Well Conditioned Pseudospectral Schemes with Tunable Basis for Fractional Delay Differential Equations

Abstract

The main purpose of this work is to develop spectrally accurate and well conditioned pseudospectral schemes for solving fractional delay differential equations (FDDEs). The essential idea is to recast FDDEs into fractional integral equations (FIEs) and then discretize the FIEs via generalized fractional pseudospectral integration matrices (GFPIMs). We construct GFPIMs by employing the basis of weighted Lagrange interpolating functions, and provide an exact, efficient, and stable approach to computing GFPIMs. The GFPIM schemes have two remarkable features: (i) the endpoint singularity of the solution to FDDEs can be effectively captured via the tunable basis, and (ii) the linear system resulting from pseudospectral discretization is well conditioned. We also provide a rigorous convergence analysis for the particular FPIM schemes via a linear FIE with any $$\gamma >0$$ where $$\gamma $$ is the order of fractional integrals. Numerical results on benchmark FDDEs with smooth/singular solutions demonstrate the spectral rate of convergence for the GFPIM schemes. For FDDEs with piecewise smooth solutions, the GFPIM schemes can obtain accurate solutions but converge slowly due to their essential feature of “global” approximation on the entire time interval.