Asymptotic expansion method with respect to a small parameter for fractional differential equations with Riemann–Liouville derivate

In this paper, we proposed an asymptotic approach with respect to a small parameter for fractional differential equations, with the small parameter linked to the fractional order derivative. This approach allows the splitting of the field variable and consequently the Riemann-Liouville Integral as the sum of two contributions. One describes the unperturbed state and the other describes the behavior of the model in the perturbed state due to the presence of the fractional derivative. This approach allows mapping time-fractional differential equations into a system of two coupled partial differential equations with integer order. By solving these partial differential equations we are able to obtain solutions of the assigned fractional differential equations. To obtain solutions we determine the approximate Lie symmetries through which the system of two coupled partial differential equations is reduced to a system of two coupled ordinary differential equations. As an application, we consider the time fractional diffusion-reaction equation with linear source term; we test the proposed procedure by comparing the profiles of the obtained approximate solutions with solutions found in the fractional context for values of the fractional parameter tending to the integer value.