Browsing by Author "BAYRAKTAR, BAHTİYAR"

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About an algorithm of function approximation by the linear splines
(Turkic World Mathematical Soc, 2016-01-01) Kudaev, V.; Bayraktar, B.; BAYRAKTAR, BAHTİYAR; Bursa Uludağ Üniversitesi/Eğitim Fakültesi.; 0000-0001-7594-8291; ABI-7823-2020
The actual application for the problem of best approximation of grid functionby linear splines was formulated. A mathematical model and a method for its solution were developed. Complexity of the problem was that it was multi - extremal and could not be solved analytically. The method was developed in order to solve the problem of dynamic programming scheme, which was extended by us. Given the application of the method to the problem of flow control in the pressure-regulating systems, the pipeline network for transport of substances (pipelines of water, oil, gas, and etc.) that minimizes the amount of substance reservoirs and reduces the discharge of substance from the system. The method and the algorithm developed here may be used in computational mathematics, optimal control and regulation system, and regressive analysis.
Generalization of hadamard-type trapezoid inequalities for fractional integral operators
(Inst Mathematics Computer Center Russia, 2021-03-01) Bayraktar, Bahtiyar; Özdemir, Muhamet Emin; BAYRAKTAR, BAHTİYAR; ÖZDEMİR, MUHAMET EMİN; 0000-0001-7594-8291; ABI-7823-2020; AAH-1091-2021
The role of convexity theory in applied problems, especially in optimization problems, is well known. The integral Hermite-Hadamard inequality has a special place in this theory since it provides an upper bound for the mean value of a function. In solving applied problems from different fields of science and technology, along with the classical integro-differential calculus, fractional calculus plays an important role. A lot of research is devoted to obtaining an upper bound in the Hermite-Hadamard inequality using operators of fractional calculus.The article formulates and proves the identity with the participation of the fractional integration operator. Based on this identity, new generalized Hadamard-type integral inequalities are obtained for functions for which the second derivatives are convex and take values at intermediate points of the integration interval. These results are obtained using the convexity property of a function and two classical integral inequalities, the Hermite-Hadamard integral inequality and its other form, the power mean inequality. It is shown that the upper limit of the absolute error of inequality decreases in approximately n(2) times, where.. is the number of intermediate points. In a particular case, the obtained estimates are consistent with known estimates in the literature. The results obtained in the article can be used in further researches in the integro-differential fractional calculus.
Grid function approximation by linear splines with minimum deviation
(Scibulcom, 2015-01-01) Bayraktar, B.; Kudaev, V.; BAYRAKTAR, BAHTİYAR; Bursa Uludağ Üniversitesi/Eğitim Fakültesi/Matematik ve Fen Bilimleri Eğitimi Bölümü.; 0000-0001-7594-8291; ABI-7823-2020
The actual application for the problem of best approximation of grid function by linear splines was formulated. A mathematical model and method of its solution were developed. Complexity of the problem was that it was multi-extremal and can not be solved analytically. This fact assumed the need to develop an efficient algorithm for solving the problem.The method was developed for solving the problem of dynamic programming scheme, which was extended by us. In some studies, a similar problem was solved locally, and their solution did not include a large number of segments. However, in this paper, the problem was solved globally with defect delta.Given the application of the method to the problem of flow control in the pressure regulating systems, the pipeline network for transport of substances (pipelines of oil, gas, water, etc.) minimises the amount of substance in reservoirs and reduces the discharge of substance from the system. The method and algorithm developed may be used in computational mathematics, optimum control and regulation system, and regressive analysis.
Integral inequalities for mappings whose derivatives are (h, m, s)-convex modified of second type via katugampola integrals
(Univ Craiova, 2022-12-01) Bayraktar, Bahtiyar; Valdas, June E. Naepoles; BAYRAKTAR, BAHTİYAR; Bursa Uludağ Üniversitesi/Eğitim Fakültesi; 0000-0001-7594-8291; ABI-7823-2020
In this paper, using the definition of functions (h, m, s)-convex modified of sec-ond type, various extensions of the classic Hermite-Hadamard Inequality are obtained using Katugampola integrals. In addition, we show that several results known are particular cases of ours.
New extensions of hermite-hadamard inequality using k-fractional caputo derivatives
(Tbilisi Centre Math Sci, 2023-06-01) Napoles, Juan E.; Bayraktar, Bahtiyar; BAYRAKTAR, BAHTİYAR; Bursa Uludağ Üniversitesi/Eğitim Fakültesi; 0000-0003-2470-1090; 0000-0001-7594-8291; ABI-7823-2020
By means of Caputo k-fractional derivatives, in this work, we obtain new extensions of the Hermite-Hadamard inequality for modified (h, m)- convex functions of the second type. At work, we show that some known results from the literature can be obtained as particular cases of the results presented here.
New hadamard-type inequalities via (S, M1, M2)-convex functions
(Udmurt State Univ, 2021-01-01) Butt, S., I; Shaokat, Sh; Napoles Valdes, J. E.; Bayraktar, Bahtiyar; BAYRAKTAR, BAHTİYAR; Bursa Uludağ Üniversitesi/Eğitim Fakültesi.; 0000-0001-7594-8291; ABI-7823-2020
The article introduces a new concept of convexity of a function: (s, m(1), m(2)) convex functions. This class of functions combines a number of convexity types found in the literature. Some properties of (s, m(1), m(2))-convexities are established and simple examples of functions belonging to this class are given. On the basis of the proved identity, new integral inequalities of the Hadamard type are obtained in terms of the fractional integral operator. It is shown that these results give us, in particular, generalizations of a number of results available in the literature.
New integral inequalities of hermite-hadamard type in a generalized context
(Univ Punjab, Dept Mathematics, 2021-01-01) Napoles Valdes, Juan Eduardo; Bayraktar, Bahtiyar; Butt, Saad İhsan; BAYRAKTAR, BAHTİYAR; Bursa Uludağ Üniversitesi/Eğitim Bilimleri Bölümü; 0000-0001-7594-8291; ABI-7823-2020
In this paper, we obtained new integral inequalities of the Hermite-Hadamard type for convex and quasi-convex functions in a gen-eralized context.
New jensen-type integral inequalities via modified (h, m)-convexity and their applications
(Comenius Univ, 2023-01-01) Korus, P.; Valdes, J. E. Napoles; Bayraktar, Bahtiyar; BAYRAKTAR, BAHTİYAR; Bursa Uludağ Üniversitesi/Eğitim Fakültesi/Matematik ve Fen Bilimleri Eğitimi Bölümü.; 0000-0001-7594-8291; 0000-0001-8540-6293; ABI-7823-2020
This article presents new developments and applications of Jensen's inequality. We explore various variations of Jensen's inequality related to modified (h, m)-convex functions. The obtained results extend the applicability of the inequality to a broader class of functions and contexts. In addition to the fact that the results presented in the article provide additional results available in the literature, we also give examples of their application.
On generalizations of integral inequalities
(Petrozavodsk State Univ, 2022-01-01) Napoles, J.; Rabossi, F.; Bayraktar, Bahtiyar; BAYRAKTAR, BAHTİYAR; Bursa Uludağ Üniversitesi/Eğitim Fakültesi.; 0000-0001-7594-8291; ABI-7823-2020
In the present study, several new generalized integral inequalities of the Hadamard and Simpson-type are obtained. The results were obtained for functions whose first and third derivatives are either convex or satisfy the Lipschitz condition or the conditions of the Lagrange theorem. In a particular case, these results not only confirm but also improve some upper bounds, well known in the literature for the Simpson and Hermite-Hadamard-type inequali-ties.
On some integral inequalities for (s,m) -convex functions
(Türk Dünyası Matematik Topluluğu, 2020-01-01) Bayraktar, Bahtiyar; Gürbüz, Mustafa; BAYRAKTAR, BAHTİYAR; Bursa Uludağ Üniversitesi/Eğitim Fakültesi; 0000-0001-7594-8291; ABI-7823-2020
A new identity has been handled in this paper. It allows to derive new inequalities referring to upper estimation of the Jensen functional in the class of (s, m)-convex functions. Also some applications for special means are given by using new inequalities.
Problem of constructing a step function fluctuating least around a given function
(Inst Applied Mathematics, 2013-01-01) Bayraktar, Bahtiyar; BAYRAKTAR, BAHTİYAR; Uludağ Üniversitesi/Matematik Eğitimi Bölümü; 0000-0001-7594-8291; ABI-7823-2020
It is well known that step functions are often used in regulation and optimal control. Nevertheless, to our knowledge, this problem has not been raised previously. Therefore, the problem of constructing a step function, least fluctuating around a given function (in the sense of integration), is the main point of this article. This is a multiextremal task. In particular, such problems mostly arise in the design.In this paper formulated a mathematical model of the problem. To solve the problem developed an effective method of shrinking neighborhoods. Moreover, for algorithm of the method was developed software.
Several new integral inequalities via k- riemann- liouville fractional integrals operators
(Petrozavodsk State, 2021-01-01) Butt, S., I; Umar, M.; Bayraktar, B.; BAYRAKTAR, BAHTİYAR; Bursa Uludağ Üniversitesi/Eğitim Fakültesi.; 0000-0001-7594-8291; ABI-7823-2020
The main objective of this paper is to establish several new integral inequalities including k-Riemann - Liouville fractional integrals for convex, s-Godunova - Levin convex functions, quasi-convex, eta-quasi-convex. In order to obtain our results, we have used classical inequalities as Holder inequality, Power mean inequality and Weighted Holder inequality. We also give some applications.
Some generalized hadamard-type inequalities via fractional integrals
(Pleiades Publishing Inc, 2021-02-01) Bayraktar, Bahtiyar; Attaev, Anatoly. Kh; Kudaev, V. Ch; BAYRAKTAR, BAHTİYAR; Bursa Uludağ Üniversitesi/Eğitim Fakültesi/Matematik ve Fen Bilimleri; 0000-0001-7594-8291; ABI-7823-2020
In this paper, we establish some generalized inequalities of the Hermite-Hadamard type using fractional Riemann-Liouville integrals for the class of s-convex functions in the first and second sense. We assume that second derivatives of these functions are convex and take on values at intermediate points of the interval under consideration. We prove that this approach reduces the absolute error of Hadamard-type inequalities by a multiple of the number of intermediate points. In a particular case, the obtained upper bounds for the Hadamard inequality coincide with those given in the literature.
Some integral inequalities of hermite-hadamard type for differentiable (s,m)-convex functions via fractional integrals
(Türk Dünyası Matematik Topluluğu, 2020-01-01) Bayraktar, Bahtiyar; BAYRAKTAR, BAHTİYAR; Bursa Uludağ Üniversitesi/Eğitim Fakültesi/Matematik ve Fen Bilimleri Eğitimi; 0000-0001-7594-8291; ABI-7823-2020
In this paper, we present new inequalities connected with fractional integrals for twice differentiable functions derivatives which are (s, m) convex functions. To obtain this, integral inequalities were used classical inequalities as Holder inequalitiy and power mean inequality.This results are related to the well-known integral inequality of the Hermite-Hadamard type. Also some applications to special means are provided.
Some new bullen-type inequalities obtained via fractional integral operators
(MDPI, 2023-07-01) Fahad, Asfand; Butt, Saad Ihsaan; Bayraktar, Bahtiyar; Anwar, Mehran; Wang, Yuanheng; BAYRAKTAR, BAHTİYAR; Uludağ Üniversitesi/Matematik ve Fen Bilimleri Eğitimi Bölümü; 0000-0001-7594-8291; ABI-7823-2020
In this paper, we establish a new auxiliary identity of the Bullen type for twice-differentiable functions in terms of fractional integral operators. Based on this new identity, some generalized Bullen-type inequalities are obtained by employing convexity properties. Concrete examples are given to illustrate the results, and the correctness is confirmed by graphical analysis. An analysis is provided on the estimations of bounds. According to calculations, improved Holder and power mean inequalities give better upper-bound results than classical inequalities. Lastly, some applications to quadrature rules, modified Bessel functions and digamma functions are provided as well.
Some new integral inequalities for (s, m)-convex and (α, m)-convex functions
(Karaganda State Univ, 2019-01-01) Bayraktar, B.; Kudaev, V. Ch.; BAYRAKTAR, BAHTİYAR; Uludağ Üniversitesi/Matematik ve Fen Bilimleri Eğitimi Bölümü; 0000-0001-7594-8291; ABI-7823-2020
The paper considers several new integral inequalities for functions the second derivatives of which, withrespect to the absolute value, are (s, m)-convex and (alpha, m)-convex functions. These results are relatedto well-known Hermite-Hadamard type integral inequality, Simpson type integral inequality, and Jensentype inequality. In other words, new upper bounds for these inequalities using the indicated classes ofconvex functions have been obtained. These estimates are obtained using a direct definition for a convexfunction, classical integral inequalities of Holder and power mean types. Along with the new outcomes, thepaper presents results confirming the existing in literature upper bound estimates for integral inequalities(in particular well known in literature results obtained by U. Kirmaci in [7] and M.Z. Sarikaya and N. Aktanin [35]). The last section presents some applications of the obtained estimates for special computing facilities(arithmetic, logarithmic, generalized logarithmic average and harmonic average for various quantities)
Some new integral inequalities for (s,m)-convex and (α,m)-convex functions
(Karaganda State Univ, 2019-01-01) Bayraktar, Bahtiyar; Kudaev, V. Ch.; BAYRAKTAR, BAHTİYAR; Bursa Uludağ Üniversitesi/Matematik ve Fen Bilimleri Eğitimi Bölümü; 0000-0001-7594-8291; ABI-7823-2020
The paper considers several new integral inequalities for functions the second derivatives of which, withrespect to the absolute value, are (s, m)-convex and (alpha, m)-convex functions. These results are relatedto well-known Hermite-Hadamard type integral inequality, Simpson type integral inequality, and Jensentype inequality. In other words, new upper bounds for these inequalities using the indicated classes ofconvex functions have been obtained. These estimates are obtained using a direct definition for a convexfunction, classical integral inequalities of Holder and power mean types. Along with the new outcomes, thepaper presents results confirming the existing in literature upper bound estimates for integral inequalities(in particular well known in literature results obtained by U. Kirmaci in [7] and M.Z. Sarikaya and N. Aktanin [35]). The last section presents some applications of the obtained estimates for special computing facilities(arithmetic, logarithmic, generalized logarithmic average and harmonic average for various quantities)
Some new jensen-mercer type integral inequalities via fractional operators
(Mdpi, 2023-06-01) Korus, Peter; Valdes, Juan Eduardo Napoles; Bayraktar, Bahtiyar; BAYRAKTAR, BAHTİYAR; Bursa Uludağ Üniversitesi/Eğitim Fakültesi.; 0000-0001-7594-8291; ABI-7823-2020
In this study, we present new variants of the Hermite-Hadamard inequality via non-conformable fractional integrals. These inequalities are proven for convex functions and differentiable functions whose derivatives in absolute value are generally convex. Our main results are established using the classical Jensen-Mercer inequality and its variants for (h,m)-convex modified functions proven in this paper. In addition to showing that our results support previously known results from the literature, we provide examples of their application.
Some refinements of the hermite-hadamard inequality with the help of weighted integrals
(Springer, 2023-11-11) Bayraktar, Bahtiyar; Napoles, J. E.; Rabossi, F.; BAYRAKTAR, BAHTİYAR; Bursa Uludağ Üniversitesi/Eğitim Fakültesi; 0000-0001-7594-8291; ABI-7823-2020
By using the definition of modified (h, m, s)-convex functions of the second type, we present various refinements of the classical Hermite-Hadamard inequality obtained within the framework of weighted integrals. Throughout the paper, we show that various known results available from the literature can be obtained as particular cases of our results.
Weighted hermite-hadamard integral inequalities for general convex functions
(Amer Inst Mathematical Sciences-aims, 2023-01-01) Korus, Peter; Valdes, Juan Eduardo Napoles; Bayraktar, Bahtiyar; BAYRAKTAR, BAHTİYAR; Bursa Uludağ Üniversitesi/Eğitim Fakültesi.; 0000-0001-7594-8291; ABI-7823-2020
In this article, starting with an equation for weighted integrals, we obtained several extensions of the well-known Hermite-Hadamard inequality. We used generalized weighted integral operators, which contain the Riemann-Liouville and the k-Riemann-Liouville fractional integral operators. The functions for which the operators were considered satisfy various conditions such as the h-convexity, modified h-convexity and s-convexity.