Abstract
In this article, we study a model problem for the advection-reaction-diffusion equation involving a new nonsingular time-fractional derivative with Rabotnov fractional-exponential (RFE) kernel. In order to solve this model numerically, we first obtain the numerical approximation of RFE fractional derivative for a simple polynomial function, which gives rise to an operational matrix of fractional differentiation. We illustrate the accuracy and validity of this operational matrix with the aid of an example. We use Legendre collocation technique together with the newly developed operational matrix to find the numerical solution of the given model. The numerical results depict the feasibility and efficacy of our method. The error estimates show that our method is valid with great accuracy and is applicable to a fractional ODE system and an integral equation with RFE kernel fractional derivative.