Browsing by Author "Aksoy, Ümit"

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A HIERARCHY OF SINGULAR INTEGRAL OPERATORS FOR MIXED BOUNDARY VALUE PROBLEMS
(2011) Aksoy, Ümit; Çelebi, Okay; Mathematics
A class of integral operators having a hierarchy of polyharmonic kernels is introduced and some properties are derived. Iterated mixed boundary value problems for complex model equations and linear elliptic complex partial differential equations are discussed in the unit disc of the complex plane.
A Survey on Boundary Value Problems for Complex Partial Differential Equations
(Advances in Dynamical Systems and Applications, 2010) Aksoy, Ümit; Çelebi, A. Okay; Mathematics
In 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.
Amerikan opsiyonlarının hesaplamalı yöntemlerle fiyatlandırılması
(2014) Aydoğan, Burcu; Aksoy, Ümit; Uğur, Ömür; Mathematics
Finansal matematikte, opsiyon fiyatlama finansal teori ve matematiksel olarak düşünüldüğünde, çok popüler bir problemdir. Opsiyon fiyatlama teorisinde, Amerikan opsiyonlarının fiyatlandırılması en önemli problemlerden biridir. Amerikan opsiyonları, finansal piyasalarda en çok işlem gören opsiyon türüdür. Son zamanlardaki birçok gelişmeye rağmen, Amerikan opsiyon fiyatlandırması hala en zor problemlerden biri olarak kalmaya devam etmektedir. Amerikan opsiyonlarının kapalı analitik çözümleri yoktur, bu sebeple bu problemle uğraşmanın en yaygın yollarından biri sayısal ve yaklaşım teknikleri geliştirmektir. Bu tezde, Amerikan opsiyonlarını fiyatlandırmak için hesaplamalı metotlardan; binom, sonlu fark ve yaklaşım metotları analiz edilmiştir. İlk olarak, uygulaması çok kolay olan ve varlık fiyatlarının geometrik Brownian hareketinden geldiğini varsayan binom yaklaşımı ele alınmıştır. Daha sonra, Amerikan opsiyonları için Black-Scholes kısmi diferansiyel denklemine dayanarak serbest sınır değer problemi verilmiştir. Bu problemi çözmek için PSOR metodu kullanılmıştır. Amerikan opsiyonlarının kapalı çözümleri olmamasına rağmen, opsiyonun değerine çok yaklaşan bazı analitik yaklaşım metotları üzerinde çalışılmıştır. Her bir metodun uygulamaları yapılmıştır ve çözümler karşılaştırılmıştır.
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: 0
Dirichlet Problem for Inhomogeneous Biharmonic Equation in Clifford Analysis
(Springer Science and Business Media Deutschland GmbH, 2022) Aksoy,Ü.; Çelebi,A.O.; Mathematics
An integral representation formula in terms of the bi-Laplacian operator is obtained and Dirichlet problem for the bi-Poisson equation is solved in Clifford analysis. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.
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: 15
Dirichlet problems for the generalized n-poisson equation
(Springer International Publishing, 2010) Aksoy,Ü.; Çelebi,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. © 2009 Birkhäuser Verlag Basel/Switzerland.
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.
Doğrusal olmayan Black-Scholes denklemi için üstel sonlu fark yöntemi
(2017) Omar, Fathıa; Aksoy, Ümit; Aydın, Ayhan; Mathematics
Bu tezde, likit olmayan bir piyasada ortaya çıkan doğrusal olmayan Black-Scholes denklemi için üstel sonlu fark yöntemi çalışılmıştır. 1. Bölüm opsiyon fiyatlandırması problemi terminolojisi, temel tanımlar ve literatür taramasına ayrılmıştır. 2. Bölümde Black-Scholes modeli ve Black-Scholes denklemi için sonlu fark yöntemleri gözden geçirilmiştir. 3. Bölümde doğrusal olmayan Black-Scholes denklemi için açık sonlu fark yöntemi, monotonluk, kararlılık ve tutarlılık sonuçları ile birlikte çalışılmıştır. 4. Bölümde doğrusal ve doğrusal olmayan Black-Scholes denklemleri için üstel sonlu fark yöntemi uygulanmıştır. Ayrıca, yöntemin tutarlılığı ve yakınsaklığı araştırılmıştır. Teorik sonuçları doğrulamak için sayısal örnekler verilmiştir. Sayısal sonuçlar, üstel sonlu fark yönteminin açık sonlu fark yönteminden daha iyi performans sergilediğini göstermiştir. 5. Bölüm sonuç kısmına ayrılmıştır.
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: 69
Fixed points of generalized α-admissible contractions on b-metric spaces with an application to boundary value problems
(Yokohama Publications, 2016) Aksoy,Ü.; Karapinar,E.; Erhan,I.M.; Mathematics
A general class of α-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. © 2016.
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: 5
Mixed boundary value problems for higher-order complex partial differential equations
(2010) Aksoy,U.; Okay Ĉedil,; elebi,A.; Mathematics
In this paper, we introduce the operators related to mixed boundary value problems for general linear elliptic partial complex differential equations in the unit disc of the complex plane. The solvability of the relevant boundary value problems will be studied by transforming them into singular integral equations. © 2010, by Oldenbourg Wissenschaftsverlag, München, Germany. All rights reserved.
Neumann problem for generalized n-Poisson equation
(Journal of Mathematical Analysis and Applications, 2009) Aksoy, Ümit; Çelebi, Okay; Mathematics
Using a hierarchy of integral operators having higher-order Neumann functions and their derivatives as kernels, the Neumann problem for a 2nth order linear partial complex differential equation is discussed. The solvability of the problem is obtained.
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.