Matematik Bölümü Yayın Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/413
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Article Citation - WoS: 39Citation - Scopus: 42A Second Order Accurate Approximation for Fractional Derivatives With Singular and Non-Singular Kernel Applied To a Hiv Model(Elsevier Science inc, 2020) Baleanu, Dumitru; Arshad, Sadia; Defterli, OzlemIn this manuscript we examine the CD4(+) T cells model of HIV infection under the consideration of two different fractional differentiation operators namely Caputo and Caputo-Fabrizio (CF). Moreover, the generalized HIV model is investigated by considering Reverse Transcriptase (RT) inhibitors as a drug treatment for HIV. The threshold values for the stability of the equilibrium point belonging to non-infected case are calculated for both models with and without treatment. For the numerical solutions of the studied model, we construct trapezoidal approximation schemes having second order accuracy for the approximation of fractional operators with singular and non-singular kernel. The stability and convergence of the proposed schemes are analyzed analytically. To illustrate the dynamics given by these two fractional operators, we perform numerical simulations of the HIV model for different biological scenarios with and without drug concentration. The studied biological cases are identified by considering different values of the parameters such as infection rate, growth rate of CD4(+) T cells, clearance rate of virus particles and also the order of the fractional derivative. (C) 2020 Elsevier Inc. All rights reserved.Article Citation - WoS: 18Citation - Scopus: 21A Spectral Collocation Method With Piecewise Trigonometric Basis Functions for Nonlinear Volterra-Fredholm Integral Equations(Elsevier Science inc, 2020) Hajipour, Mojtaba; Baleanu, Dumitru; Amiri, SadeghThe aim of this paper is to investigate an efficient numerical method based on a novel shifted piecewise cosine basis for solving Volterra-Fredholm integral equations of the second kind. Using operational matrices of integration for the proposed basis functions, this integral equation is transformed into a system of nonlinear algebraic equations. The convergence and error analysis of the proposed method are studied. Some comparative results are provided to verify the efficiency of the presented method. (C) 2019 Elsevier Inc. All rights reserved.Article Citation - WoS: 76Citation - Scopus: 82On Shifted Jacobi Spectral Approximations for Solving Fractional Differential Equations(Elsevier Science inc, 2013) Bhrawy, A. H.; Baleanu, D.; Ezz-Eldien, S. S.; Doha, E. H.In this paper, a new formula of Caputo fractional-order derivatives of shifted Jacobi polynomials of any degree in terms of shifted Jacobi polynomials themselves is proved. We discuss a direct solution technique for linear multi-order fractional differential equations (FDEs) subject to nonhomogeneous initial conditions using a shifted Jacobi tau approximation. A quadrature shifted Jacobi tau (Q-SJT) approximation is introduced for the solution of linear multi-order FDEs with variable coefficients. We also propose a shifted Jacobi collocation technique for solving nonlinear multi-order fractional initial value. problems. The advantages of using the proposed techniques are discussed and we compare them with other existing methods. We investigate some illustrative examples of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques. (C) 2013 Elsevier Inc. All rights reserved.Article Citation - WoS: 26Citation - Scopus: 26Investigation of the Fractional Diffusion Equation Based on Generalized Integral Quadrature Technique(Elsevier Science inc, 2015) Razminia, Abolhassan; Baleanu, Dumitru; Razminia, KambizNowadays, the conventional Euclidean models are mostly used to describe the behavior of fluid flow through porous media. These models assume the homogeneity of the reservoir, and in naturally fractured reservoir, the fractures are distributed uniformly and use the interconnected fractures assumption. However, several cases such as core, log, outcrop data, production behavior of reservoirs, and the dynamic behavior of reservoirs indicate that the reservoirs have a different behavior other than these assumptions in most cases. According to the fractal theory and the concept of fractional derivative, a generalized diffusion equation is presented to analyze the transport in fractal reservoirs. Three outer boundary conditions are investigated. Using exact analytical or semi-analytical solutions for generalized diffusion equation with fractional order differential equation and a fractal physical form, under the usual assumptions, requires large amounts of computation time and may produce inaccurate and fake results for some combinations of parameters. Because of fractionality, fractal shape, and therefore the existence of infinite series, large computation times occur, which is sometimes slowly convergent. This paper provides a computationally efficient and accurate method via differential quadrature (DQ) and generalized integral quadrature (GIQ) analyses of diffusion equation to overcome these difficulties. The presented method would overcome the imperfections in boundary conditions' implementations of second-order partial differential equation (PDE) encountered in such problems. (C) 2014 Elsevier Inc. All rights reserved.Article Citation - WoS: 137Citation - Scopus: 151Variational Iteration Method for the Burgers' Flow With Fractional Derivatives-New Lagrange Multipliers(Elsevier Science inc, 2013) Baleanu, Dumitru; Wu, Guo-ChengThe flow through porous media can be better described by fractional models than the classical ones since they include inherently memory effects caused by obstacles in the structures. The variational iteration method was extended to find approximate solutions of fractional differential equations with the Caputo derivatives, but the Lagrange multipliers of the method were not identified explicitly. In this paper, the Lagrange multiplier is determined in a more accurate way and some new variational iteration formulae are presented. (C) 2013 Elsevier Inc. All rights reserved.Article Citation - WoS: 12Citation - Scopus: 12A Kamenev-Type Oscillation Result for a Linear (1+α)-Order Fractional Differential Equation(Elsevier Science inc, 2015) Mustafa, Octavian G.; O'Regan, Donal; Baleanu, Dumitru; Bəleanu, DumitruWe investigate the eventual sign changing for the solutions of the linear equation (x((alpha)))' + q(t)x = t >= 0, when the functional coefficient q satisfies the Kamenev-type restriction lim sup 1/t epsilon integral(t)(to) (t - s)epsilon q(s)ds = +infinity for some epsilon > 2; t(0) > 0. The operator x((alpha)) is the Caputo differential operator and alpha is an element of (0, 1). (C) 2015 Elsevier Inc. All rights reserved.Article Citation - WoS: 30Citation - Scopus: 46On Non-Homogeneous Singular Systems of Fractional Nabla Difference Equations(Elsevier Science inc, 2014) Baleanu, Dumitru I.; Kalogeropoulos, Grigoris I.; Dassios, Ioannis K.In this article we study the initial value problem of a class of non-homogeneous singular systems of fractional nabla difference equations whose coefficients are constant matrices. By taking into consideration the cases that the matrices are square with the leading coefficient singular, non-square and square with a matrix pencil which has an identically zero determinant, we provide necessary and sufficient conditions for the existence and uniqueness of solutions. More analytically we study the conditions under which the system has unique, infinite and no solutions. Furthermore, we provide a formula for the case of the unique solution. Finally, numerical examples are given to justify our theory. Crown Copyright (C) 2013 Published by Elsevier Inc. All rights reserved.Article Citation - WoS: 35Citation - Scopus: 35On Existence of a Globally Attractive Periodic Solution of Impulsive Delay Logarithmic Population Model(Elsevier Science inc, 2008) Alzabut, Jehad O.; Abdeljawad, ThabetIn this paper, it is shown that a logarithmic population model which is governed by impulsive delay differential equation has a globally attractive periodic solution. (c) 2007 Elsevier Inc. All rights reserved.Article Citation - WoS: 3Citation - Scopus: 3Third Order Differential Equations With Fixed Critical Points(Elsevier Science inc, 2009) Jarad, Fahd; Kessi, Arezki; Mugan, Ugurhan; Adjabi, Yasin; Jrad, FahdThe singular point analysis of third order ordinary differential equations which are algebraic in y and y' is presented. Some new third order ordinary differential equations that pass the Painleve test as well as the known ones are found. (C) 2008 Elsevier Inc. All rights reserved.Article Citation - WoS: 63Citation - Scopus: 72A New Hybrid Algorithm for Continuous Optimization Problem(Elsevier Science inc, 2018) Jafarian, Ahmad; Baleanu, Dumitru; Farnad, BehnamThis paper applies a new hybrid method by a combination of three population base algorithms such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO) and Symbiotic Organisms Search (SOS). The proposed method has been inspired from natural selection process and it completes this process in GA by using the PSO and SOS. It tends to minimize the execution time and in addition to reduce the complexity. Symbiotic organisms search is a robust and powerful metaheuristic algorithm which has attracted increasing attention in recent decades. There are three alternative phases in the proposed algorithm: GA, which develops and selects best population for the next phases, PSO, which gets experiences for each appropriate solution and updates them as well and SOS, which benefits from previous phases and performs symbiotic interaction update phases in the real-world population. The proposed algorithm was tested on the set of best known unimodal and multimodal benchmark functions in various dimensions. It has further been evaluated in, the experiment on the clustering of benchmark datasets. The obtained results from basic and non-parametric statistical tests confirmed that this hybrid method dominates in terms of convergence, execution time, success rate. It optimizes the high dimensional and complex functions Rosenbrock and Griewank up to 10(-330) accuracy in less than 3 s, outperforming other known algorithms. It had also applied clustering datasets with minimum intra-cluster distance and error rate. (C) 2017 Elsevier Inc. 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