Generalized Exponential Time Differencing Schemes for Stiff Fractional Systems with Nonsmooth Source Term | Journal of Scientific Computing
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Generalized Exponential Time Differencing Schemes for Stiff Fractional Systems with Nonsmooth Source Term

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Abstract

Many processes in science and engineering are described by fractional systems which may in general be stiff and involve a nonsmooth source term. In this paper, we develop robust first, second, and third order accurate exponential time differencing schemes for solving such systems. Rather than imposing regularity requirements on the solution to account for the singularity caused by the fractional derivative, we only consider regularity requirements on the source term for preserving the optimal order of accuracy of the proposed schemes. Optimal convergence rates are proved for both smooth and nonsmooth source terms using uniform and graded meshes, respectively. For efficient implementation, high-order global Padé approximations together with their fractional decompositions are developed for Mittag–Leffler functions. We present numerical experiments involving a typical stiff system, a fractional two-compartment pharmacokinetics model, a two-term fractional Kelvin–Viogt model of viscoelasticity, and a large system obtained by spatial discretization of a sub-diffusion problem. Demonstrations of the efficiency of the rational approximation implementation technique and the newly constructed high-order schemes are provided.

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Acknowledgements

The authors would like to acknowledge the support provided by the Deanship of Scientific Research at King Fahd University of Petroleum and Minerals via the Project SB191001. We thank the anonymous reviewers for their valuable suggestions and comments.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Ibrahim O. Sarumi, Khaled M. Furati & Kassem Mustapha

  2. Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN, 37132-0001, USA

    Abdul Q. M. Khaliq

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  1. Ibrahim O. Sarumi
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  2. Khaled M. Furati
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  3. Abdul Q. M. Khaliq
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  4. Kassem Mustapha
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Correspondence to Khaled M. Furati.

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Sarumi, I.O., Furati, K.M., Khaliq, A.Q.M. et al. Generalized Exponential Time Differencing Schemes for Stiff Fractional Systems with Nonsmooth Source Term. J Sci Comput 86, 23 (2021). https://doi.org/10.1007/s10915-020-01374-y

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  • DOI: https://doi.org/10.1007/s10915-020-01374-y

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