WoS İndeksli Yayınlar Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8653
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Article Citation - WoS: 26Citation - Scopus: 32On a More General Fractional Integration by Parts Formulae and Applications(Elsevier, 2019) Gomez-Aguilar, J. F.; Jarad, Fahd; Abdeljawad, Thabet; Atangana, AbdonThe integration by part comes from the product rule of classical differentiation and integration. The concept was adapted in fractional differential and integration and has several applications in control theory. However, the formulation in fractional calculus is the classical integral of a fractional derivative of a product of a fractional derivative of a given function f and a function g. We argue that, this formulation could be done using only fractional operators: thus, we develop fractional integration by parts for fractional integrals, Riemann-Liouville, Liouville-Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives. We allow the left and right fractional integrals of order alpha > 0 to act on the integrated terms instead of the usual integral and then make use of the fractional type Leibniz rules to formulate the integration by parts by means of new generalized type fractional operators with binomial coefficients defined for analytic functions. In the case alpha = 1, our formulae of fractional integration by parts results in previously obtained integration by parts in fractional calculus. The two disciplines or branches of mathematics are built differently, while classical differentiation is built with the concept of rate of change of a given function, a fractional differential operator is a convolution. (C) 2019 Elsevier B.V. All rights reserved.Article Citation - WoS: 23Citation - Scopus: 25Modeling the Fractional Non-Linear Schrodinger Equation Via Liouville-Caputo Fractional Derivative(Elsevier Gmbh, 2018) Morales-Delgado, V. F.; Gomez-Aguilar, J. F.; Taneco-Hernandez, M. A.; Baleanu, DumitruIn this paper the modified homotopy analysis transform method is applied to obtain approximate analytical solutions of the time-fractional non-linear Schrodinger equation. The fractional derivative is described in the Liouville-Caputo sense. The solutions are obtained in the form of rapidly convergent infinite series with easily computable terms. New exact solutions are constructed under constraint conditions. Employing theoretical parameters, we present some numerical simulations. (C) 2018 Elsevier GmbH. All rights reserved.Article Citation - WoS: 105Citation - Scopus: 107Beta-Derivative and Sub-Equation Method Applied To the Optical Solitons in Medium With Parabolic Law Nonlinearity and Higher Order Dispersion(Elsevier Gmbh, Urban & Fischer verlag, 2018) Gomez-Aguilar, J. F.; Baleanu, Dumitru; Yepez-Martinez, H.By using the sub-equation method, we construct the analytical solutions of the space-time generalized nonlinear Schrodinger equation involving the beta-derivative. This equation describing the propagation of ultra-short optical solitons through parabolic law medium. Nonlinear perturbations of higher-order and self-steepening terms are taken into account. As a result, some new exact solutions are constructed under constraint conditions. (C) 2017 Elsevier GmbH. All rights reserved.Article Citation - WoS: 11Citation - Scopus: 18A New Approach To Exact Optical Soliton Solutions for the Nonlinear Schrodinger Equation(Springer Heidelberg, 2018) Gomez-Aguilar, J. F.; Baleanu, Dumitru; Morales-Delgado, V. F.By using the modified homotopy analysis transform method, we construct the analytical solutions of the space-time generalized nonlinear Schrodinger equation involving a new fractional conformable derivative in the Liouville-Caputo sense and the fractional-order derivative with the Mittag-Leffler law. Employing theoretical parameters, we present some numerical simulations and compare the solutions obtained.