The unified theory of shifted convolution quadrature for fractional calculus
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The convolution quadrature theory is a systematic approach to analyse the approximation of the Riemann-Liouville fractional operator I^α at node x_n. In this paper, we develop the shifted convolution quadrature (SCQ) theory which generalizes the theory of convolution quadrature by introducing a shifted parameter θ to cover as many numerical schemes that approximate the operator I^α with an integer convergence rate as possible. The constraint on the parameter θ is discussed in detail and the phenomenon of superconvergence for some schemes is examined from a new perspective. For some technique purposes when analysing the stability or convergence estimates of a method applied to PDEs, we design some novel formulas with desired properties under the framework of the SCQ. Finally, we conduct some numerical tests with nonsmooth solutions to further confirm our theory.
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