Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation

Here, we provide a new method to solve the time-fractional diffusion equation (TFDE) following the spectral tau approach. Our proposed numerical solution is expressed in terms of a double Lucas expansion. The discretization of the technique is based on several formulas about Lucas polynomials, such as those for explicit integer and fractional derivatives, products, and certain definite integrals of these polynomials. These formulas aid in transforming the TFDE and its conditions into a matrix system that can be treated through a suitable numerical procedure. We conduct a study on the convergence analysis of the double Lucas expansion. In addition, we provide a few examples to ensure that the proposed numerical approach is applicable and efficient.