This communication addresses a comparison of newly presented non-integer order derivatives with and without singular kernel, namely Michele Caputo-Mauro Fabrizio (CF) CF ∂ β /∂t β and Atangana-Baleanu (AB) AB (∂ α /∂t α ) fractional derivatives. For this purpose, second grade fluids flow with combined gradients of mass concentration and temperature distribution over a vertical flat plate is considered. The problem is first written in non-dimensional form and then based on AB and CF fractional derivatives, it is developed in fractional form, and then using the Laplace transform technique, exact solutions are established for both cases of AB and CF derivatives. They are then expressed in terms of newly defined M-function M p q (z) and generalized Hyper-geometric function p Ψ q (z). The obtained exact solutions are plotted graphically for several pertinent parameters and an interesting comparison is made between AB and CF derivatives results with various similarities and differences.
Atangana-Baleanu fractional derivative has been applied to study heat transfer problem of magnetohydrodynamic (MHD) Maxwell fluid over a vertical plate embedded in a porous medium. The analytical solutions have been obtained for temperature distribution and velocity field by employing Laplace transforms technique for both sine and cosine oscillations of the plate. The general solutions have been expressed in terms of Fox-H function satisfying imposed conditions. The results are plotted graphically and discussed for embedded parameters such as magnetic field, Maxwell parameter, porous medium, Prandtl number and fractional parameter.
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