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In this paper we construct a new difference analog of the Caputo fractional derivative (called the L 2 - 1 σ formula). The basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and fourth order in space and the second order in time for the time fractional diffusion equation with variable coefficients are considered. Stability of the suggested schemes and also their convergence in the grid L 2 -norm with the rate equal to the order of the approximation error are proved. The obtained results are supported by the numerical calculations carried out for some test problems. [ABSTRACT FROM AUTHOR]

Fourier extensions have been shown to be an effective means for the approximation of smooth, nonperiodic functions on bounded intervals given their values on an equispaced, or in general, scattered grid. Related to this method are two parameters. These are the extension parameter T (the ratio of the size of the extended domain to the physical domain) and the oversampling ratio η (the number of sampling nodes per Fourier mode). The purpose of this paper is to investigate how the choice of these parameters affects the accuracy and stability of the approximation. Our main contribution is to document the following interesting phenomenon: namely, if the desired condition number of the algorithm is fixed in advance, then the particular choice of such parameters makes little difference to the algorithm's accuracy. As a result, one is free to choose T without concern that it is suboptimal. In particular, one may use the value T=2 - which corresponds to the case where the extended domain is precisely twice the size of the physical domain - for which there is known to be a fast algorithm for computing the approximation. In addition, we determine the resolution power (points-per-wavelength) of the approximation to be equal to Tη, and address the trade-off between resolution power and stability. [ABSTRACT FROM AUTHOR]

Abstract: High-order compact finite difference scheme with operator splitting technique for solving two-dimensional time fractional diffusion equation is considered in this paper. A Grünwald–Letnikov approximation is used for the Riemann–Liouville time derivative, and the second order spatial derivatives are approximated by the compact finite differences to obtain a fully discrete implicit scheme. Alternating direction implicit (ADI) method is used to split the original problem into two separate one-dimensional problems. The local truncation error is analyzed and the stability is discussed by the Fourier method. The proposed scheme is suitable when the order of the time fractional derivative γ lies in the interval . A correction term is added to maintain high accuracy when . Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, a compact finite difference scheme for the fractional sub-diffusion equations is derived. After a transformation of the original problem, the L1 discretization is applied for the time-fractional part and fourth-order accuracy compact approximation for the second-order space derivative. The unique solvability of the difference solution is discussed. The stability and convergence of the finite difference scheme in maximum norm are proved using the energy method, where a new inner product is introduced for the theoretical analysis. The technique is quite novel and different from previous analytical methods. Finally, a numerical example is provided to show the effectiveness and accuracy of the method. [ABSTRACT FROM AUTHOR]

Abstract: High-order compact finite difference scheme for solving one-dimensional fractional diffusion equation is considered in this paper. After approximating the second-order derivative with respect to space by the compact finite difference, we use the Grünwald–Letnikov discretization of the Riemann–Liouville derivative to obtain a fully discrete implicit scheme. We analyze the local truncation error and discuss the stability using the Fourier method, then we prove that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis. Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm. [Copyright &y& Elsevier]