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Generalizations of the derivative
About: Generalizations of the derivative is a research topic. Over the lifetime, 653 publications have been published within this topic receiving 14601 citations.
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TL;DR: In this paper, a general theory of non-commutative differential geometry on quantum groups is developed, where bicovariant bimodules as objects analogous to tensor bundles over Lie groups are studied.
Abstract: The paper deals with non-commutative differential geometry. The general theory of differential calculus on quantum groups is developed. Bicovariant bimodules as objects analogous to tensor bundles over Lie groups are studied. Tensor algebra and external algebra constructions are described. It is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra, so the external derivative of higher order differential forms is well defined and obeys the usual properties. The proper form of the Cartan Maurer formula is found. The vector space dual to the space of left-invariant differential forms is endowed with a bilinear operation playing the role of the Lie bracket (commutator). Generalized antisymmetry relation and Jacobi identity are proved.
1,248 citations
TL;DR: In this paper, the topological derivative for an arbitrary shape functional is defined and examples are provided for elliptic equations and the elasticity system in the plane, which can be used for solving shape optimization problems in structural mechanics.
Abstract: In this paper the topological derivative for an arbitrary shape functional is defined. Examples are provided for elliptic equations and the elasticity system in the plane. The topological derivative can be used for solving shape optimization problems in structural mechanics.
871 citations
TL;DR: In this paper, Atangana and Baleanu proposed a derivative with fractional order to answer some outstanding questions that were posed by many researchers within the field of fractional calculus.
Abstract: Recently, Atangana and Baleanu proposed a derivative with fractional order to answer some outstanding questions that were posed by many researchers within the field of fractional calculus. Their derivative has a non-singular and nonlocal kernel. In this paper, we presented further relationship of their derivatives with some integral transform operators. New results are presented. We applied this derivative to a simple nonlinear system. We show in detail the existence and uniqueness of the system solutions of the fractional system. We obtain a chaotic behavior which was not obtained by local derivative.
683 citations
TL;DR: A numerical method, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented and the efficiency and applicability of the method are shown.
617 citations
TL;DR: In this paper, the authors presented an efficient and reliable treatment of the homotopy perturbation method (HPM) for nonlinear partial differential equations with fractional time derivative, described in the Caputo sense.
368 citations