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Generalized Jacobi functions and their applications to fractional differential equations


Authors: Sheng Chen, Jie Shen and Li-Lian Wang
Journal: Math. Comp. 85 (2016), 1603-1638
MSC (2010): Primary 65N35, 65E05, 65M70, 41A05, 41A10, 41A25
Published electronically: October 1, 2015
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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 functions (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 problems (FIVPs) and fractional boundary value problems (FBVPs) of general order, and we show that with an appropriate choice of the parameters in GJFs, the resulting linear systems are sparse and well-conditioned. Moreover, we derive error estimates with convergence rates only depending on the smoothness of data, so true spectral accuracy can be attained if the data are smooth enough. The ideas and results presented in this paper will be useful in dealing with more general FDEs involving Riemann-Liouville or Caputo fractional derivatives.


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Additional Information

Sheng Chen
Affiliation: School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Xiamen, Fujian 361005, People’s Republic of China

Jie Shen
Affiliation: School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Xiamen, Fujian 361005, People’s Republic of China – and Department of Mathematics, Purdue University, West Lafayette, IN 47907-1957

Li-Lian Wang
Affiliation: Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore

DOI: https://doi.org/10.1090/mcom3035
Keywords: Fractional differential equations, singularity, Jacobi polynomials with real parameters, generalized Jacobi functions, weighted Sobolev spaces, approximation results, spectral accuracy
Received by editor(s): July 30, 2014
Received by editor(s) in revised form: December 26, 2014, and January 4, 2015
Published electronically: October 1, 2015
Additional Notes: The first author was partially supported by NSFC grants 91130002 and 11371298.
The second author was partially supported by AFOSR grant FA9550-11-1-0328 and NSF grant DMS-1217066.
The third author was partially supported by Singapore MOE AcRF Tier 1 Grant (RG 15/12), Singapore MOE AcRF Tier 2 Grant (MOE 2013-T2-1-095, ARC 44/13) and Singapore A$^{∗}$STAR-SERC-PSF Grant (122-PSF-007)
Article copyright: © Copyright 2015 American Mathematical Society