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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