Matematik Bölümü Yayın Koleksiyonu

Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/413

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Citation - WoS: 4
Citation - Scopus: 3
2D gravity and the Hamilton-Jacobi formalism
(Soc Italiana Fisica, 2002) Baleanu, D; Baleanu, Dumitru; Güler, Y; 56389; Matematik
Hamilton-Jacobi formalism is used to study 2D gravity and its SL(2, R) hidden symmetry. If the contribution of the surface term is considered, the obtained results coincide with those given by the Dirac and Faddeev-Jackiw approaches.
Citation - WoS: 0
Citation - Scopus: 0
A 6-point subdivision scheme and its applications for the solution of 2nd order nonlinear singularly perturbed boundary value problems
(Amer inst Mathematical Sciences-aims, 2020) Mustafa, Ghulam; Baleanu, Dumitru; Baleanu, Dumitru; Ejaz, Syeda Tehmina; Anju, Kaweeta; Ahmadian, Ali; Salahshour, Soheil; Ferrara, Massimiliano; 56389; Matematik
In this paper, we first present a 6-point binary interpolating subdivision scheme (BISS) which produces a C-2 continuous curve and 4th order of approximation. Then as an application of the scheme, we develop an iterative algorithm for the solution of 2nd order nonlinear singularly perturbed boundary value problems (NSPBVP). The convergence of an iterative algorithm has also been presented. The 2nd order NSPBVP arising from combustion, chemical reactor theory, nuclear engineering, control theory, elasticity, and fluid mechanics can be solved by an iterative algorithm with 4th order of approximation.
Citation - WoS: 7
Citation - Scopus: 5
A Brief Overview and Survey of the Scientific Work by Feng Qi
(Mdpi, 2022) Agarwal, Ravi Prakash; Karapinar, Erdal; Kostic, Marko; Cao, Jian; Du, Wei-Shih; 19184
In the paper, the authors present a brief overview and survey of the scientific work by Chinese mathematician Feng Qi and his coauthors.
Citation - WoS: 6
Citation - Scopus: 6
A caputo fractional order boundary value problem with integral boundary conditions
(Eudoxus Press, Llc, 2013) Babakhani, Azizollah; Abdeljawad, Thabet; Abdeljawad, Thabet; Matematik
In this paper, we discuss existence and uniqueness of solutions to nonlinear fractional order ordinary differential equations with integral boundary conditions in an ordered Banach space. We use the Caputo fractional differential operator and the nonlinearity depends on the fractional derivative of an unknown function. The nonlinear alternative of the Leray- Schauder type Theorem is the main tool used here to establish the existence and the Banach contraction principle to show the uniqueness of the solution under certain conditions. The compactness of solutions set is also investigated and an example is included to show the applicability of our results.
Citation - Scopus: 23
A Caputo-Fabrizio Fractional-Order Cholera Model And İts Sensitivity Analysis
(Mehmet Yavuz, 2023) Ahmed, I.; Jarad, Fahd; Akgül, A.; Jarad, F.; Kumam, P.; Nonlaopon, K.; 234808; Matematik
In recent years, the availability of advanced computational techniques has led to a growing emphasis on fractional-order derivatives. This development has enabled researchers to explore the intricate dynamics of various biological models by employing fractional-order derivatives instead of traditional integer-order derivatives. This paper proposes a Caputo-Fabrizio fractional-order cholera epidemic model. Fixed-point theorems are utilized to investigate the existence and uniqueness of solutions. A recent and effective numerical scheme is employed to demonstrate the model’s complex behaviors and highlight the advantages of fractional-order derivatives. Additionally, a sensitivity analysis is conducted to identify the most influential parameters. © 2023 by the authors.
Citation - WoS: 165
A central difference numerical scheme for fractional optimal control problems
(Sage Publications Ltd, 2009) Baleanu, Dumitru; Baleanu, Dumitru; Defterli, Ozlem; Defterli, Özlem; Agrawal, Om P.; 56389; 31401; Matematik
This paper presents a modified numerical scheme for a class of fractional optimal control problems where a fractional derivative (FD) is defined in the Riemann-Liouville sense. In this scheme, the entire time domain is divided into several sub-domains, and a FD at a time node point is approximated using a modified Grunwald-Letnikov approach. For the first-order derivative, the proposed modified Grunwald-Letnikov definition leads to a central difference scheme. When the approximations are substituted into the fractional optimal control equations, it leads to a set of algebraic equations which are solved using a direct numerical technique. Two examples, one time-invariant and the other time-variant, are considered to study the performance of the numerical scheme. Results show that 1) as the order of the derivative approaches an integer value, these formulations lead to solutions for the integer-order system, and 2) as the sizes of the sub-domains are reduced, the solutions converge. It is hoped that the present scheme would lead to stable numerical methods for fractional differential equations and optimal control problems.
Citation - WoS: 72
Citation - Scopus: 99
A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel
(Springer, 2018) Baleanu, D.; Baleanu, Dumitru; Shiri, B.; Srivastava, H. M.; Al Qurashi, M.; 56389; Matematik
In this paper, we solve a system of fractional differential equations within a fractional derivative involving the Mittag-Leffler kernel by using the spectral methods. We apply the Chebyshev polynomials as a base and obtain the necessary operational matrix of fractional integral using the Clenshaw-Curtis formula. By applying the operational matrix, we obtain a system of linear algebraic equations. The approximate solution is computed by solving this system. The regularity of the solution investigated and a convergence analysis is provided. Numerical examples are provided to show the effectiveness and efficiency of the method.
Citation - WoS: 24
A chebyshev-laguerre-gauss-radau collocation scheme for solving a time fractional sub-diffusion equation on a semi-infinite domain
(Editura Acad Romane, 2015) Bhrawy, A. H.; Baleanu, Dumitru; Abdelkawy, M. A.; Alzahrani, A. A.; Baleanu, D.; Alzahrani, E. O.; Matematik
We propose a new efficient spectral collocation method for solving a time fractional sub-diffusion equation on a semi-infinite domain. The shifted Chebyshev-Gauss-Radau interpolation method is adapted for time discretization along with the Laguerre-Gauss-Radau collocation scheme that is used for space discretization on a semi-infinite domain. The main advantage of the proposed approach is that a spectral method is implemented for both time and space discretizations, which allows us to present a new efficient algorithm for solving time fractional sub-diffusion equations.
Citation - WoS: 9
Citation - Scopus: 9
A class of fractal Hilbert-type inequalities obtained via Cantor-type spherical coordinates
(Wiley, 2021) Baleanu, Dumitru; Baleanu, Dumitru; Krnic, Mario; Vukovic, Predrag; 56389; Matematik
We present a class of higher dimensional Hilbert-type inequalities on a fractal set (Double-struck capital R+alpha n)k. The crucial step in establishing our results are higher dimensional spherical coordinates on a fractal space. Further, we impose the corresponding conditions under which the constants appearing in the established Hilbert-type inequalities are the best possible. As an application, our results are compared with the previous results known from the literature.
Citation - WoS: 4
Citation - Scopus: 4
A Class of Refinement Schemes With Two Shape Control Parameters
(Ieee-inst Electrical Electronics Engineers inc, 2020) Mustafa, Ghulam; Baleanu, Dumitru; Hameed, Rabia; Baleanu, Dumitru; Mahmood, Ayesha; 56389; Matematik
A subdivision scheme defines a smooth curve or surface as the limit of a sequence of successive refinements of given polygon or mesh. These schemes take polygons or meshes as inputs and produce smooth curves or surfaces as outputs. In this paper, a class of combine refinement schemes with two shape control parameters is presented. These even and odd rules of these schemes have complexity three and four respectively. The even rule is designed to modify the vertices of the given polygon, whereas the odd rule is designed to insert a new point between every edge of the given polygon. These schemes can produce high order of continuous shapes than existing combine binary and ternary family of schemes. It has been observed that the schemes have interpolating and approximating behaviors depending on the values of parameters. These schemes have an interproximate behavior in the case of non-uniform setting of the parameters. These schemes can be considered as the generalized version of some of the interpolating and B-spline schemes. The theoretical as well as the numerical and graphical analysis of the shapes produced by these schemes are also presented.
Citation - Scopus: 3
A close look at Newton–Cotes integration rules
(Cankaya University, 2019) Sermutlu, E.; Sermutlu, Emre; 17647; Matematik
Newton–Cotes integration rules are the simplest methods in numerical integration. The main advantage of using these rules in quadrature software is ease of programming. In practice, only the lower orders are implemented or tested, because of the negative coefficients of higher orders. Most textbooks state it is not necessary to go beyond Boole’s 5-point rule. Explicit coefficients and error terms for higher orders are seldom given literature. Higher-order rules include negative coefficients therefore roundoff error increases while truncation error decreases as we increase the number of points. But is the optimal one really Simpson or Boole? In this paper, we list coefficients up to 19-points for both open and closed rules, derive the error terms using an elementary and intuitive method, and test the rules on a battery of functions to find the optimum all-round one. © 2019, Cankaya University. All rights reserved.
A common fixed point theorem of a Greguš type on convex cone metric spaces
(2011) Abdeljawad, Thabet; Karapinar, Erdal; 19184; Matematik
The result of Ćirić [1] on a common fixed point theorem of Greguš type on metric spaces is extended to the class of cone metric spaces. Namely, a common fixed point theorem is proved in s-convex cone metric spaces under the normality of the cone and another common fixed point theorem is proved in convex cone metric spaces under the assumption that the cone is strongly minihedral.
A common fixed point theorem of a Gregus type on convex cone metric spaces
(Eudoxus Press, 2011) Abdeljawad, Thabet; Karapınar, Erdal; 19184; Matematik
The result of Ciric [1] on a common fixed point theorem of Gregus-type on metric spaces is extended to the class of cone metric spaces. Namely, a common fixed point theorem is proved in s-convex cone metric spaces under the normality of the cone and another common fixed point theorem is proved in convex cone metric spaces under the assumption that the cone is strongly minihedral
Citation - WoS: 16
A comparative study of silicon nitride and SiAlON ceramics against E. coli
(Elsevier Sci Ltd, 2021) Akin, Seniz R. Kushan; Garcia, Caterina Bartomeu; Webster, Thomas J.; 224219
In recent decades, due to some limitations from alumina (Al2O3) and zirconia (ZrO2), silicon nitride (Si3N4) has been investigated as a novel bioceramic material, mainly in situations where a bone replacement is required. Si3N4 ceramics and its derivative form, SiAlON, possess advantages in orthopedics due to their mechanical properties and biologically acceptable chemistry, which accelerates bone repair. However, biological applications require additional properties, enabling stronger chemical bonding to the surrounding tissue for better fixation and the prevention of bacteria biofilm formation. Therefore, two commercial Si3N4 and SiAlON ceramics were investigated in this study and compared to each other according to their material properties (like wetting angles and surface chemistry) and their antibacterial behaviors using E. coli. Results provided evidence of a 15% reduction in E. coli colonization after just 24 h on Si3N4 compared to SiAlON which is impressive considering no antibiotics were used. Further, a mechanism of action is provided. In this manner, this study provides evidence that Si3N4 should be further studied for a wide range of antibacterial orthopedic, or other suitable biomaterial applications.
Citation - WoS: 29
Citation - Scopus: 30
A comparative study on biodegradation and mechanical properties of pressureless infiltrated Ti/Ti6Al4V-Mg composites
(Elsevier Science Bv, 2016) Esen, Ziya; Esen, Ziya; Butev, Ezgi; Karakaş, Mustafa Serdar; Karakas, M. Serdar; 52373; 47423; Ortak Dersler Bölümü; Malzeme Bilimi ve Mühendisliği
The mechanical response and biodegradation behavior of pressureless Mg-infiltrated Ti-Mg and Ti6Al4V-Mg composites were investigated by compression and simulated body fluid immersion tests, respectively. Prior porous preforms were surrounded uniformly with magnesium as a result of infiltration and the resultant composites were free of secondary phases and intermetallics. Although the composites' compressive strengths were superior compared to bone, both displayed elastic moduli similar to that of cortical bone and had higher ductility with respect to their starting porous forms. However, Ti-Mg composites were unable to preserve their mechanical stabilities during in-vitro tests such that they fractured in multiple locations within 15 days of immersion. The pressure generated by H-2 due to rapid corrosion of magnesium caused failure of the Ti-Mg composites through sintering necks. On the other hand, the galvanic effect seen in Ti6Al4V-Mg was less severe compared to that of Ti-Mg. The degradation rate of magnesium in Ti6Al4V-Mg was slower, and the composites were observed to be mechanically stable and preserved their integrities over the entire 25-day immersion test. Both composites showed bioinert and biodegradable characteristics during immersion tests and magnesium preferentially corroded leaving porosity behind while Ti/Ti6Al4V remained as a permanent scaffold. The porosity created by degradation of magnesium was refilled by new globular agglomerates. Mg(OH)(2) and CaHPO4 phases were encountered during immersion tests while MgCl2 was detected during only the first 5 days. Both composites were classified as bioactive since the precipitation of CaHPO4 phase is known to be precursor of hydroxyapatite formation, an essential requirement for an artificial material to bond to living bone. (C) 2016 Elsevier Ltd. All rights reserved.
Citation - WoS: 11
Citation - Scopus: 13
A Composition Formula of the Pathway Integral Transform Operator
(Aracne Editrice, 2014) Baleanu, Dumitru; Baleanu, Dumitru; Agarwal, Praveen; 56389; Matematik
In the present paper, we aim at presenting composition formula of integral transform operator due to Nair, which is expressed in terms of the generalized Wright hypergeometric function, by inserting the generalized Bessel function of the first kind w(v) z). Furthermore the special cases for the product of trigonometric functions are also consider.
Citation - WoS: 17
Citation - Scopus: 19
A comprehensive analysis of the stochastic fractal–fractional tuberculosis model via Mittag-Leffler kernel and white noise
(Elsevier, 2022) Rashid, Saima; Jarad, Fahd; Iqbal, Muhammad Kashif; Alshehri, Ahmed M.; Ashraf, Rehana; Jarad, Fahd; 234808; Matematik
In this research, we develop a stochastic framework for analysing tuberculosis (TB) evolution that includes new-born immunization via the fractal-fractional (F-F) derivative in the Atangana-Baleanu sense. The population is divided into four groups by this system: susceptibility S(xi), infectious I(xi), immunized infants V(xi), and restored R(xi). The stochastic technique is used to describe and assess the invariant region, basic reproduction number, and local stability for disease-free equilibrium. This strategy has significant modelling difficulties since it ignores the unpredictability of the system phenomena. To prevent such problems, we convert the deterministic strategy to a randomized one, which seems recognized to have a vital influence by adding an element of authenticity and fractional approach. Owing to the model intricacies, we established the existence-uniqueness of the model and the extinction of infection was carried out. We conducted a number of experimental tests using the F-F derivative approach and obtained some intriguing modelling findings in terms of (i) varying fractional-order (phi) and fixing fractal-dimension (omega), (ii) varying omega and fixing phi, and (iii) varying both phi and omega, indicating that a combination of such a scheme can enhance infant vaccination and adequate intervention of infectious patients can give a significant boost.
Citation - WoS: 11
Citation - Scopus: 11
A Computational Approach Based On The Fractional Euler Functions And Chebyshev Cardinal Functions For Distributed-Order Time Fractional 2D Diffusion Equation
(Elsevier, 2023) Heydari, M. H.; Hosseininia, M.; Baleanu, D.; 56389
In this paper, the distributed-order time fractional diffusion equation is introduced and studied. The Caputo fractional derivative is utilized to define this distributed-order fractional derivative. A hybrid approach based on the fractional Euler functions and 2D Chebyshev cardinal functions is proposed to derive a numerical solution for the problem under consideration. It should be noted that the Chebyshev cardinal functions process many useful properties, such as orthogonal-ity, cardinality and spectral accuracy. To construct the hybrid method, fractional derivative oper-ational matrix of the fractional Euler functions and partial derivatives operational matrices of the 2D Chebyshev cardinal functions are obtained. Using the obtained operational matrices and the Gauss-Legendre quadrature formula as well as the collocation approach, an algebraic system of equations is derived instead of the main problem that can be solved easily. The accuracy of the approach is tested numerically by solving three examples. The reported results confirm that the established hybrid scheme is highly accurate in providing acceptable results.(c) 2023 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/ 4.0/).
Citation - WoS: 10
Citation - Scopus: 9
A computational study of a stochastic fractal-fractional hepatitis B virus infection incorporating delayed immune reactions via the exponential decay
(Amer inst Mathematical Sciences-aims, 2022) Al Qurashi, Maysaa; Jarad, Fahd; Rashid, Saima; Jarad, Fahd; 234808; Matematik
Recently, researchers have become interested in modelling, monitoring, and treatment of hepatitis B virus infection. Understanding the various connections between pathogens, immune systems, and general liver function is crucial. In this study, we propose a higher-order stochastically modified delay differential model for the evolution of hepatitis B virus transmission involving defensive cells. Taking into account environmental stimuli and ambiguities, we presented numerical solutions of the fractal-fractional hepatitis B virus model based on the exponential decay kernel that reviewed the hepatitis B virus immune system involving cytotoxic T lymphocyte immunological mechanisms. Furthermore, qualitative aspects of the system are analyzed such as the existence-uniqueness of the non-negative solution, where the infection endures stochastically as a result of the solution evolving within the predetermined system's equilibrium state. In certain settings, infection-free can be determined, where the illness settles down tremendously with unit probability. To predict the viability of the fractal-fractional derivative outcomes, a novel numerical approach is used, resulting in several remarkable modelling results, including a change in fractional-order delta with constant fractal-dimension pi, delta with changing pi, and delta with changing both delta and pi. White noise concentration has a significant impact on how bacterial infections are treated.
A Computationally Efficient Method For a Class of Fractional Variational and Optimal Control Problems Using Fractional Gegenbauer Functions
(Editura Academiei Romane, 2018) Baleanu, Dumitru; Doha, Eid H.; Ezz-Eldien, Samer S.; Abdelkawy, M. A.; Hafez, R. M.; Amin, A. Z. M.; Baleanu, Dumitru; Zaky, M. A.; 56389; Matematik
This paper is devoted to investigate, from the numerical point of view, fractional-order Gegenbauer functions to solve fractional variational problems and fractional optimal control problems. We first introduce an orthonormal system of fractional-order Gegenbauer functions. Then, a formulation for the fractional-order Gegenbauer operational matrix of fractional integration is constructed. An error upper bound for the operational matrix of the fractional integration is also given. The properties of the fractional-order Gegenbauer functions are utilized to reduce the given optimization problems to systems of algebraic equations. Some numerical examples are included to demonstrate the efficiency and the accuracy of the proposed approach.

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