### Exact solutions of time fractional free convection flows of viscous fluid over an isothermal vertical plate with caputo and caputo-fabrizio derivatives

**Nehad Ali Shah, M. A. Imran, Fizza Miraj**

**Abstract:**The unsteady time fractional free convection flow of an incompressible Newtonian fluid over an infinite vertical plate due to an impulsive motion of the plate and constant temperature at the boundary is analyzed. The old (Caputo) and new (Caputo-Fabrizio) fractional derivative approaches have been used to develop a physical model and a comparison has been drawn between their solutions. Boundary layers equations in non dimensional form are solved analytically by the Laplace transform technique. Exact solutions for velocity and temperature are obtained in terms of Wrights function. The expressions for rate of heat transfer in both cases are also determined. Solutions for integer order derivatives are obtained as limiting case. Numerical computations were made through software Mathcad and observed some physical aspects of fractional and material parameters are presented. It is found that the rate of heat transfer of Caputo Fabrizio model have higher values than Caputo one as we increased the value of fractional parameter and fractional fluids tend to superpose to that of ordinary fluid.