Browsing by Author "Baleanu, D."

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Citation - Scopus: 5
About Fractional Calculus of Singular Lagrangians
(Institute of Electrical and Electronics Engineers Inc., 2004) Baleanu, D.; Baleanu, D.; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
In this paper the solutions of the fractional Euler-Lagrange equations corresponding to singular fractional Lagrangians were examined. Despite of the complexity of solutions in the fractional case the gauge classical symmetry was preserved. Four examples of fractional singular Lagrangians were analyzed in details. © 2004 IEEE.
Citation - WoS: 10
Citation - Scopus: 10
About Lagrangian Formulation of Classical Fields Within Riemann-Liouville Fractional Derivatives
(American Society of Mechanical Engineers, 2005) Baleanu, D.; Muslih, S.I.; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
Recently, an extension of the simplest fractional problem and the fractional variational problem of Lagrange was obtained by Agrawal. The first part of this study presents the fractional Lagrangian formulation of mechanical systems and introduce the Levy path integral. The second part is an extension to Agrawal's approach to classical fields with fractional derivatives. The classical fields with fractional derivatives are investigated by using the Lagrangian formulation. The case of the fractional Schrödinger equation is presented. Copyright © 2005 by ASME.
Citation - Scopus: 2
About the Fn Approximation To Fractional Neutron Transport Equation in Slab Geometry
(Amer Soc Mechanical Engineers, 2011) Baleanu, D.; Kadem, A.; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
The neutron transport denotes the study of the motions and interactions of neutrons with materials. In given applications we need to know where neutrons are in an apparatus, what direction they are moving, and how fast they are going. In this manuscript the Legendre polynomial approximation method F N was applied to the one dimensional slab geometry neutron transport equation. © 2011 by ASME.
Citation - WoS: 14
Citation - Scopus: 16
An Accurate Approximate-Analytical Technique for Solving Time-Fractional Partial Differential Equations
(Wiley-hindawi, 2017) Salahshour, S.; Ahmadian, A.; Ismail, F.; Baleanu, D.; Bishehniasar, M.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
The demand of many scientific areas for the usage of fractional partial differential equations (FPDEs) to explain their real-world systems has been broadly identified. The solutions may portray dynamical behaviors of various particles such as chemicals and cells. The desire of obtaining approximate solutions to treat these equations aims to overcome the mathematical complexity of modeling the relevant phenomena in nature. This research proposes a promising approximate-analytical scheme that is an accurate technique for solving a variety of noninteger partial differential equations (PDEs). The proposed strategy is based on approximating the derivative of fractional-order and reducing the problem to the corresponding partial differential equation (PDE). Afterwards, the approximating PDE is solved by using a separation-variables technique. The method can be simply applied to nonhomogeneous problems and is proficient to diminish the span of computational cost as well as achieving an approximate-analytical solution that is in excellent concurrence with the exact solution of the original problem. In addition and to demonstrate the efficiency of the method, it compares with two finite difference methods including a nonstandard finite difference (NSFD) method and standard finite difference (SFD) technique, which are popular in the literature for solving engineering problems.
Citation - WoS: 13
Citation - Scopus: 14
An Accurate Legendre Collocation Scheme for Coupled Hyperbolic Equations With Variable Coefficients
(Editura Acad Romane, 2014) Doha, E. H.; Baleanu, Dumitru; Bhrawy, A. H.; Baleanu, D.; Abdelkawy, M. A.; 56389; Matematik; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
The study of numerical solutions of nonlinear coupled hyperbolic partial differential equations (PDEs) with variable coefficients subject to initial-boundary conditions continues to be a major research area with widespread applications in modern physics and technology. One of the most important advantages of collocation method is the possibility of dealing with nonlinear partial differential equations (NPDEs) as well as PDEs with variable coefficients. A numerical solution based on a Legendre collocation method is extended to solve nonlinear coupled hyperbolic PDEs with variable coefficients. This approach, which is based on Legendre polynomials and Gauss-Lobatto quadrature integration, reduces the solving of nonlinear coupled hyperbolic PDEs with variable coefficients to a system of nonlinear ordinary differential equations that is far easier to solve. The obtained results show that the proposed numerical algorithm is efficient and very accurate.
Citation - WoS: 64
Citation - Scopus: 63
An Accurate Numerical Technique for Solving Fractional Optimal Control Problems
(Editura Acad Romane, 2015) Bhrawy, A. H.; Baleanu, Dumitru; Doha, E. H.; Baleanu, D.; Ezz-Eldien, S. S.; Abdelkawy, M. A.; 56389; Matematik; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
In this article, we propose the shifted Legendre orthonormal polynomials for the numerical solution of the fractional optimal control problems that appear in several branches of physics and engineering. The Rayleigh-Ritz method for the necessary conditions of optimization and the operational matrix of fractional derivatives are used together with the help of the properties of the shifted Legendre orthonormal polynomials to reduce the fractional optimal control problem to solving a system of algebraic equations that greatly simplifies the problem. For confirming the efficiency and accuracy of the proposed technique, an illustrative numerical example is introduced with its approximate solution.
Citation - Scopus: 8
Advanced Analysis of Local Fractional Calculus Applied To the Rice Theory in Fractal Fracture Mechanics
(Springer Science and Business Media Deutschland GmbH, 2022) Baleanu, D.; Srivastava, H.M.; Yang, X.-J.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
In this chapter, the recent results for the analysis of local fractional calculus are considered for the first time. The local fractional derivative (LFD) and the local fractional integral (LFI) in the fractional (real and complex) sets, the series and transforms involving the Mittag-Leffler function defined on Cantor sets are introduced and reviewed. The uniqueness of the solutions of the local fractional differential and integral equations and the local fractional inequalities are considered in detail. The local fractional vector calculus is applied to describe the Rice theory in fractal fracture mechanics. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.
Advances on Integrodifferential Equations and Transforms
(Hindawi Publishing Corporation, 2015) Yang, X.-J.; Baleanu, D.; Nieto, J.J.; Hristov, J.; Srivastava, H.M.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
Citation - Scopus: 3
Algorithmic Complexity-Based Fractional-Order Derivatives in Computational Biology
(Springer Science and Business Media Deutschland GmbH, 2023) Baleanu, D.; Karaca, Y.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
Fractional calculus approach, providing novel models through the introduction of fractional-order calculus to optimization methods, is employed in machine learning algorithms. This scheme aims to attain optimized solutions by maximizing the accuracy of the model and minimizing the functions like the computational burden. Mathematical-informed frameworks are to be employed to enable reliable, accurate, and robust understanding of various complex biological processes that involve a variety of spatial and temporal scales. This complexity requires a holistic understanding of different biological processes through multi-stage integrative models that are capable of capturing the significant attributes on the related scales. Fractional-order differential and integral equations can provide the generalization of traditional integral and differential equations through the extension of the conceptions with respect to biological processes. In addition, algorithmic complexity (computational complexity), as a way of comparing the efficiency of an algorithm, can enable a better grasping and designing of efficient algorithms in computational biology as well as other related areas of science. It also enables the classification of the computational problems based on their algorithmic complexity, as defined according to the way the resources are required for the solution of the problem, including the execution time and scale with the problem size. Based on a novel mathematical informed framework and multi-staged integrative method concerning algorithmic complexity, this study aims at establishing a robust and accurate model reliant on the combination of fractional-order derivative and Artificial Neural Network (ANN) for the diagnostic and differentiability predictive purposes for the disease, (diabetes, as a metabolic disorder, in our case) which may display various and transient biological properties. Another aim of this study is benefitting from the concept of algorithmic complexity to obtain the fractional-order derivative with the least complexity in order that it would be possible to achieve the optimized solution. To this end, the following steps were applied and integrated. Firstly, the Caputo fractional-order derivative with three-parametric Mittag-Leffler function (α,β,γ) was applied to the diabetes dataset. Thus, new fractional models with varying degrees were established by ensuring data fitting through the fitting algorithm Mittag-Leffler function with three parameters (α,β,γ) based on heavy-tailed distributions. Following this application, the new dataset, named the mfc_diabetes, was obtained. Secondly, classical derivative (calculus) was applied to the diabetes dataset, which yielded the cd_diabetes dataset. Subsequently, the performance of the new dataset as obtained from the first step and of the dataset obtained from the second step as well as of the diabetes dataset was compared through the application of the feed forward back propagation (FFBP) algorithm, which is one of the ANN algorithms. Next, the fractional order derivative model which would be the most optimal for the disease was generated. Finally, algorithmic complexity was employed to attain the Caputo fractional-order derivative with the least complexity, or to achieve the optimized solution. This approach through the application of fractional-order calculus to optimization methods and the experimental results have revealed the advantage of maximizing the model’s accuracy and minimizing the cost functions like the computational costs, which points to the applicability of the method proposed in different domains characterized by complex, dynamic and transient components. © 2023, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
Citation - WoS: 13
Citation - Scopus: 15
Analysis and Numerical Solution of the Generalized Proportional Fractional Cauchy Problem
(Elsevier, 2021) Baleanu, D.; Makhlouf, Abdellatif Ben; Nagy, A. M.; Boucenna, Djalal; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
In this paper, we explore the existence and uniqueness theorem for a problem of the fractional Cauchy form, with dependence on the generalized proportional Caputo derivative. Furthermore, a new numerical technique is presented based on a decomposition formula for the generalized proportional Caputo derivative. Convergence analysis of the proposed technique is proved. Finally, numerical results are obtained to confirm the validity of the proposed method. (C) 2021 IMACS. Published by Elsevier reserved.
Citation - WoS: 6
Citation - Scopus: 7
Analysis of Diffusivity Equation Using Differential Quadrature Method
(Editura Acad Romane, 2014) Razminia, K.; Baleanu, Dumitru; Razminia, A.; Kharrat, R.; Baleanu, D.; 56389; Matematik; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
Evaluation of exact analytical solution for flow to a well, under the assumptions made in its development commonly requires large amounts of computation time and can produce inaccurate results for selected combinations of parameters. Large computation times occur because the solution involves the infinite series. Each term of the series requires evaluation of exponentials and Bessel functions, and the series itself is sometimes slowly convergent. Inaccuracies can result from lack of computer precision or from the use of improper methods of numerical computation. This paper presents a computationally efficient and an accurate new methodology in differential quadrature analysis of diffusivity equation to overcome these difficulties. The methodology would overcome the difficulties in boundary conditions implementations of second order partial differential equations encountered in such problems. The weighting coefficients employed are not exclusive, and any accurate and efficient method such as the generalized differential quadrature method may be used to produce the method's weighting coefficients. By solving finite and infinite boundary condition in diffusivity equation and by comparing the results with those of existing solutions and/or those of other methodologies, accuracy, convergences, reduction of computation time, and efficiency of the methodology are asserted.
Citation - WoS: 5
Citation - Scopus: 6
Analytic Solution for a Nonlinear Problem of Magneto-Thermoelasticity
(Pergamon-elsevier Science Ltd, 2013) Ghaderi, P.; Golmankhaneh, Alireza K.; Baleanu, D.; Jafarian, A.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
In this paper, we present a comparative study of the homotopy analysis method (HAM), the variational iteration method (VIM) and the iterative method (He's polynomials). The approximate solution of the coupled harmonic waves nonlinear magneto-thermoelasticity equations under influence of rotation is obtained. In order to control and adjust the convergence region and the rate of solution series, we show that it is possible to choose a valid auxiliary parameter h of HAM. Using the boundary and the initial conditions we select a suitable initial approximation. The results show that these methods are very efficient, convenient and applicable to a large class of problems.
Citation - Scopus: 9
Analytic Solution of Space Time Fractional Advection Dispersion Equation With Retardation for Contaminant Transport in Porous Media
(Natural Sciences Publishing, 2019) Yadav, M.P.; Agarwal, R.P.; Baleanu, D.; Agarwal, R.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
Motivated by recent applications of fractional calculus, in this paper, we derive analytical solutions of fractional advection-dispersion equation with retardation by replacing the integer order partial derivatives with fractional Riesz-Feller derivative for space variable and Caputo fractional derivative for time variable. The Laplace and Fourier transforms are applied to obtain the solution in terms of the Mittag-Leffler function. Some interesting special cases of the time-space fractional advection-dispersion equation with retardation are also considered. The composition formulas for Green function has been evaluated which enables us to express the solution of the space time fractional advection dispersion equation in terms of the solution of space fractional advection dispersion equation and time fractional advection dispersion equation. Furthermore, from this representation we derive explicit formulae, which enable us to plot the probability densities in space for the different values of the relevant parameters. © 2019 NSP Natural Sciences Publishing Cor.
Citation - WoS: 3
Citation - Scopus: 5
Analytical and Numerical Simulations for the Kinetics of Phase Separation in Iron (fe-Cr (X = Mo, Cu)) Based on Ternary Alloys (Vol 537c, 122634, 2019)
(Elsevier, 2021) Lu, D.; Osman, M. S.; Khater, M. M. A.; Attia, R. A. M.; Baleanu, D.; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
Citation - WoS: 78
Citation - Scopus: 77
Analytical and Numerical Simulations for the Kinetics of Phase Separation in Iron (fe-Cr (X=mo, Cu)) Based on Ternary Alloys
(Elsevier, 2020) Osman, M. S.; Khater, M. M. A.; Attia, R. A. M.; Baleanu, D.; Lu, D.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
In this paper, we investigate the physical behavior of the basic elements that related to phase decomposition in ternary alloys of (Fe-Cr-Mo) and (Fe-Cr-Cu) according to analytical and approximate simulation. We study the dynamic of the separation phase for the ternary alloys of iron. The dynamical process of this separation has been described in a mathematical model called the Cahn-Hilliard equation. The minor element behavior in the process has been described by the Cahn-Hilliard equation. It describes the process of phase separation for two components of a binary fluid in ternary alloys of (Fe-Cr-Mo) and (Fe-Cr-Cu). We implement a modified auxiliary equation method and the cubic B-spline scheme on this mathematical model to show the dynamical process of phase separation and the concentration of one of two components in a system. We try obtaining the solitary and approximate solutions of this model to show the relation between the components in this phase. We discuss our solutions in view of a Stefan, Thomas-Windle, and Navier-Stokes models. Whereas, these models describe the motion of viscous fluid substance. (C) 2019 Elsevier B.V. All rights reserved.
Citation - Scopus: 1
Analytical and Numerical Solutions for Time-Fractional New Coupled Mkdv Equation Arising in Interaction of Two Long Waves
(Asia Pacific Academic, 2019) Şenol, M.; Kurt, A.; Baleanu, D.; Tasbozan, O.; 56389; 06.04. Endüstri Mühendisliği; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 06. Mühendislik Fakültesi; 01. Çankaya Üniversitesi; 09. Rektörlük; 09.02. Yabancı Diller Bölümü
The aim of this paper is to present new exact solution sets of nonlinear conformable time-fractional new coupled mKdV equations which arise in interaction of two long waves with different dispersion relations by means of sub-equation method. In addition, we also propose an analytical-approximate method namely residual power series method (RPSM) for the system. The fractional derivatives have been explained in newly defined conformable type, during the solution procedure. The exact solutions of the system obtained by the sub-equation method have been compared to approximate solutions derived by RPSM. The results showed that both methods are robust, dependable, easy to apply and a good alternative for seeking solutions of fractional partial differential equations. © 2019 Asia Pacific Journal of Mathematics.
Citation - WoS: 20
Citation - Scopus: 21
Analytical Approximate Solutions of the Zakharov-Kuznetsov Equations
(Editura Acad Romane, 2014) Jafarian, A.; Baleanu, Dumitru; Ghaderi, P.; Golmankhaneh, Alireza K.; Baleanu, D.; 56389; Matematik; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
In this paper, analytical approximate solutions for the Zakharov-Kuznetsov equations by homotopy analysis method (HAM) and the He's polynomials iterative method (HPIM) are presented. Our results indicate the remarkable efficiency of HAM as compared to HPIM. The convergence of these two methods is also analyzed.
Citation - WoS: 31
Citation - Scopus: 24
An Analytical Study of (2 + 1)-Dimensional Physical Models Embedded Entirely in Fractal Space
(Editura Academiei Romane, 2019) Alquran, M.; Baleanu, Dumitru; Jaradat, I.; Baleanu, D.; Abdel-Muhsen, R.; 56389; Matematik; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
In this article, we analytically furnish the solution of (2+1)-dimension-al fractional differential equations, with distinct fractal-memory indices in all coordinates, as a trivariate (α, β, γ)−fractional power series representation. The method is tested on several physical models with inherited memories. Moreover, a version of Taylor’s theorem in fractal three-dimensional space is presented. As a special case, the solutions of the corresponding integer-order cases are extracted by letting α, β, γ → 1, which indicates to some extent for a sequential memory. © 2019, Editura Academiei Romane. All rights reserved.
Citation - WoS: 20
Citation - Scopus: 22
Analytical Treatment of System of Abel Integral Equations by Homotopy Analysis Method
(Editura Acad Romane, 2014) Jafarian, A.; Baleanu, Dumitru; Ghaderi, P.; Golmankhaneh, Alireza K.; Baleanu, D.; 56389; Matematik; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
Abel equation has important applications in describing the least time for an object which is sliding on surface without friction in uniform gravity, and the classical theory of elasticity of materials is modeled by a system of Abel integral equations. In this manuscript, the homotopy analysis method is presented for obtaining analytical solutions of a system of Abel integral equations as fractional equations. The applied method has lessened the size of calculation and improved the accuracy of solution in the case of the singular Abel integral equation. The illustrated examples and numerical results have proved the assertion.
Citation - WoS: 81
Citation - Scopus: 90
Application of a Homogeneous Balance Method To Exact Solutions of Nonlinear Fractional Evolution Equations
(Asme, 2014) Tajadodi, H.; Baleanu, D.; Jafari, H.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
The fractional Fan subequation method of the fractional Riccati equation is applied to construct the exact solutions of some nonlinear fractional evolution equations. In this paper, a powerful algorithm is developed for the exact solutions of the modified equal width equation, the Fisher equation, the nonlinear Telegraph equation, and the Cahn-Allen equation of fractional order. Fractional derivatives are described in the sense of the modified Riemann-Liouville derivative. Some relevant examples are investigated.