Scopus İndeksli Yayınlar Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8651
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Article Citation - Scopus: 61Solving Multi-Term Orders Fractional Differential Equations by Operational Matrices of Bps With Convergence Analysis(2013) Rostamy, D.; Baleanu, Dumitru; Alipour, M.; Jafari, H.; Baleanu, D.; MatematikIn this paper, we present a numerical method for solving a class of fractional differential equations (FDEs). Based on Bernstein Polynomials (BPs) basis, new matrices are utilized to reduce the multi-term orders fractional differential equation to a system of algebraic equations. Convergence analysis is shown by several theorems. Illustrative examples are included to demonstrate the validity and applicability of this method.Article Citation - Scopus: 3On Some Impulsive Fractional Neutral Differential Systems With Nonlocal Condition Through Fractional Operators(Cambridge Scientific Publishers, 2017) Anuradha, A.; Baleanu, Dumitru; Baleanu, D.; Suganya, S.; Arjunan, M.M.; MatematikAccording to semigroup theories, fractional calculus, Banach contraction principle and Schaefer's fixed point theorem, this paper is fundamentally involved with the existence of mild solutions for an impulsive fractional neutral differential systems (abbreviated, IFNDS) with nonlocal conditions (abbreviated, NLC) in Banach space X. At last, an illustration is also presented to exhibit the use of our theoretical results. © CSP - Cambridge, UK; I & S - Florida, USA, 2017.Article Citation - WoS: 29Citation - Scopus: 31A Novel Spectral Approximation for the Two-Dimensional Fractional Sub-Diffusion Problems(Editura Acad Romane, 2015) Bhrawy, A. H.; Baleanu, Dumitru; Zaky, M. A.; Baleanu, D.; Abdelkawy, M. A.; MatematikThis paper reports a new numerical method that enables easy and convenient discretization of a two-dimensional sub-diffusion equation with fractional derivatives of any order. The suggested method is based on Jacobi tau spectral procedure together with the Jacobi operational matrix for fractional derivatives, described in the Caputo sense. Such approach has the advantage of reducing the problem to the solution of a system of algebraic equations, which may then be solved by any standard numerical technique. The validity and effectiveness of the method are demonstrated by solving two numerical examples, which are presented in the form of tables and graphs to make more easier comparisons with the exact solutions and the results obtained by other methods.Article Citation - WoS: 89Citation - Scopus: 124New Numerical Approximations for Space-Time Fractional Burgers' Equations Via a Legendre Spectral-Collocation Method(Editura Acad Romane, 2015) Bhrawy, A. H.; Zaky, M. A.; Baleanu, D.Burgers' equation is a fundamental partial differential equation in fluid mechanics. This paper reports a new space-time spectral algorithm for obtaining an approximate solution for the space-time fractional Burgers' equation (FBE) based on spectral shifted Legendre collocation (SLC) method in combination with the shifted Legendre operational matrix of fractional derivatives. The fractional derivatives are described in the Caputo sense. We propose a spectral shifted Legendre collocation method in both temporal and spatial discretizations for the space-time FBE. The main characteristic behind this approach is that it reduces such problem to that of solving a system of nonlinear algebraic equations that can then be solved using Newton's iterative method. Numerical results with comparisons are given to confirm the reliability of the proposed method for FBE.Article Citation - Scopus: 16Lucas Wavelet Scheme for Fractional Bagley–torvik Equations: Gauss–jacobi Approach(Springer, 2022) Koundal, R.; Kumar, R.; Srivastava, K.; Baleanu, D.A novel technique called as Lucas wavelet scheme (LWS) is prepared for the treatment of Bagley–Torvik equations (BTEs). Lucas wavelets for the approximation of unknown functions appearing in BTEs are introduced. Fractional derivatives are evaluated by employing Gauss–Jacobi quadrature formula. Further, well-known least square method (LSM) is adopted to compute the residual function, and the system of algebraic equation is obtained. Convergence criterion is derived and error bounds are obtained for the established technique. The scheme is investigated by choosing some reliable test problems through tables and figures, which ensures the convenience, validity and applicability of LWS. © 2021, The Author(s), under exclusive licence to Springer Nature India Private Limited.Article Citation - Scopus: 22Solitary Wave Solution for a Generalized Hirota-Satsuma Coupled Kdv and Mkdv Equations: a Semi-Analytical Approach(Elsevier B.V., 2020) Chakraverty, S.; Baleanu, D.; Jena, R.M.Nonlinear fractional differential equations (NFDEs) offer an effective model of numerous phenomena in applied sciences such as ocean engineering, fluid mechanics, quantum mechanics, plasma physics, nonlinear optics. Some studies in control theory, biology, economy, and electrodynamics, etc. demonstrate that NFDEs play the primary role in explaining various phenomena arising in real-life. Now-a-day NFDEs in various scientific fields in particular optical fibers, chemical physics, solid-state physics, and so forth have the most important subjects for study. Finding exact responses to these equations will help us to a better understanding of our environmental nonlinear physical phenomena. In this regard, in the present study, we have applied fractional reduced differential transform method (FRDTM) to obtain the solution of nonlinear time-fractional Hirota-Satsuma coupled KdV and MKdV equations. The novelty of the FRDTM is that it does not require any discretization, transformation, perturbation, or any restrictive conditions. Moreover, this method requires less computation compared to other methods. Computed results are compared with the existing results for the special cases of integer order. The present results are in good agreement with the existing solutions. Here, the fractional derivatives are considered in the Caputo sense. The presented method is a semi-analytical method based on the generalized Taylor series expansion and yields an analytical solution in the form of a polynomial. © 2020 Faculty of Engineering, Alexandria UniversityArticle Citation - WoS: 40Citation - Scopus: 53Efficient Generalized Laguerre-Spectral Methods for Solving Multi-Term Fractional Differential Equations on the Half Line(Sage Publications Ltd, 2014) Baleanu, D.; Assas, L. M.; Bhrawy, A. H.The main purpose of this paper is to provide an efficient numerical approach for the fractional differential equations (FDEs) on the half line with constant coefficients using a generalized Laguerre tau (GLT) method. The fractional derivatives are described in the Caputo sense. We state and prove a new formula expressing explicitly the derivatives of generalized Laguerre polynomials of any degree and for any fractional order in terms of generalized Laguerre polynomials themselves. We develop also a direct solution technique for solving the linear multi-order FDEs with constant coefficients using a spectral tau method. The spatial approximation with its fractional-order derivatives described in the Caputo sense are based on generalized Laguerre polynomials L-i((alpha))(x) with x is an element of Lambda = (0,infinity) and i denoting the polynomial degree.Article Citation - Scopus: 11Analytic 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.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.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: 139Citation - Scopus: 156Solving Differential Equations of Fractional Order Using an Optimization Technique Based on Training Artificial Neural Network(Elsevier Science inc, 2017) Ahmadian, A.; Effati, S.; Salahshour, S.; Baleanu, D.; Pakdaman, M.The current study aims to approximate the solution of fractional differential equations (FDEs) by using the fundamental properties of artificial neural networks (ANNs) for function approximation. In the first step, we derive an approximate solution of fractional differential equation (FDE) by using ANNs. In the second step, an optimization approach is exploited to adjust the weights of ANNs such that the approximated solution satisfies the FDE. Different types of FDEs including linear and nonlinear terms are solved to illustrate the ability of the method. In addition, the present scheme is compared with the analytical solution and a number of existing numerical techniques to show the efficiency of ANNs with high accuracy, fast convergence and low use of memory for solving the FDEs. (C) 2016 Elsevier Inc. All rights reserved.