Generalized Jacobi functions and their applications to fractional differential equations

Abstract. In this paper, we consider spectral approximation of fractional differential equations (FDEs). A main ingredient of our approach is to define a new class of generalized Jacobi func-tions (GJFs), which is intrinsically related to fractional calculus, and can serve as natural basis functions for properly designed spectral methods for FDEs. We establish spectral approximation results for these GJFs in weighted Sobolev spaces involving fractional derivatives. We construct efficient GJF-Petrov-Galerkin methods for a class of prototypical fractional initial value prob-lems (FIVPs) and fractional boundary value problems (FBVPs) of general order, and show that with an appropriate choice of the parameters in GJFs, the resulted linear systems are sparse and well-conditioned. Moreover, we derive error estimates with convergence rate only depending on the smoothness of data, so truly spectral accuracy can be attained if the data are smooth enough. The ideas and results presented in this paper will be useful to deal with more general FDEs involving Riemann-Liouville or Caputo fractional derivatives. 1