Scopus İndeksli Yayınlar Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8651
Browse
6 results
Search Results
Article Citation - Scopus: 10Solving System of Fractional Differential Equations Via Vieta-Lucas Operational Matrix Method(Springer, 2024) Aeri, S.; Bala, A.; Kumar, R.; Baleanu, D.; Chaudhary, R.Vieta-Lucas polynomials (VLPs) belong to the class of weighted orthogonal polynomials, which can be used to effectively handle various natural and engineered problems. The classical fractional derivative due to Caputo is used to write the emerging operational matrices. These matrices are developed and evaluated by using the properties of VLPs. The residuated functions are mapped to zero by the tools of the Tau algorithm. Convergence and error analysis are thoroughly explored. Test examples for a fractional system of differential equations are borrowed from literature. The theoretical and simulated exercise on these examples authenticate the relevance of this scheme. Here, novel inclusion of Vieta-Lucas polynomials has been ensured in combination with the Tau approach. The operational matrix approach which provides extensive information about the fractional derivatives of different terms of Vieta-Lucas polynomial expansion, is ensured to operate to reduce the problem into an algebraic setup. The novelty is further enhanced by comparing the present scheme with the fourth-order Runge–Kutta method. © 2023, The Author(s), under exclusive licence to Springer Nature India Private Limited.Article Citation - WoS: 33Citation - Scopus: 36Shifted Ultraspherical Pseudo-Galerkin Method for Approximating the Solutions of Some Types of Ordinary Fractional Problems(Springer, 2021) Mahmoud, Doha; Baleanu, Dumitru; El-kady, Mamdouh; Abdelhakem, MohamedIn this work, a technique for finding approximate solutions for ordinary fraction differential equations (OFDEs) of any order has been proposed. The method is a hybrid between Galerkin and collocation methods. Also, this method can be extended to approximate fractional integro-differential equations (FIDEs) and fractional optimal control problems (FOCPs). The spatial approximations with their derivatives are based on shifted ultraspherical polynomials (SUPs). Modified Galerkin spectral method has been used to create direct approximate solutions of linear/nonlinear ordinary fractional differential equations, a system of ordinary fraction differential equations, fractional integro-differential equations, or fractional optimal control problems. The aim is to transform those problems into a system of algebraic equations. That system will be efficiently solved by any solver. Three spaces of collocation nodes have been used through that transformation. Finally, numerical examples show the accuracy and efficiency of the investigated method.Article Citation - WoS: 114Citation - Scopus: 104Fractional Calculus in the Sky(Springer, 2021) Agarwal, Ravi P.; Baleanu, DumitruFractional calculus was born in 1695 on September 30 due to a very deep question raised in a letter of L'Hospital to Leibniz. The prophetical answer of Leibniz to that deep question encapsulated a huge inspiration for all generations of scientists and is continuing to stimulate the minds of contemporary researchers. During 325 years of existence, fractional calculus has kept the attention of top level mathematicians, and during the last period of time it has become a very useful tool for tackling the dynamics of complex systems from various branches of science and engineering. In this short manuscript, we briefly review the tremendous effect that the main ideas of fractional calculus had in science and engineering and briefly present just a point of view for some of the crucial problems of this interdisciplinary field.Article Citation - WoS: 30Citation - Scopus: 31Existence, Uniqueness and Stability Analysis of a Coupled Fractional-Order Differential Systems Involving Hadamard Derivatives and Associated With Multi-Point Boundary Conditions(Springer, 2021) Baleanu, Dumitru; Samei, Mohammad Esmael; Zada, Akbar; Subramanian, Muthaiah; Alzabut, JehadIn this paper, we examine the consequences of existence, uniqueness and stability of a multi-point boundary value problem defined by a system of coupled fractional differential equations involving Hadamard derivatives. To prove the existence and uniqueness, we use the techniques of fixed point theory. Stability of Hyers-Ulam type is also discussed. Furthermore, we investigate variations of the problem in the context of different boundary conditions. The current results are verified by illustrative examples.Article Citation - WoS: 109Citation - Scopus: 113Applying New Fixed Point Theorems on Fractional and Ordinary Differential Equations(Springer, 2019) Karapinar, Erdal; Abdeljawad, Thabet; Jarad, FahdIn this paper, we consider a fixed point theorem that extends and unifies several existing results in the literature. We apply the proven fixed point results on the existence of solution of ordinary boundary value problems and fractional boundary value problems with integral type boundary conditions in the frame of some Caputo type fractional operators.Article Citation - WoS: 22Citation - Scopus: 28On Existence Results for Impulsive Fractional Neutral Stochastic Integro-Differential Equations With Nonlocal and State-Dependent Delay Conditions(Springer, 2016) Baleanu, Dumitru; Selvarasu, Siva; Arjunan, Mani Mallika; Kalamani, Palaniyappan; Mallika Arjunan, ManiThis manuscript deals with a new set of sufficient conditions for the existence of solutions for a class of impulsive fractional neutral stochastic integro-differential systems (IFNSIDS) with nonlocal conditions (NLCs) and state-dependent delay (SDD) in Hilbert spaces. An example is provided to illustrate the obtained theory.