Matematik Bölümü Yayın Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/413
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Article Controlled Forced Fractional Vibrating System(Editura Acad Romane, 2019) Agila, Adel; Baleanu, Dumitru; Baleanu, Dumitru; MatematikThe reliability of dynamic systems is enhanced by vibration control. Many types of controllers are used to control the dynamic systems' vibrations. The integer and fractional PID controllers are used to control the fractional and integer dynamic systems. Different techniques are utilized to model the controlled systems. In this study, the discrete integer proportional integral derivative (PID) controller is used to control a forced damped variable-order fractional oscillatory systems. The objectives of this study are the analysis of controlled fractional system responses, and the investigation of controller gains' effects on system response characteristics. The Caputo-Fabrizio fractional derivative is used to model the system fractional dissipating force. The system responses are approximated by numerical and time discretization techniques. In order to verify the feasibility and effectiveness of the introduced methods, the fractional system response and integer system response are compared at fractional order close to one. The controlled responses of the fractional system are obtained for different fractional derivative order values. The results demonstrate same effects of PID gains on the fractional and integer oscillatory system responses' metrics. However, the system responses are varying based on the fractional derivative order values. The study shows that the integer response and the fractional responses have same behaviors and different instantaneous values.Article Citation - WoS: 12Citation - Scopus: 21Generalized Invexity and Duality in Multiobjective Variational Problems Involving Non-Singular Fractional Derivative(de Gruyter Poland Sp Z O O, 2022) Kumar, Devendra; Alshehri, Hashim M.; Singh, Jagdev; Baleanu, Dumitru; Dubey, Ved PrakashIn this article, we extend the generalized invexity and duality results for multiobjective variational problems with fractional derivative pertaining to an exponential kernel by using the concept of weak minima. Multiobjective variational problems find their applications in economic planning, flight control design, industrial process control, control of space structures, control of production and inventory, advertising investment, impulsive control problems, mechanics, and several other engineering and scientific problems. The proposed work considers the newly derived Caputo-Fabrizio (CF) fractional derivative operator. It is actually a convolution of the exponential function and the first-order derivative. The significant characteristic of this fractional derivative operator is that it provides a non-singular exponential kernel, which describes the dynamics of a system in a better way. Moreover, the proposed work also presents various weak, strong, and converse duality theorems under the diverse generalized invexity conditions in view of the CF fractional derivative operator.Article Citation - WoS: 25Citation - Scopus: 27Solution of Modified Bergman Minimal Blood Glucose-Insulin Model Using Caputo-Fabrizio Fractional Derivative(Tech Science Press, 2021) Baleanu, Dumitru; Mishra, Manvendra Narayan; Goswami, Pranay; Dubey, Ravi ShankerDiabetes is a burning issue in the whole world. It is the imbalance between body glucose and insulin. The study of this imbalance is very much needed from a research point of view. For this reason, Bergman gave an important model named-Bergman minimal model. In the present work, using Caputo-Fabrizio (CF) fractional derivative, we generalize Bergman's minimal blood glucose-insulin model. Further, we modify the old model by including one more component known as diet D(t), which is also essential for the blood glucose model. We solve the modified model with the help of Sumudu transform and fixed-point iteration procedures. Also, using the fixed point theorem, we examine the existence and uniqueness of the results along with their numerical and graphical representation. Furthermore, the comparison between the values of parameters obtained by calculating different values of t with experimental data is also studied. Finally, we draw the graphs of G(t), X(t), I(t), and D(t) for different values of tau. It is also clear from the obtained results and their graphical representation that the obtained results of modified Bergman's minimal model are better than Bergman's model.Article Citation - WoS: 6Citation - Scopus: 8Recovering the Space Source Term for the Fractional-Diffusion Equation With Caputo-Fabrizio Derivative(Springer, 2021) Nguyen Hoang Luc; Baleanu, Dumitru; Le Dinh Long; Le Nhat Huynh; Long, Le Dinh; Huynh, Le Nhat; Luc, Nguyen HoangThis article is devoted to the study of the source function for the Caputo-Fabrizio time fractional diffusion equation. This new definition of the fractional derivative has no singularity. In other words, the new derivative has a smooth kernel. Here, we investigate the existence of the source term. Through an example, we show that this problem is ill-posed (in the sense of Hadamard), and the fractional Landweber method and the modified quasi-boundary value method are used to deal with this inverse problem and the regularized solution is also obtained. The convergence estimates are addressed for the regularized solution to the exact solution by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. In addition, we give a numerical example to illustrate the proposed method.Article Citation - WoS: 13Citation - Scopus: 14A New Numerical Method for Time Fractional Non-Linear Sharma-Tasso Equation and Klein-Gordon Equation With Exponential Kernel Law(Frontiers Media Sa, 2020) Baleanu, Dumitru; Kumar, SachinIn this work, we derived a novel numerical scheme to find out the numerical solution of fractional PDEs having Caputo-Fabrizio (C-F) fractional derivatives. We first find out the formula of approximation for the C-F derivative of the function f(t) = t(k). We approximate the C-F derivative in time direction with the help of Legendre spectral method and approximation formula of t(k). The unknown function and their derivatives in spatial direction are approximated with the help of the method which is based on a quasi wavelet. We implement this newly derived method to solve the non-linear Sharma-Tasso-Oliver equation and non-linear Klein-Gordon equation in which time-fractional derivative is of C-F type. The accuracy and validity of this new method are depicted by giving the numerical solution of some numerical examples. The numerical results for the particular cases of Klein-Gordon equation are compared with the existing exact solutions and from the obtained error we can conclude that our proposed numerical method achieves accurate results. The effect of time-fractional exponent alpha on the solution profile is characterized by figures. The comparison of solution profile u(x, t) for different type time-fractional derivative (C-F vs. Caputo) is depicted by figures.Article Citation - WoS: 102Citation - Scopus: 120A Fractional Model for the Dynamics of Tuberculosis Infection Using Caputo-Fabrizio Derivative(Amer inst Mathematical Sciences-aims, 2020) Khan, Muhammad Altaf; Farooq, Muhammad; Hammouch, Zakia; Baleanu, Dumitru; Ullah, SaifIn 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.Article Citation - WoS: 48Citation - Scopus: 57Solving Fdes With Caputo-Fabrizio Derivative by Operational Matrix Based on Genocchi Polynomials(Wiley, 2018) Roshan, Sedighe Sadeghi; Jafari, Hossein; Baleanu, Dumitru; Sadeghi Roshan, SedigheWe introduce a new approach to solve a type of fractional order differential equations without singularity. For fractional integration, we obtain the operational matrix through Genocchi polynomials. Some examples are presented to test the applicability and efficiency of the technique.Article Citation - WoS: 268Citation - Scopus: 281Caputo-Fabrizio Derivative Applied To Groundwater Flow Within Confined Aquifer(Asce-amer Soc Civil Engineers, 2017) Baleanu, Dumitru; Atangana, AbdonThe model of the movement of subsurface water via the geological formation called aquifer was extended using a newly proposed derivative with fractional order. An alternative derivative to that of Caputo-Fabrizio with fractional order was presented. The relationship between both derivatives was presented. The new equation was solved analytically using some integral transforms. The exact solution is therefore compared to experimental data obtained from the settlement of the University of the Free State in South Africa. The numerical simulation shows the agreement of the experimental data with an analytical solution for some values of fractional order. (C) 2016 American Society of Civil Engineers.Article Citation - WoS: 47Citation - Scopus: 57Nonlocal Cauchy Problem Via a Fractional Operator Involving Power Kernel in Banach Spaces(Mdpi, 2019) Yavuz, Mehmet; Baleanu, Dumitru; Keten, AysegulWe investigated existence and uniqueness conditions of solutions of a nonlinear differential equation containing the Caputo-Fabrizio operator in Banach spaces. The mentioned derivative has been proposed by using the exponential decay law and hence it removed the computational complexities arising from the singular kernel functions inherit in the conventional fractional derivatives. The method used in this study is based on the Banach contraction mapping principle. Moreover, we gave a numerical example which shows the applicability of the obtained results.Article Citation - WoS: 30Citation - Scopus: 38Modified Kawahara Equation Within a Fractional Derivative With Non-Singular Kernel(Vinca inst Nuclear Sci, 2018) Singh, Jagdev; Baleanu, Dumitru; Kumar, DevendraThe article addresses a time fractional modified Kawahara equation through a fractional derivative with exponential kernel. The Kawahara equation describes the generation of non-linear water-waves in the long-wavelength regime. The numerical solution of the fractional model of modified version of Kawahara equation is derived with the help of iterative scheme and the stability of applied technique is established. In order to demonstrate the usability and effectiveness of the new fractional derivative to describe water-waves in the long-wavelength regime, numerical results are presented graphically.