A generalized dusty Brinkman type fluid of MHD free convection two phase flow between parallel plates

•Non-uniform velocity and temperature conditions are imposed for Brinkman-type dusty fluid.•The Caputo-Fabrizio time-derivative is used to construct the fractional model.•Exact solutions are computed with the application of join Laplace and Fourier sin transform.•The fractional Brinkman type dusty fluid is more realistic feature as compared to the classical one.•The rate of heat transfer is boosting with the greater value of Pe and t.