Scopus İndeksli Yayınlar Koleksiyonu

Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8651

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Author: search.filters.author.Baleanu, DumitruSubject: search.filters.subject.Matematik

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Now showing 1 - 7 of 7
  • Article
    On Solutions of Variable-Order Fractional Differential Equations
    (Elsevier B.V., 2017) Akgül, Ali; Inc, Mustafa; Baleanu, Dumitru; Abdalla, Bahaaeldin; Jarad, Fahd; Bouchelaghem, Faycal; Abdeljawad, Thabet; Ardjouni, Abdelouaheb; Boulares, Hamid; Shah, Kamal
    Numerical calculation of the fractional integrals and derivatives is the code tosearch fractional calculus and solve fractional differential equations. The exactsolutions to fractional differential equations are compelling to get in real ap-plications, due to the nonlocality and complexity of the fractional differentialoperators, especially for variable-order fractional differential equations. There-fore, it is significant to enhance numerical methods for fractional differentialequations. In this work, we consider variable-order fractional differential equa-tions by reproducing kernel method. There has been much attention in theuse of reproducing kernels for the solutions to many problems in the recentyears. We give an example to demonstrate how efficiently our theory can beimplemented in practice.
  • Article
    Citation - Scopus: 20
    Evolutionary Mathematical Science, Fractional Modeling and Artificial Intelligence of Nonlinear Dynamics in Complex Systems
    (Akif AKGUL, 2022) Karaca, Yeliz; Baleanu, Dumitru
    Complex problems in nonlinear dynamics foreground the critical support of artificial phenomena so that each domain of complex systems can generate applicable answers and solutions to the pressing challenges. This sort of view is capable of serving the needs of different aspects of complexity by minimizing the problems of complexity whose solutions are based on advanced mathematical foundations and analogous algorithmic models consisting of numerous applied aspects of complexity. Evolutionary processes, nonlinearity and all the other dimensions of complexity lie at the pedestal of time, reveal time and occur within time. In the ever-evolving landscape and variations, with causality breaking down, the idea of complexity can be stated to be a part of unifying and revolutionary scientific framework to expound complex systems whose behavior is perplexing to predict and control with the ultimate goal of attaining a global understanding related to many branches of possible states as well as high-dimensional manifolds, while at the same time keeping abreast with actuality along the evolutionary and historical path, which itself, has also been through different critical points on the manifold. In view of these, we put forth the features of complexity of varying phenomena, properties of evolution and adaptation, memory effects, nonlinear dynamic system qualities, the importance of chaos theory and applications of related aspects in this study. In addition, processes of fractional dynamics, differentiation and systems in complex systems as well as the dynamical processes and dynamical systems of fractional order with respect to natural and artificial phenomena are discussed in terms of their mathematical modeling. Fractional calculus and fractional-order calculus approach to provide novel models with fractional-order calculus as employed in machine learning algorithms to be able to attain optimized solutions are also set forth besides the justification of the need to develop analytical and numerical methods. Subsequently, algorithmic complexity and its goal towards ensuring a more effective handling of efficient algorithms in computational sciences is stated with regard to the classification of computational problems. We further point out the neural networks, as descriptive models, for providing the means to gather, store and use experiential knowledge as well as Artificial Neural Networks (ANNs) in relation to their employment for handling experimental data in different complex domains. Furthermore, the importance of generating applicable solutions to problems for various engineering areas, medicine, biology, mathematical science, applied disciplines and data science, among many others, is discussed in detail along with an emphasis on power of predictability, relying on mathematical sciences, with Artificial Intelligence (AI) and machine learning being at the pedestal and intersection with different fields which are characterized by complex, chaotic, nonlinear, dynamic and transient components to validate the significance of optimized approaches both in real systems and in related realms.
  • Article
    Citation - WoS: 17
    Citation - Scopus: 16
    A Spectral Technique for Solving Two-Dimensional Fractional Integral Equations With Weakly Singular Kernel
    (Hacettepe Univ, Fac Sci, 2018) Abdelkawy, Mohamed A.; Baleanu, Dumitru; Amin, Ahmed Z. M.; Bhrawy, Ali H.; Amink, Ahmed Z. M.; Abdelkawyy, Mohamed A.
    This paper adapts a new numerical technique for solving twodimensional fractional integral equations with weakly singular. Using the spectral collocation method, the fractional operators of Legendre and Chebyshev polynomials, and Gauss-quadrature formula, we achieve a reduction of given problems into those of a system of algebraic equations. We apply the reported numerical method to solve several numerical examples in order to test the accuracy and validity. Thus, the novel algorithm is more responsible for solving two-dimensional fractional integral equations with weakly singular.
  • Article
    Citation - WoS: 7
    Citation - Scopus: 8
    Representation for the Reproducing Kernel Hilbert Space Method for a Nonlinear System
    (Hacettepe Univ, Fac Sci, 2019) Akgul, Ali; Khan, Yasir; Baleanu, Dumitru; Akgul, Esra Karatas; Karatas Akgül, Esra
    We apply the reproducing kernel Hilbert space method to a nonlinear system in this work. We utilize this technique to overcome the nonlinearity of the problem. We obtain accurate results. We demonstrate our results by tables and figures. We prove the efficiency of the method.
  • Article
    On the solutions of a fractional boundary value problem
    (Scientific Technical Research Council Turkey-Tubitak, 2018) Uğurlu, Ekin; Baleanu, Dumitru; Taş, Kenan
    This paper is devoted to showing the existence and uniqueness of solution of a regular second-order nonlinear fractional differential equation subject to the ordinary boundary conditions. The Banach fixed point theorem is used to prove the results.
  • Article
    Citation - WoS: 2
    Citation - Scopus: 3
    Analysis of Mixed-Order Caputo Fractional System With Nonlocal Integral Boundary Condition
    (Tubitak Scientific & Technological Research Council Turkey, 2018) Khodabakhshi, Neda; Baleanu, Dumitru; Akman Yildiz, Tugba; Yıldız, Tuğba Akman
    This paper deals with a mixed-order Caputo fractional system with nonlocal integral boundary conditions. This study can be considered as an extension of previous studies, since the orders of the equations lie on different intervals. We discuss the existence and uniqueness of the solution using fixed point methods. We enrich the study with an example.
  • Article
    Citation - WoS: 10
    Citation - Scopus: 9
    On the Existence Interval for the Initial Value Problem of a Fractional Differential Equation
    (Hacettepe Univ, Fac Sci, 2011) Mustafa, Octavian G.; Baleanu, Dumitru
    We compute via a comparison function technique, a new bound for the existence interval of the initial value problem for a fractional differential equation given by means of Caputo derivatives. We improve in this way the estimate of the existence interval obtained very recently in the literature.