A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation

This study considers the solution of a class of linear systems related with the fractional Poisson equation (FPE) (−∇2)α/2φ=g(x,y) with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to generate a linear system of equations with its matrix A raised to the fractional power α/2. The solution of the linear system then requires the action of the matrix function f(A)=A−α/2 on a vector b. For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f(A)b≈β0Vmf(Tm)e1. This method works well when both the analytic grade of A with respect to b and the residual for the linear system are sufficiently small. Memory constraints often require restarting the Lanczos decomposition; however this is not straightforward in the context of matrix function approximation. In this paper, we use the idea of thick-restart and adaptive preconditioning for solving linear systems to improve convergence of the Lanczos approximation. We give an error bound for the new method and illustrate its role in solving FPE. Numerical results are provided to gauge the performance of the proposed method relative to exact analytic solutions.

A novel nonrepudiable threshold multi-proxy multi-signature scheme with shared verification

Tzeng et al. proposed a new threshold multi-proxy multi-signature scheme with threshold verification. In their scheme, a subset of original signers authenticates a designated proxy group to sign on behalf of the original group. A message m has to be signed by a subset of proxy signers who can represent the proxy group. Then, the proxy signature is sent to the verifier group. A subset of verifiers in the verifier group can also represent the group to authenticate the proxy signature. Subsequently, there are two improved schemes to eliminate the security leak of Tzeng et al.’s scheme. In this paper, we have pointed out the security leakage of the three schemes and further proposed a novel threshold multi-proxy multi-signature scheme with threshold verification.

Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term

In this paper, we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term. A new implicit difference method is constructed. The stability and convergence are discussed using a new energy method. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis

Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term

In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moreover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.

Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation

Anomalous dynamics in complex systems have gained much interest in recent years. In this paper, a two-dimensional anomalous subdiffusion equation (2D-ASDE) is considered. Two numerical methods for solving the 2D-ASDE are presented. Their stability, convergence and solvability are discussed. A new multivariate extrapolation is introduced to improve the accuracy. Finally, numerical examples are given to demonstrate the effectiveness of the schemes and confirm the theoretical analysis.

Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process

In this paper, we consider the following non-linear fractional reaction–subdiffusion process (NFR-SubDP): Formula where f(u, x, t) is a linear function of u, the function g(u, x, t) satisfies the Lipschitz condition and 0Dt1–{gamma} is the Riemann–Liouville time fractional partial derivative of order 1 – {gamma}. We propose a new computationally efficient numerical technique to simulate the process. Firstly, the NFR-SubDP is decoupled, which is equivalent to solving a non-linear fractional reaction–subdiffusion equation (NFR-SubDE). Secondly, we propose an implicit numerical method to approximate the NFR-SubDE. Thirdly, the stability and convergence of the method are discussed using a new energy method. Finally, some numerical examples are presented to show the application of the present technique. This method and supporting theoretical results can also be applied to fractional integrodifferential equations.

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order αset membership, variant(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order βset membership, variant(0,1) and of order αset membership, variant(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.

Numerical investigations of linear least squares methods for derivative estimation

The results of a numerical investigation into the errors for least squares estimates of function gradients are presented. The underlying algorithm is obtained by constructing a least squares problem using a truncated Taylor expansion. An error bound associated with this method contains in its numerator terms related to the Taylor series remainder, while its denominator contains the smallest singular value of the least squares matrix. Perhaps for this reason the error bounds are often found to be pessimistic by several orders of magnitude. The circumstance under which these poor estimates arise is elucidated and an empirical correction of the theoretical error bounds is conjectured and investigated numerically. This is followed by an indication of how the conjecture is supported by a rigorous argument.

Simplified numerical models of LiteSteel Beam floor joists with Web openings for the prediction of section moment capacities

This paper is aimed at investigating the effect of web openings on the plastic bending behaviour and section moment capacity of a new cold-formed steel beam known as LiteSteel beam (LSB) using numerical modelling. Different LSB sections with varying circular hole diameter and spacing were considered. A simplified but appropriate numerical modelling technique was developed for the modelling of monosymmetric sections such as LSBs subject to bending, and was used to simulate a series of section moment capacity tests of LSB flexural members with web openings. The buckling and ultimate strength behaviour was investigated in detail and the modeling technique was further improved through a comparison of numerical and experimental results. This paper describes the simplified finite element modeling technique used in this study that includes all the significant behavioural effects affecting the plastic bending behaviour and section moment capacity of LSB sections with web holes. Numerical and test results and associated findings are also presented.

Numerical and experimental investigation of wedge tip radius effect on wedge plasmons

We report numerical analysis and experimental observation of strongly localized plasmons guided by triangular metal wedges and pay special attention to the effect of smooth (nonzero radius) tips. Dispersion, dissipation, and field structure of such wedge plasmons are analyzed using the compact two-dimensional finite-difference time-domain algorithm. Experimental observation is conducted by the end-fire excitation and near-field scanning optical microscope detection of the predicted plasmons on 40°silver nanowedges with the wedge tip radii of 20, 85, and 125 nm that were fabricated by the focused-ion beam method. The effect of smoothing wedge tips is shown to be similar to that of increasing wedge angle. Increasing wedge angle or wedge tip radius results in increasing propagation distance at the same time as decreasing field localization (decreasing wave number). Quantitative differences between the theoretical and experimental propagation distances are suggested to be due to a contribution of scattered bulk and surface waves near the excitation region as well as the addition of losses due to surface roughness. The theoretical and measured propagation distances are several plasmon wavelengths and are useful for a range of nano-optical applications