Spectral and Discontinuous Spectral Element Methods for Fractional Delay Equations

We first develop a spectrally accurate Petrov--Galerkin spectral method for fractional delay differential equations (FDDEs). This scheme is developed based on a new spectral theory for fractional Sturm--Liouville problems (FSLPs), which has been recently presented in [M. Zayernouri and G. E. Karniadakis, J. Comput. Phys. , 252 (2013), pp. 495--517]. Specifically, we obtain solutions to FDDEs in terms of new nonpolynomial basis functions, called Jacobi polyfractonomials , which are the eigenfunctions of the FSLP of the first kind (FSLP-I). Correspondingly, we employ another space of test functions as the span of polyfractonomial eigenfunctions of the FSLP of the second kind (FSLP-II). We prove the wellposedness of the problem and carry out the corresponding stability and error analysis of the PG spectral method. In contrast to standard (nondelay) fractional differential equations, the delay character of FDDEs might induce solutions, which are either nonsmooth or piecewise smooth. In order to effectively treat such cases, we first develop a discontinuous spectral method (DSM) of Petrov--Galerkin type for FDDEs, where the basis functions do not satisfy the initial conditions. Consequently, we extend the DSM scheme to a discontinuous spectral element method (DSEM) for possible adaptive refinement and long time-integration. In DSM and DSEM schemes, we employ the asymptotic eigensolutions to FSLP-I and FSLP-II, which are of Jacobi polynomial form, both as basis and test functions. Our numerical tests demonstrate spectral convergence for a wide range of FDDE model problems with different benchmark solutions.