Fen - Edebiyat Fakültesi

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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: 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: 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 - 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: 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.
Citation - WoS: 57
Citation - Scopus: 62
A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets
(Sage Publications Ltd, 2019) Moradi, L.; Baleanu, Dumitru; Mohammadi, F.; Baleanu, D.; 56389; Matematik
The aim of the present study is to present a numerical algorithm for solving time-delay fractional optimal control problems (TDFOCPs). First, a new orthonormal wavelet basis, called Chelyshkov wavelet, is constructed from a class of orthonormal polynomials. These wavelet functions and their properties are implemented to derive some operational matrices. Then, the fractional derivative of the state function in the dynamic constraint of TDFOCPs is approximated by means of the Chelyshkov wavelets. The operational matrix of fractional integration together with the dynamical constraints is used to approximate the control function directly as a function of the state function. Finally, these approximations were put in the performance index and necessary conditions for optimality transform the under consideration TDFOCPs into an algebraic system. Moreover, some illustrative examples are considered and the obtained numerical results were compared with those previously published in the literature.
Citation - WoS: 9
Citation - Scopus: 10
A Discussion On the Existence of Best Proximity Points That Belong to the Zero Set
(Mdpi, 2020) Karapinar, Erdal; Karapınar, Erdal; Abbas, Mujahid; Farooq, Sadia; 19184; Matematik
In this paper, we investigate the existence of best proximity points that belong to the zero set for the alpha p -admissible weak (F,phi) -proximal contraction in the setting of M-metric spaces. For this purpose, we establish phi -best proximity point results for such mappings in the setting of a complete M-metric space. Some examples are also presented to support the concepts and results proved herein. Our results extend, improve and generalize several comparable results on the topic in the related literature.
Citation - WoS: 2
Citation - Scopus: 3
A Drone-Based Blood Donation Approach Using an Ant Colony Optimization Algorithm
(Tech Science Press, 2023) Abbas, Sana; Jarad, Fahd; Ashraf, Faraha; Jarad, Fahd; Sardar, Muhammad Shoaib; Siddique, Imran; 234808; Matematik
This article presents an optimized approach of mathematical techniques in the medical domain by manoeuvring the phenomenon of ant colony optimization algorithm (also known as ACO). A complete graph of blood banks and a path that covers all the blood banks without repeating any link is required by applying the Travelling Salesman Problem (often TSP). The wide use promises to accelerate and offers the opportunity to cultivate health care, particularly in remote or unmerited environments by shrinking lab testing reversal times, empowering just-in-time lifesaving medical supply.
Citation - WoS: 35
Citation - Scopus: 44
A fractional derivative with non-singular kernel for interval-valued functions under uncertainty
(Elsevier Gmbh, Urban & Fischer verlag, 2017) Salahshour, S.; Baleanu, Dumitru; Ahmadian, A.; Ismail, F.; Baleanu, D.; 56389; Matematik
The purpose of the current investigation is to generalize the concept of fractional derivative in the sense of Caputo Fabrizio derivative (CF-derivative) for interval-valued function under uncertainty. The reason to choose this new approach is originated from the non singularity property of the kernel that is critical to interpret the memory aftermath of the system, which was not precisely illustrated in the previous definitions. We study the properties of CF-derivative for interval-valued functions under generalized Hukuhara-differentiability. Then, the fractional differential equations under this notion are presented in details. We also study three real-world systems such as the falling body problem, Basset and Decay problem under interval-valued CF-differentiability. Our cases involve a demonstration that this new notion is accurately applicable for the mechanical and viscoelastic models based on the interval CF-derivative equations. (C) 2016 Elsevier GmbH. All rights reserved.
Citation - WoS: 95
Citation - Scopus: 108
A Fractional Model for the Dynamics of Tuberculosis Infection Using Caputo-Fabrizio Derivative
(Amer inst Mathematical Sciences-aims, 2020) Ullah, Saif; Baleanu, Dumitru; Khan, Muhammad Altaf; Farooq, Muhammad; Hammouch, Zakia; Baleanu, Dumitru; 56389; Matematik
In the present paper, we study the dynamics of tuberculosis model using fractional order derivative in Caputo-Fabrizio sense. The number of confirmed notified cases reported by national TB program Khyber Pakhtunkhwa, Pakistan, from the year 2002 to 2017 are used for our analysis and estimation of the model biological parameters. The threshold quantity R-0 and equilibria of the model are determined. We prove the existence of the solution via fixed-point theory and further examine the uniqueness of the model variables. An iterative solution of the model is computed using fractional Adams-Bashforth technique. Finally, the numerical results are presented by using the estimated values of model parameters to justify the significance of the arbitrary fractional order derivative. The graphical results show that the fractional model of TB in Caputo-Fabrizio sense gives useful information about the complexity of the model and one can get reliable information about the model at any integer or non-integer case.
Citation - WoS: 61
Citation - Scopus: 76
A fractional model of convective radial fins with temperature-dependent thermal conductivity
(Editura Acad Romane, 2017) Kumar, Devendra; Baleanu, Dumitru; Singh, Jagdev; Baleanu, Dumitru; Matematik
The principal purpose of the present article is to examine a fractional model of convective radial fins having constant and temperature-dependent thermal conductivity. In order to solve fractional order energy balance equation, a numerical algorithm namely homotopy analysis transform method is considered. The fin temperature is derived in terms of thermo-geometric fin parameter. Our method is not limited to the use of a small parameter, such as in the standard perturbation technique. The numerical simulation for temperature and fin tip temperature are presented graphically. The results can be used in thermal design to consider radial fins having both constant and temperature-dependent thermal conductivity.
Citation - WoS: 14
Citation - Scopus: 18
A Freely Damped Oscillating Fractional Dynamic System Modeled By Fractional Euler-Lagrange Equations
(Sage Publications Ltd, 2018) Agila, Adel; Baleanu, Dumitru; Baleanu, Dumitru; Eid, Rajeh; Irfanoglu, Bulent; 56389; Matematik
The behaviors of some vibrating dynamic systems cannot be modeled precisely by means of integer representation models. Fractional representation looks like it is more accurate to model such systems. In this study, the fractional Euler-Lagrange equations model is introduced to model a fractional damped oscillating system. In this model, the fractional inertia force and the fractional damping force are proportional to the fractional derivative of the displacement. The fractional derivative orders in both forces are considered to be variable fractional orders. A numerical approximation technique is utilized to obtain the system responses. The discretization of the Coimbra fractional derivative and the finite difference technique are used to accomplish this approximation. The response of the system is verified by a comparison to a classical integer representation and is obtained based on different values of system parameters.
Citation - WoS: 13
Citation - Scopus: 14
A generalisation of the Malgrange-Ehrenpreis theorem to find fundamental solutions to fractional PDEs
(Univ Szeged, Bolyai institute, 2017) Baleanu, Dumitru; Baleanu, Dumitru; Fernandez, Arran; 56389; Matematik
We present and prove a new generalisation of the Malgrange-Ehrenpreis theorem to fractional partial differential equations, which can be used to construct fundamental solutions to all partial differential operators of rational order and many of arbitrary real order. We demonstrate with some examples and mention a few possible applications.
Citation - WoS: 5
Citation - Scopus: 5
A halanay-type inequality on time scales in higher dimensional spaces
(Element, 2014) Jia, Baoguo; Mert, Raziye; Erbe, Lynn; Mert, Raziye; 19485; Matematik
In this paper, we investigate a certain class of Halanay-type inequalities on time scales in higher dimensional spaces. By means of the obtained inequality, we derive some new global stability conditions for linear delay dynamic systems on time scales. An example is given to illustrate the results.
Citation - WoS: 228
Citation - Scopus: 261
A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems
(Sage Publications Ltd, 2007) Agrawal, Om P.; Baleanu, Dumitru; Baleanu, Dumitru; 56389; Matematik
This paper deals with a direct numerical technique for Fractional Optimal Control Problems (FOCPs). In this paper, we formulate the FOCPs in terms of Riemann-Liouville Fractional Derivatives (RLFDs). It is demonstrated that right RLFDs automatically arise in the formulation even when the dynamics of the system is described using left RLFDs only. For numerical computation, the FDs are approximated using the Grunwald-Letnikov definition. This leads to a set of algebraic equations that can be solved using numerical techniques. Two examples, one time-invariant and the other time-variant, are considered to demonstrate the effectiveness of the formulation. Results show that as the order of the derivative approaches an integer value, these formulations lead to solutions for integer order system. The approach requires dividing of the entire time domain into several sub-domains. Further, as the sizes of the sub-domains are reduced, the solutions converge to unique solutions. However, the convergence is slow. A scheme that improves the convergence rate will be considered in a future paper. Other issues to be considered in the future include formulations using other types of derivatives, nonlinear and stochastic fractional optimal controls, existence and uniqueness of the solutions, and the error analysis.
Citation - WoS: 17
Citation - Scopus: 20
A high-order unconditionally stable numerical method for a class of multi-term time-fractional diffusion equation arising in the solute transport models
(Taylor & Francis Ltd, 2023) Alam, Mohammad Prawesh; Baleanu, Dumitru; Khan, Arshad; Baleanu, Dumitru; 56389; Matematik
In this paper, we study a high-order unconditionally stable numerical method to approximate the class of multi-term time-fractional diffusion equations. This type of problem appears in the modelling of transport of certain quantities such as heat, mass, energy, solutes in ground water and soils. The multi-term time-fractional derivative is approximated by using the Crank-Nicolson method for the Caputo's time derivative. The space derivative is approximated by using the collocation method based on quintic B-spline basis functions. We have established the stability and convergence analysis of the proposed numerical scheme thoroughly, and it is shown that the order of convergence in space variable is almost four and in the time variable is O (Delta t(2-max{gamma,gamma i})). To prove the accuracy and efficiency of the developed method, we consider four numerical examples and perform the numerical simulation. The developed algorithm works well andvalidate the theoretical results. The developed method is fourth-order convergent in the space variable, which is almost two orders of magnitude higher than the other spline collocation methods.
Citation - WoS: 6
Citation - Scopus: 7
A hybrid computing approach to design the novel second order singular perturbed delay differential Lane-Emden model
(Iop Publishing Ltd, 2022) Sabir, Zulqurnain; Baleanu, Dumitru; Baleanu, Dumitru; Raja, Muhammad Asif Zahoor; Hincal, Evren; 56389; Matematik
In this study, the mathematical form of the second order perturbed singular delay differential system is presented. The comprehensive features using the singular-point, perturbed factor and pantograph term are provided together with the shape factor of the second order perturbed singular delay differential system. The novel model is simulated numerically through the artificial neural networks (ANNs) based on the global/local optimization procedures, i.e., genetic algorithm (GA) and sequential quadratic programming (SQP). An activation function is constructed by using the differential model based on the second order perturbed singular delay differential system. The optimization of fitness function is performed through the hybrid computing strength of the ANNs-GA-SQP to solve the second order perturbed singular delay differential system. The exactness, substantiation, and authentication of the novel system is observed to solve three different variants of the novel model. The convergence, robustness, correctness, and stability of the numerical approach is performed using the comparison procedures of the available exact solutions. For the reliability, the statistical performances with necessary processes are provided using the ANNs-GA-SQP.
Citation - WoS: 108
Citation - Scopus: 120
A hybrid functions numerical scheme for fractional optimal control problems: Application to nonanalytic dynamic systems
(Sage Publications Ltd, 2018) Mohammadi, F.; Baleanu, Dumitru; Moradi, L.; Baleanu, D.; Jajarmi, A.; 56389; Matematik
In this paper, a numerical scheme based on hybrid Chelyshkov functions (HCFs) is presented to solve a class of fractional optimal control problems (FOCPs). To this end, by using the orthogonal Chelyshkov polynomials, the HCFs are constructed and a general formulation for their operational matrix of the fractional integration, in the Riemann-Liouville sense, is derived. This operational matrix together with HCFs are used to reduce the FOCP to a system of algebraic equations, which can be solved by any standard iterative algorithm. Moreover, the application of presented method to the problems with a nonanalytic dynamic system is investigated. Numerical results confirm that the proposed HCFs method can achieve spectral accuracy to approximate the solution of FOCPs.
Citation - WoS: 3
Citation - Scopus: 5
A mathematical model to optimize the available control measures of
(Elsevier, 2021) Baba, Isa Abdullahi; Baleanu, Dumitru; Nasidi, Bashir Ahmad; Baleanu, Dumitru; Saadi, Sultan Hamed; 56389; Matematik
In the absence of valid medicine or vaccine for treating the pandemic Coronavirus (COVID-19) infection, other control strategies like; quarantine, social distancing, self- isolation, sanitation and use of personal protective equipment are effective tool used to prevent and curtail the spread of the disease. In this paper, we present a mathematical model to study the dynamics of COVID-19. We then formulate an optimal control problem with the aim to study the most effective control strategies to prevent the proliferation of the disease. The existence of an optimal control function is established and the Pontryagin maximum principle is applied for the characterization of the controller. The equilibrium solutions (DFE & endemic) are found to be locally asymptotically stable and subsequently the basic reproduction number is obtained. Numerical simulations are carried out to support the analytic results and to explicitly show the significance of the control. It is shown that Quarantine/isolating those infected with the disease is the best control measure at the moment.

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