A numerical method for space-time fractional advection-reaction-diffusion problems with generalized Caputo-type derivative

In this study, the generalized Caputo-type fractional derivative is applied to construct a family of fractional advection-reaction-diffusion equations. The orthonormal discrete Chebyshev polynomials are considered to construct a computational method for these equations. The proposed method uses the approximation of the unknown solution by these polynomials and it employs the fractional derivative of these polynomials (which is obtained in this study) along with the collocation method to derive an algebraic system of equations. A smooth solution in terms of the mentioned polynomials is obtained for the introduced fractional problem by solving this system. The implementation and numerical simulations of the method are very simple. Three test problems have been studied to investigate the numerical accuracy of the method.