Chaos, Solitons and Fractals 117 (2018) 68–75 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos Stokes’ first problem for heated flat plate with Atangana–Baleanu fractional derivative Ndolane Sene Laboratoire Lmdan, Département de Mathématiques de la Décision, Université Cheikh Anta Diop de Dakar, Faculté des Sciences Economiques et Gestion, Dakar Fann BP 5683, Senegal a r t i c l e i n f o Article history: Received 20 July 2018 Revised 7 October 2018 Accepted 8 October 2018 Keywords: Stokes’ first problem Atangana–Baleanu fractional derivative Newtonian fluid a b s t r a c t In this paper, we investigate on the exact solutions of the Stokes’ first problem for generalized second grade fluid with a new fractional derivative operator. The Riemann–Liouville and the Caputo fractional derivative are substituted by the Atangana–Baleanu fractional derivative in the Stokes’ first fractional dif- ferential equation. With the Laplace transform given by the Atangana–Baleanu fractional derivative oper- ator, we give for the Stokes’ first fractional differential equations the exact solutions for the velocity and temperature field. The Fourier sine transform and the Laplace transform will be used to get the exact solutions of the Stokes’ first fractional differential equations. The solutions of the Stokes’ first differen- tial equations for a viscous Newtonian fluid, as well as those corresponding to a second grade fluid, are obtained in limiting cases, and an approach with the graphical surfaces representations for the exact solutions is proposed. © 2018 Elsevier Ltd. All rights reserved. 1. Introduction Fractional calculus has attracted many researchers in this last decade due to its applications in science and engineering. Recently, Sun and al. have given a complete collection of real world appli- cations of fractional calculus in science and engineering, see in [33]. There exists in the literature various types of fractional dif- ferential equations. In recent years many authors were interested by the Stokes’ and Rayleigh–Stokes problems in first and second grade fluid. The Rayleigh–Stokes fractional differential equation is a special case of the Navier–Stokes problems. The viscoelastic fluids continue to attract many mathematicians due to its applications in various fields in science and engineering. Some examples of ap- plications: in extraction of polymer fluid, in cooling of a metallic plate in a bath, in exotic lubricants, in artificial and material gels, colloidal and suspension solutions [6,35–37]. Maxwell was the first to introduce the viscoelastic model on fluids to describe the responses of some polymers liquids. The con- structive equations used in the viscoelastic models use ordinary derivative [11]. Recently, fractional derivatives operators such as the Riemann–Liouville fractional derivative and the Caputo frac- tional derivative [26] were introduced in viscoelastic constructive equations [6,9,15,30,32]. And many investigations to get the exact solutions of the Rayleigh and Stokes’ fractional differential equa- E-mail address: ndolane.sene@ucad.edu.sn tions in first and second grade fluids were done, see the papers enumerated as follows [9,30,31]. For Newtonian fluids: the exact solutions were studied by Stokes’ in 1851 and later by Rayleigh in 1911. the exact solutions to Stokes’ first problem for Maxwell fluid were studied by Tanner in 1962 in [10,34]. Rayleigh–Stokes’s problem is non-Newtonian problem and has received much interest due to its applications in industry, chemical and petroleum engineering, see illustration in [11,30,39]. The flat plate viscoelastic fluid was studied by Taipel et al. in 1981, Stokes first problem for viscoelastic fluid studied with Joseph et al. in [21], Stokes’ first problem for Non-Newtonian fluid with different constructive models studied by Bandelli et al. in [7], see also in [24]. The exact solutions for some Rayleigh– Stokes problems were investigated by Khan in [19], Salim et al. in [25], Fatecau et al. in [12–14], Fan Shen et al. in [30], Xue et al. [38], and others. We introduce the Atangana–Baleanu frac- tional derivative [3] in constructive viscoelastic equations to study the Stokes’ problem for generalized second grade fluid. In partic- ular, we particularly investigate to find the exact solutions of the Stokes’ first fractional differential equation. The Resultant motions of the fluid are studied by many authors for various types of fluid. Other investigations on the exact solutions exist in the literature, see above references. Many advancements with the Atangana–Baleanu fractional derivative were recently done. This fractional derivative opera- tor has several applications. In [18] Giusti has done some com- ments related to the Atangana–Baleanu fractional derivative, in https://doi.org/10.1016/j.chaos.2018.10.014 0960-0779/© 2018 Elsevier Ltd. All rights reserved.