A Petrov-Galerkin Spectral Method Of Linear Multi-Term Fractional ODE

We present a new Petrov-Galerkin (PG) spectral method for solving linear multi-term fractional initial value problems with derivative orders at most one and constant coefficients. In this paper, we use Riemann-Liouville fractional derivatives. These schemes are based on a new spectral theory for fractional Sturm-Liouville problems (FSLPs). We obtain solutions to linear multi-term fractional initial value problems with new basis functions (non-polynomial) called Jacobi polyfractonomials, which are the eigenfunctions of the FSLP of first kind (FSLP-I). We introduce another space of test functions as the span of polyfractonomial eigenfunctions of the FSLP of second kind (FSLP-II). We extended this method for linear multi-term fractional initial value problems by Jacobi polyfractonomial basis on the closed interval. Finally, the resulting spectral method is used to solve some multi-term fractional differential equations‎.