High-order BDF convolution quadrature for fractional evolution equations with hyper-singular source term
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Anomalous diffusion in the presence or absence of an external force field is often modelled in terms of the fractional evolution equations, which can involve the hyper-singular source term. For this case, conventional time stepping methods may exhibit a severe order reduction. Although a second-order numerical algorithm is provided for the subdiffusion model with a simple hyper-singular source term t^μ, -2<μ<-1 in [arXiv:2207.08447], the convergence analysis remain to be proved. To fill in these gaps, we present a simple and robust smoothing method for the hyper-singular source term, where the Hadamard finite-part integral is introduced. This method is based on the smoothing/IDm-BDFk method proposed by the authors [Shi and Chen, SIAM J. Numer. Anal., to appear] for subdiffusion equation with a weakly singular source term. We prove that the kth-order convergence rate can be restored for the diffusion-wave case γ∈ (1,2) and sketch the proof for the subdiffusion case γ∈ (0,1), even if the source term is hyper-singular and the initial data is not compatible. Numerical experiments are provided to confirm the theoretical results.
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