We clarify its physical content and apply the results in a toy model. Finally we point out the possibility of violation of CPT in this framework.