Abstract
In a recent paper we showed numerically and theoretically that a straightforward generalisation of Alikhanov’s “L2-\(1_\sigma \)” scheme is \(O(M^{-2})\) accurate on suitably chosen graded meshes (with M time intervals) for initial-value problems (IVPs) and initial-boundary value problems (IBVPs) with a Caputo fractional time derivative of order \(\alpha \), whose solutions typically exhibit a weak singularity at the initial time \(t=0\). The present paper constructs a better generalisation of Alikhanov’s scheme that is demonstrated numerically to be \(O(M^{-(3-\alpha )})\) accurate for these classes of IVPs and IBVPs, but its rigorous analysis remains an open problem.
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References
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Acknowledgements
The work of the first author was funded by the Chinese Postdoc Foundation Grant 2018M631316 and the National Natural Science Foundation of China young scientists fund Grant 11801026. The work of the second author was funded by the National Natural Science Foundation of China under grants 91430216 and NSAF-U1530401.
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Chen, H., Stynes, M. (2019). A High Order Method on Graded Meshes for a Time-Fractional Diffusion Problem. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods. Theory and Applications. FDM 2018. Lecture Notes in Computer Science(), vol 11386. Springer, Cham. https://doi.org/10.1007/978-3-030-11539-5_2
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DOI: https://doi.org/10.1007/978-3-030-11539-5_2
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