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Permanent URI for this collectionhttps://ada.atilim.edu.tr/handle/123456789/18

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Citation Count: 8
AV Bitsadze's observation on bianalytic functions and the Schwarz problem
(Taylor & Francis Ltd, 2019) Aksoy, Umit; Begehr, Heinrich; Celebi, A. Okay; Mathematics
According to an observation of A.V. Bitsadze from 1948 the Dirichlet problem for bianalytic functions is ill-posed. A natural boundary condition for the polyanalytic operator, however, is the Schwarz condition. An integral representation for the solutions in the unit disc to the inhomogeneous polyanalytic equation satisfying Schwarz boundary conditions is known. This representation is extended here to any simply connected plane domain having a harmonic Green function. Some other boundary value problems are investigated with some Dirichlet and Neumann conditions illuminating that just the Schwarz problem is a natural boundary condition for the Bitsadze operator.
Citation Count: 7
AV Bitsadze's observation on bianalytic functions and the Schwarz problem revisited
(Taylor & Francis Ltd, 2021) Aksoy, Umit; Begehr, Heinrich; Celebi, A. Okay; Mathematics
The extension of the Schwarz representation formula to simply connected domains with harmonic Green function and its polyanalytic generalization is not valid in general. They do hold only for certain domains.
Citation Count: 3
Dirichlet problem for a generalized inhomogeneous polyharmonic equation in an annular domain
(Taylor & Francis Ltd, 2012) Aksoy, U.; Celebi, A. O.; Mathematics
In this article, we investigate the solvability of the Dirichlet problems in ring domains for elliptic linear complex partial differential equations having polyharmonic operators as main parts. First, we give higher order Green functions as fundamental solutions of the homogeneous problems using the iteration of harmonic Green functions for ring domains. Second, we introduce some classes of operators related to Dirichlet problems together with their basic properties. Next, we transform the original problems into equivalent singular integral equations. Finally, solvability of the problems is discussed by defining the adjoint problems and using Fredholm alternative.
Citation Count: 11
Dirichlet Problems for the Generalized n-Poisson Equation
(Birkhauser verlag Ag, 2010) Aksoy, U.; Celebi, A. O.; Mathematics
Polyharmonic hybrid Green functions, obtained by convoluting polyharmonic Green and Almansi Green functions, are taken as kernels to define a hierarchy of integral operators. They are used to investigate the solvability of some types of Dirichlet problems for linear complex partial differential equations with leading term as the polyharmonic operator.
Citation Count: 0
Dirichlet-type problems for n-Poisson equation in Clifford analysis
(Taylor & Francis Ltd, 2022) Aksoy, Umit; Celebi, A. Okay; Mathematics
Iterated Dirichlet problem, also called as Riquier or Navier problem, and polyharmonic Dirichlet problem are studied for n-Poisson equation in Clifford analysis using iterated polyharmonic Green function and polyharmonic Green-Almansi type function appropriate for the boundary conditions of the problems.
Citation Count: 37
F-contraction mappings on metric-like spaces in connection with integral equations on time scales
(Springer-verlag Italia Srl, 2020) Agarwal, Ravi P.; Aksoy, Umit; Karapinar, Erdal; Erhan, Inci M.; Mathematics
In this paper we investigate the existence and uniqueness of fixed points of certain (phi,F)-type contractions in the frame of metric-like spaces. As an application of the theorem we consider the existence and uniqueness of solutions of nonlinear Fredholm integral equations of the second kind on time scales. We also present a particular example which demonstrates our theoretical results.
Citation Count: 0
FIXED POINT THEOREMS FOR MAPPINGS WITH A CONTRACTIVE ITERATE AT A POINT IN MODULAR METRIC SPACES
(House Book Science-casa Cartii Stiinta, 2022) Karapinar, Erdal; Aksoy, Umit; Fulga, Andreea; Erhan, Inci M.; Mathematics
In this manuscript, we introduce two new types of contraction, namely, w-contraction and strong Sehgal w-contraction, in the framework of modular metric spaces. We indicate that under certain assumptions, such mappings possess a unique fixed point. For the sake of completeness, we consider examples and an application to matrix equations.
Citation Count: 23
Fixed Point Theorems in Complete Modular Metric Spaces and an Application to Anti-periodic Boundary Value Problems
(Univ Nis, Fac Sci Math, 2017) Aksoy, Umit; Karapinar, Erdal; Erhan, Inci M.; Mathematics
In this paper existence and uniqueness of fixed points for a general class of contractive and nonexpansive mappings on modular metric spaces is discussed. As an application of the theoretical results, the existence of a solution of anti-periodic boundary value problems for nonlinear first order differential equations of Caratheodory's type is considered in the framework of modular metric spaces.
Citation Count: 67
FIXED POINTS OF GENERALIZED α-ADMISSIBLE CONTRACTIONS ON b-METRIC SPACES WITH AN APPLICATION TO BOUNDARY VALUE PROBLEMS
(Yokohama Publ, 2016) Aksoy, Umit; Karapinar, Erdal; Erhan, Inci M.; Mathematics
A general class of alpha-admissible contractions defined via (b)-comparison functions on b-metric spaces is discussed. Existence and uniqueness of the fixed point for this class of contractions is studied. Some consequences are presented. The results are employed in the discussion of existence and uniqueness of solutions of first order boundary value problems for ordinary differential equations.
Citation Count: 23
Meir-Keeler Type Contractions on Modular Metric Spaces
(Univ Nis, Fac Sci Math, 2018) Aksoy, Umit; Karapinar, Erdal; Erhan, Inci M.; Rakocevic, Vladimir; Mathematics
In this paper we introduce contraction mappings of Meir-Keeler types on modular metric spaces and investigate the existence and uniqueness of their fixed points. We give an example which demonstrates our theoretical results.
Citation Count: 7
Norm estimates of a class of Calderon-Zygmund type strongly singular integral operators
(Taylor & Francis Ltd, 2007) Aksoy, Ue; Celebi, A. O.; Mathematics
In this article, we prove the L-p boundedness of a class of Calderon - Zygmund type strongly singular operators. In particular, we give an estimate for the L-2 norm of these operators in the unit disc of the complex plane.
Citation Count: 1
A Normal Distribution on Time Scales with Application
(Univ Nis, Fac Sci Math, 2022) Aksoy, Umit; Cuchta, Tom; Georgiev, Svetlin; Okur, Yeliz Yolcu; Mathematics
We introduce a new normal distribution on time scales. Based on this generalized normal distribution, a Brownian motion is introduced and its quadratic variation is derived.
Citation Count: 0
On a boundary value problem for a class of second-order complex partial differential equations
(Univ Simon Bolivar, 2023) Aksoy, Umit; Celebi, Ahmet Okay; Mathematics
In this article, a boundary value problem for a second-order complex partial differential equation whose main part is the Laplacian, is introduced and its solvability is discussed by reduction of the problem into the Schwarz problem for a first-order equation. The condition for solvability is presented and an estimate for the unique solution is provided.
Citation Count: 1
On the fixed points of iterative contractive mappings defined via implicit relation
(Taylor & Francis Ltd, 2021) Aksoy, Umit; Erhan, Inci M.; Fulga, Andreea; Karapinar, Erdal; Mathematics
In this paper, we consider an implicit relation to generalize iterative fixed point results in the literature in the context of metric spaces. We conclude that several existing results are immediate consequences of our main results.
Citation Count: 1
On the methods of pricing American options: case study
(Springer, 2018) Aydogan, Burcu; Aksoy, Umit; Ugur, Omur; Mathematics
In this study, a comparative analysis of numerical and approximation methods for pricing American options is performed. Binomial and finite difference approximations are discussed; furthermore, Roll-Geske-Whaley, Barone-Adesi and Whaley and Bjerksund-Stensland analytical approximations as well as the least-squares Monte Carlo method of Longstaff and Schwartz are presented. Applicability and efficiency in almost all circumstances, numerical solutions of the corresponding free boundary problem is emphasized. Methods used in pricing American options are also compared on dividend and non-dividend paying assets; and their pros and cons are discussed along with numerical experiments.
Citation Count: 154
On the solution of a boundary value problem associated with a fractional differential equation
(Wiley, 2020) Sevinik Adiguzel, Rezan; Aksoy, Umit; Karapinar, Erdal; Erhan, Inci M.; Mathematics
The problem of the existence and uniqueness of solutions of boundary value problems (BVPs) for a nonlinear fractional differential equation of order 2
Citation Count: 123
ON THE SOLUTIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS VIA GERAGHTY TYPE HYBRID CONTRACTIONS
(Ministry Communications & High Technologies Republic Azerbaijan, 2021) Adiguzel, Rezan Sevinik; Aksoy, Umit; Karapinar, Erdal; Erhan, Inci M.; Mathematics
The aim of this article is twofold. Firstly, to study fixed points of mappings on b metric spaces satisfying a general contractive condition called Geraghty type hybrid contraction. Secondly, to apply the theoretical results to the problem of existence and uniqueness of solutions of boundary value problems with integral boundary conditions associated with a certain type of nonlinear fractional differential equations. The conditions for the existence of fixed points for Geraghty type hybrid contractions are determined and several consequences of the main results are deduced. Some examples on boundary value problems for nonlinear fractional differential equations of order 3 < alpha <= 4 are provided, where the existence and uniqueness of solutions are shown by using Geraghty type contractions.
Citation Count: 0
Optimal Limit Order Book Trading Strategies with Stochastic Volatility in the Underlying Asset
(Springer, 2023) Aydogan, Burcu; Ugur, Omur; Aksoy, Umit; Mathematics
In quantitative finance, there have been numerous new aspects and developments related with the stochastic control and optimization problems which handle the controlled variables of performing the behavior of a dynamical system to achieve certain objectives. In this paper, we address the optimal trading strategies via price impact models using Heston stochastic volatility framework including jump processes either in price or in volatility of the price dynamics with the aim of maximizing expected return of the trader by controlling the inventories. Two types of utility functions are considered: quadratic and exponential. In both cases, the remaining inventories of the market maker are charged with a liquidation cost. In order to achieve the optimal quotes, we control the inventory risk and follow the influence of each parameter in the model to the best bid and ask prices. We show that the risk metrics including profit and loss distribution (PnL), standard deviation and Sharpe ratio play important roles for the trader to make decisions on the strategies. We apply finite differences and linear interpolation as well as extrapolation techniques to obtain a solution of the nonlinear Hamilton-Jacobi-Bellman (HJB) equation. Moreover, we consider different cases on the modeling to carry out the numerical simulations.
Citation Count: 3
Polyharmonic Robin problem for complex linear partial differential equations
(Taylor & Francis Ltd, 2014) Aksoy, U.; Celebi, A. O.; Mathematics
In this article, generalized polyharmonic Robin functions are introduced together with some of their properties. A hierarchy of integral operators with relevant kernel functions are investigated. These operators are used to transform the Robin problem for a 2nth order linear partial complex differential equation with polyharmonic leading term (generalized n-Poisson equation) into a singular integral equation having Fredholm property.
Citation Count: 0
Polynomial logistic distribution associated with a cubic polynomial
(Taylor & Francis inc, 2017) Aksoy, Umit; Ostrovska, Sofiya; Ozban, Ahmet Yasar; Mathematics
Let P(x) be a polynomial monotone increasing on ( - , +). The probability distribution possessing the distribution function is called the polynomial logistic distribution with associated polynomial P. This has recently been introduced by Koutras etal., who have also demonstrated its importance for modeling financial data. In this article, the properties of the polynomial logistic distribution with an associated polynomial of degree 3 have been investigated in detail. An example of polynomial logistic distribution describing daily exchange rate fluctuations for the US dollar versus the Turkish lira is provided.

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