Matematik Bölümü Yayın Koleksiyonu

Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/413

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  • Article
    Citation - WoS: 35
    Citation - Scopus: 31
    Optimal Solutions for Singular Linear Systems of Caputo Fractional Differential Equations
    (Wiley, 2021) Baleanu, Dumitru; Dassios, Ioannis
    In this article, we focus on a class of singular linear systems of fractional differential equations with given nonconsistent initial conditions (IC). Because the nonconsistency of the IC can not lead to a unique solution for the singular system, we use two optimization techniques to provide an optimal solution for the system. We use two optimization techniques to provide the optimal solution for the system because a unique solution for the singular system cannot be obtained due to the non-consistency of the IC. These two optimization techniques involve perturbations to the non-consistent IC, specifically, an l(2) perturbation (which seeks an optimal solution for the system in terms of least squares), and a second-order optimization technique at an l(1) minimum perturbation, (which includes an appropriate smoothing). Numerical examples are given to justify our theory. We use the Caputo (C) fractional derivative and two recently defined alternative versions of this derivative, the Caputo-Fabrizio (CF) and the Atangana-Baleanu (AB) fractional derivative.
  • Article
    Citation - WoS: 48
    Citation - Scopus: 56
    Caputo and Related Fractional Derivatives in Singular Systems
    (Elsevier Science inc, 2018) Baleanu, Dumitru, I; Dassios, Ioannis K.
    By using the Caputo (C) fractional derivative and two recently defined alternative versions of this derivative, the Caputo-Fabrizio (CF) and the Atangana-Baleanu (AB) fractional derivative, firstly we focus on singular linear systems of fractional differential equations with constant coefficients that can be non-square matrices, or square & singular. We study existence of solutions and provide formulas for the case that there do exist solutions. Then, we study the existence of unique solution for given initial conditions. Several numerical examples are given to justify our theory. (C) 2018 Elsevier Inc. All rights reserved.
  • Article
    Citation - WoS: 23
    Citation - Scopus: 32
    Duality of Singular Linear Systems of Fractional Nabla Difference Equations
    (Elsevier Science inc, 2015) Baleanu, Dumitru I.; Dassios, Ioannis K.
    The main objective of this article is to provide a link between the solutions of an initial value problem of a linear singular system of fractional nabla difference equations, its proper dual system and its transposed dual system. By taking into consideration the case that the coefficients are square constant matrices with the leading coefficient singular, we study the prime system and by using the invariants of its pencil we give necessary and sufficient conditions for existence and uniqueness of solutions. After we prove that by using the pencil of the prime system we can study the existence and uniqueness of solutions of the proper dual system and the transposed dual system. Moreover their solutions, when they exist, can be explicitly represented without resorting to further processes of computations for each one separately. Finally, numerical examples are given based on a singular fractional nabla real dynamical system to justify our theory. (C) 2014 Elsevier Inc. All rights reserved.