DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2020021 DYNAMICAL SYSTEMS SERIES S MHD FLOW OF FRACTIONAL NEWTONIAN FLUID EMBEDDED IN A POROUS MEDIUM VIA ATANGANA-BALEANU FRACTIONAL DERIVATIVES Kashif Ali Abro Department of Basic Sciences and Related Studies Mehran University of Engineering and Technology Jamshoro, Pakistan Ilyas Khan ∗ Faculty of Mathematics and Statistics Ton Duc Thang University Ho Chi Minh City, Vietnam Abstract. The novelty of this research is to utilize the modern approach of Atangana-Baleanu fractional derivative to electrically conducting viscous fluid embedded in porous medium. The mathematical modeling of the governing partial differential equations is characterized via non-singular and non-local kernel. The set of governing fractional partial differential equations is solved by employing Laplace transform technique. The analytic solutions are investigated for the velocity field corresponding with shear stress and expressed in term of special function namely Fox-H function, moreover a comparative study with an ordinary and Atangana-Baleanu fractional models is analyzed for viscous flow in presence and absence of magnetic field and porous medium. The Atangana- Baleanu fractional derivative is observed more reliable and appropriate for handling mathematical calculations of obtained solutions. Finally, graphical illustration is depicted via embedded rheological parameters and comparison of models plotted for smaller and larger time on the fluid flow. 1. Introduction. It is well established fact that the idea of fractional calculus has diverted the attention of scientists and researchers because of its crucial applica- tions for the descriptions of complex system and in different academic disciplines. The modeling via integer-order derivatives does not provide better prediction in comparison to modeling via fractional-order derivatives in the real-world problems. The modelling of physical phenomena via fractional-order derivatives is significant for the control theory [44], pharmacokinetics [42], electrical engineering [22], [25], anomalous diffusion [14], [2], [19], [23], fluids [10], [26], [40], [41], electromagnetism [11], [27], [32], [28], [33], heat transfer [13], [3], [34]. Although fractional derivatives are more suitable than ordinary derivatives to describe a physical phenomenon yet 2010 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35. Key words and phrases. Atangana-Baleanu fractional derivative, magnetohydrodynamics, porous medium, rheological effects. The author Kashif Ali Abro is highly thankful and grateful to Mehran University of Engineering and Technology, Jamshoro, Pakistan, for generous support and facilities of this research work. * Corresponding author:Ilyas Khan (ilyaskhan@tdt.edu.vn). 377