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Article Weak Ψ -Contractions on Partially Ordered Metric Spaces and Applications To Boundary Value Problems(2015) Aksoy, ÜmitRecent developments in fixed point theory have been encouraged by the applicability of the results in the area of boundary value problems for differential and integral equations. Especially in the last few years, a lot of publications in fixed point theory have presented results directly related to specific initial or boundary value problems. These problems include not only ordinary and partial differential equations, but also fractional differential equations.Article Norm Estimates of a Class of Calderon–zygmund Type Strongly Singular Integral Operators(Integral Transforms and Special Functions, 2007) Aksoy, Ümit; Çelebi, OkayIn this article, we prove the Lp boundedness of a class of Calderon–Zygmund type strongly singular operators. In particular, we give an estimate for the L2 norm of these operators in the unit disc of the complex plane.Article Some Remarks About the Existence of Coupled G-Coincidence Points(Journal of Inequalities and Applications, 2015) Erhan, İnci M.; Shahzad, NaseerVery recently, in a series of subsequent papers, Nan and Charoensawan introduced the notion of g-coincidence point of two mappings in different settings (metric spaces and G-metric spaces) and proved some theorems in order to guarantee the existence and uniqueness of such kind of points. Although their notion seems to be attractive, in this paper, we show how this concept can be reduced to the unidimensional notion of coincidence point, and how their main theorems can be seen as particular cases of existing results. Moreover, we prove that the proofs of their main statements have some gaps.Article Remarks on Some Coupled Fixed Point Theorems in G-Metric Spaces(2014) Agarwal, Ravi P.; Karapınar, ErdalIn this paper, we show that, unexpectedly, most of the coupled fixed point theorems in the context of (ordered) G-metric spaces are in fact immediate consequences of usual fixed point theorems that are either well known in the literature or can be obtained easily.Article A Survey on Boundary Value Problems for Complex Partial Differential Equations(Advances in Dynamical Systems and Applications, 2010) Aksoy, Ümit; Çelebi, A. OkayIn this article, the recent results on basic boundary value problems of complex analysis are surveyed for complex model equations and linear elliptic complex par tial differential equations of arbitrary order on simply connected bounded domains, particularly in the unit disc, on unbounded domains such as upper half plane and upper right quarter plane and on multiply connected domains containing circular rings.Article Coincidence Point Theorems in Quasi-Metric Spaces Without Assuming the Mixed Monotone Property and Consequences in G-Metric Spaces(2014) Roldán, Antonio-francisco; Karapınar, Erdal; De La Sen, ManuelIn this paper, we present some coincidence point theorems in the setting of quasi-metric spaces that can be applied to operators which not necessarily have the mixed monotone property. As a consequence, we particularize our results to the field of metric spaces, partially ordered metric spaces and G-metric spaces, obtaining some very recent results. Finally, we show how to use our main theorems to obtain coupled, tripled, quadrupled and multidimensional coincidence point results.Article Romberg Integration: a Symbolic Approach With Mathematica(2003) Yazıcı, Ali; Ergenç, Tanıl; Altaş, İrfanHigher order approximations of an integral can be obtained from lower order ones in a systematic way. For 1-D integrals Romberg Integration is an example which is based upon the composite trapezoidal rule and the well-known Euler-Maclaurin expansion of the error. In this work, Mathematica is utilized to illustrate the method and the under lying theory in a symbolic fashion. This approach seems plausible for discussing integration in a numerical computing laboratory environment.Article Conservative Finite Difference Schemes for the Chiral Nonlinear Schrödinger Equation(Boundary Value Problems, 2015) Ismaıl, Mohammed S.; Al-basyounı, Khalil S.; Aydın, AyhanIn this paper, we derive three finite difference schemes for the chiral nonlinear Schrödinger equation (CNLS). The CNLS equation has two kinds of progressive wave solutions: bright and dark soliton. The proposed methods are implicit, unconditionally stable and of second order in space and time directions. The exact solutions and the conserved quantities are used to assess the efficiency of these methods. Numerical simulations of single bright and dark solitons are given. The interactions of two bright solitons are also displayed.Article Multi-Symplectic Integration of Coupled Non-Linear Schrödinger System With Soliton Solutions(International Journal of Computer Mathematics, 2009) Aydın, Ayhan; Aydın, Ayhan; Karasözen, Bülent; Aydın, Ayhan; Mathematics; MathematicsSystems of coupled non-linear Schrödinger equations with soliton solutions are integrated using the six-point scheme which is equivalent to the multi-symplectic Preissman scheme. The numerical dispersion relations are studied for the linearized equation. Numerical results for elastic and inelastic soliton collisions are presented. Numerical experiments confirm the excellent conservation of energy, momentum and norm in long-term computations and their relations to the qualitative behaviour of the soliton solutions.Article Symplectic and multi-symplectic methods for coupled nonlinear Schrödinger equations with periodic solutions(Computer Physics Communications, 2007) Aydın, Ayhan; Karasözen, BülentWe consider for the integration of coupled nonlinear Schrödinger equations with periodic plane wave solutions a splitting method from the class of symplectic integrators and the multi-symplectic six-point scheme which is equivalent to the Preissman scheme. The numerical experiments show that both methods preserve very well the mass, energy and momentum in long-time evolution. The local errors in the energy are computed according to the discretizations in time and space for both methods. Due to its local nature, the multi-symplectic six-point scheme preserves the local invariants more accurately than the symplectic splitting method, but the global errors for conservation laws are almost the same.
