• An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data

      Yan, Yubin; Khan, Monzorul; Ford, Neville J.; University of Chester (Society for Industrial and Applied Mathematics, 2018-01-11)
      We introduce a modified L1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Jin \et (2016, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. of Numer. Anal., 36, 197-221) established an $O(k)$ convergence rate for the L1 scheme for smooth and nonsmooth initial data for the homogeneous problem, where $k$ denotes the time step size. We show that the modified L1 scheme has convergence rate $O(k^{2-\alpha}), 0< \alpha <1$ for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the numerical results are consistent with the theoretical results.
    • Combining Kronecker product approximation with discrete wavelet transforms to solve dense, function-related linear systems

      Ford, Judith M.; Tyrtyshnikov, Eugene E.; Chester College of Higher Education ; Russian Academy of Sciences (Society for Industrial and Applied Mathematics, 2003-11)
    • An improved discrete wavelet transform preconditioner for dense matrix problems

      Ford, Judith M.; Chester College of Higher Education (Society for Industrial and Applied Mathematics, 2003-12)
    • On the decay of the elements of inverse triangular Toeplitz matrices

      Ford, Neville J.; Savostyanov, Dmitry V.; Zamarashkin, Nickolai L.; University of Chester ; University of Chester ; Institute of Numerical Mathematics, Russian Academy of Sciences (Society for Industrial and Applied Mathematics, 2014-10-28)
      We consider half–infinite triangular Toeplitz matrices with slow decay of the elements and prove under a monotonicity condition that the elements of the inverse matrix, as well as the elements of the fundamental matrix, decay to zero. We provide a quantitative description of the decay of the fundamental matrix in terms of p–norms. The results add to the classical results of Jaffard and Vecchio, and are illustrated by numerical examples.