Well Conditioned Pseudospectral Schemes with Tunable Basis for Fractional Delay Differential Equations
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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.
KeywordsDelay differential equations Pseudospectral integration matrices Convergence analysis Fractional derivatives
Mathematics Subject Classification26A33 65L05 65L60 65L70
The authors would like to express their gratitude to the associate editor and the anonymous reviewers for their constructive comments, which shaped the paper into its final form. The work of the first author was partially supported by the Fundamental Research Funds for the Central Universities (Grant No. 3102016ZY003). The work of the second author was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC, RGPIN-2016-05386), and the National Natural Science Foundation of China (Grant No. 61473116). The work of the third author was partially supported by the National Natural Science Foundation of China (Grant No. 11472223), the Natural Science Foundation of Shaanxi Province (Grant No. 2016JM1015), and the Science and Technology Program of Shenzhen Government (Grant No. JCYJ20160331142601031).
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