In this paper, we introduce the nabla fractional derivative and fractional integral on time scales in the Riemann–Liouville sense. We also introduce the nabla fractional derivative in Grünwald–Letnikov sense. Some of the basic properties and theorems related to nabla fractional calculus are discussed.
In this manuscript we investigate the existence and uniqueness of an impulsive fractional dynamic equation on time scales involving non-local initial condition with help of Caputo nabla derivative. The existency is based on the Scheafer's fixed point theorem along with the Arzela-Ascoli theorem and Banach contraction theorem. The comparison of the Caputo nabla derivative and Riemann-Liouvile nabla derivative of fractional order are also discussed in the context of time scale.
In this paper, a numerical investigation is presented for non-integer order derivatives with Atangana-Baleanu (AB) and Caputo-Fabrizio (CF) fractional derivatives for the variable viscosity and thermal conductivity over a moving vertical plate in a porous medium two dimensional free convection unsteady MHD flow. The effects of radiation have also been considered. The governing partial differential equations along with the boundary conditions are changed to ordinary form by similarity transformations. Hence physical parameters show up in the equations and interpretations on these parameters can be achieved suitably.By using ordinary finite difference scheme the equations are discritized and developed in fractional form. These discritized equations are numerically solved by the approach based on Gauss-seidel iteration scheme. Some numerical strategies are used to find the values of AB and CF approaches on time by developing programming code in MATLAB. The effects of all the physical parameters involved in the problem on velocity, temperature and concentration distribution are compared graphically as well as in tabular form. The effects of each parameter are found to be prominent. We have observed a significant variation of values under different parameters using AB and CF approaches on velocity, temperature and concentration distribution with respect to time.
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