Abstract
In this paper, we are concerned with the fractional sub-diffusion equation with a non-linear source term. The Legendre spectral collocation method is introduced together with the operational matrix of fractional derivatives (described in the Caputo sense) to solve the fractional sub-diffusion equation with a non-linear source term. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. In addition, the Legendre spectral collocation methods applied also for a solution of the fractional reaction sub-diffusion equation with a non-linear source term. For confirming the validity and accuracy of the numerical scheme proposed, two numerical examples with their approximate solutions are presented with comparisons between our numerical results and those obtained by other methods.