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As is well-known, the advantage of the high-order compact difference scheme (H-OCD) is that it is unconditionally stable and convergent on the order O ( &tau, 2 + h 4 ) (where &tau, is the time step size and h is the mesh size), under the maximum norm for a class of nonlinear delay partial differential equations with initial and Dirichlet boundary conditions. In this article, a new numerical gradient scheme based on the collocation polynomial and Hermite interpolation is presented. The convergence order of this kind of method is also O ( &tau, 2 + h 4 ) under the discrete maximum norm when the spatial step size is twice the one of H-OCD, which accelerates the computational process. In addition, some corresponding analyses are made and the Richardson extrapolation technique is also considered in the time direction. The results of numerical experiments are consistent with the theoretical analysis.

Fractional diffusion equations (FDEs) have recently been paid much attention. Finding accurate and efficient methods for solving FDEs has become an active research undertaking. In this paper, an efficient method based on the shifted Chebyshev-tau idea is presented to solve an initial-boundary value problem for the FDEs. The method is derived by expanding the required approximate solution as the elements of shifted Chebyshev polynomials. Using the operational matrix of the fractional derivative, the problem can be reduced to a set of linear algebraic equations. From a computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and only a small number of shifted Chebyshev polynomials is needed.

In the paper, some new upper bounds for the spectral radius of the Hadamard product of nonnegative matrices, and the low bounds for the minimum eigenvalue of the Fan product of nonsingular M-matrices are given. These new bounds improve existing results, and the estimating formulas are easier to calculate since they only depend on the entries of matrices. Finally, some examples are also given to show that the bounds are better than some previous results.

A general algorithm to derive the transfer functions of M-D discrete systems has been presented. The given M-D discrete system with arbitrary order and arbitrary topology structure can be analyzed. We construct the algorithm by nodal equations, M-D DFT, M-D signal flow graph. Resorting to the algorithm, we develop computer aided analysis software for deriving the transfer function of a M-D discrete system. An example for analyzing a 2-D lattice digital filter shows our algorithm to be valid.