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Optimal Petrov–Galerkin Spectral Approximation Method for the Fractional Diffusion, Advection, Reaction Equation on a Bounded Interval

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Abstract

In this paper we investigate the numerical approximation of the fractional diffusion, advection, reaction equation on a bounded interval. Recently the explicit form of the solution to this equation was obtained. Using the explicit form of the boundary behavior of the solution and Jacobi polynomials, a Petrov–Galerkin approximation scheme is proposed and analyzed. Numerical experiments are presented which support the theoretical results, and demonstrate the accuracy and optimal convergence of the approximation method.

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Acknowledgements

This work was partially funded by the ARO MURI Grant W911NF-15-1-0562, by the National Science Foundation under Grants DMS-1620194 and DMS-2012291, and by a SPARC Graduate Research Grant from the Office of the Vice President for Research at the University of South Carolina. All data generated or analyzed during this study are included in this published article.

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Authors and Affiliations

  1. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Xiangcheng Zheng & Hong Wang

  2. School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, 29634-0975, USA

    V. J. Ervin

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  1. Xiangcheng Zheng
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  2. V. J. Ervin
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  3. Hong Wang
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Correspondence to V. J. Ervin.

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Zheng, X., Ervin, V.J. & Wang, H. Optimal Petrov–Galerkin Spectral Approximation Method for the Fractional Diffusion, Advection, Reaction Equation on a Bounded Interval. J Sci Comput 86, 29 (2021). https://doi.org/10.1007/s10915-020-01366-y

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  • DOI: https://doi.org/10.1007/s10915-020-01366-y

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