Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations

We consider numerical approximation of the Riesz Fractional Differential Equations (FDEs), and construct a new set of generalized Jacobi functions, Jn−α,−α(x), which are tailored to the Riesz fractional PDEs. We develop optimal approximation results in non-uniformly weighted Sobolev spaces, and construct spectral Petrov-Galerkin algorithms to solve the Riesz FDEs with two kinds of boundary conditions (BCs): (i) homogeneous Dirichlet boundary conditions, and (ii) Integral BCs. We provide rigorous error analysis for our spectral Petrov-Galerkin methods, which show that the errors decay exponentially fast as long as the data (right-hand side function) is smooth, despite that fact that the solution has singularities at the endpoints. We also present some numerical results to validate our error analysis.