A Fast Preconditioned Semi-Implicit Difference Scheme for Strongly Nonlinear Space-Fractional Diffusion Equations

In this paper, we propose a semi-implicit difference scheme for solving one-dimensional nonlinear space-fractional diffusion equations. The method is first-order accurate in time and second-order accurate in space. It uses a fractional central difference formula and the backward Euler method to approximate its space and time derivatives, respectively. Stability and convergence properties of the proposed scheme are proved with the help of a discrete Grönwall inequality. Moreover, we extend the method to the solution of two-dimensional nonlinear models. A fast matrix-free implementation based on preconditioned Krylov subspace methods is presented for solving the discretized linear systems. The resulting fast preconditioned semi-implicit difference scheme reduces the memory requirement of conventional semi-implicit difference schemes from O(Ns2) to O(Ns) and the computational complexity from O(Ns3) to O(NslogNs) in each iterative step, where Ns is the number of space grid points. Experiments with two numerical examples are shown to support the theoretical findings and to illustrate the efficiency of our proposed method.

On a discrete fractional stochastic Grönwall inequality and its application in the numerical analysis of stochastic FDEs involving a martingale

The aim of this paper is to derive a novel discrete form of stochastic fractional Grönwall lemma involving a martingale. The proof of the derived inequality is accomplished by a corresponding no randomness form of the discrete fractional Grönwall inequality and an upper bound for discrete-time martingales representing the supremum in terms of the infimum. The release of a martingale term on the right-hand side of the given inequality and the graded L1 difference formula for the time Caputo fractional derivative of order 0 < α < 1 on the left-hand side are the main challenges of the stated and proved main theorem. As an example of application, the constructed theorem is used to derive an a priori estimate for a discrete stochastic fractional model at the end of the paper.

Improvement on modal curvatures methods for multiple damage identification in beams

PurposeThe purpose of this paper is to present a study on the application of four damage factors to several single and multiple damage scenarios of aluminium beams. Each one of these damage factors is defined by the information given by modal curvatures of the beams.Design/methodology/approachThe methodology consisted of a first experimental stage in which the modal rotations were measured with shearography and a subsequent numerical analysis in order to obtain the modal curvatures. To this end, three finite difference formulae were applied. The modal curvatures were then used to calculate the damage factors.FindingsIt was found that the profile of the damage factors varies according to the finite difference formula used. In view of the findings, the differences among the damage factors analysed are highlighted and some final recommendations to improve damage identifications via modal curvature-based are presented.Originality/valueTo the best of the authors’ knowledge, the application and comparison of several finite difference formulae and corresponding optimal sampling has not been carried out before. With the proposed approach, it is possible to identify multiple damages, which is still a great challenge. The post-processing of shearography measurements with a numerical method, which is inherently a multidisciplinary approach, is also a substantial improvement upon other type of approaches found in the literature.

Optimization of a Class of n-Sub-Step Time Integration Methods for Structural Dynamics

This paper develops a family of optimized [Formula: see text]-sub-step time integration methods for structural dynamics, in which the generalized trapezoidal rule is used in the first [Formula: see text] sub-steps, and the last sub-step employs [Formula: see text]-point backward difference formula. The proposed methods can achieve second-order accuracy and unconditional stability, and their degree of numerical dissipation can range from zero to one. Also, the proposed methods can achieve the identical effective stiffness matrices for all sub-steps, reducing computational costs in the analysis of linear systems. Using the spectral analysis, optimized algorithmic parameters are presented, ensuring that the proposed methods can accurately calculate different types of dynamic problems such as wave propagation, stiff and nonlinear systems. Besides, with the increase in the number of sub-steps, the accuracy of the proposed methods can be enhanced without extra workload compared with single-step methods. Numerical experiments show that the proposed methods perform better in different dynamic systems.

The Effects of Munsell Neutral Grey Backgrounds on the Colour of Chrysoprase and the Application of AP Clustering to Chrysoprase Colour Grading

Chrysoprase is a popular gemstone with consumers because of its charming apple green colour but a scientific classification of its colour has not yet been achieved. In this research, we determined the most effective background of the Munsell Chart for chrysoprase colour grading under a 6504 K fluorescent lamp and applied an affinity propagation (AP) clustering algorithm to the colour grading of coloured gems for the first time. Forty gem-quality chrysoprase samples from Australia were studied using a UV-VIS spectrophotometer and Munsell neutral grey backgrounds. The results determined the effects of a Munsell neutral grey background on the observed colour. It was found that the Munsell N9.5 background was the most effective for colour grading in this case. The observed chrysoprase colours were classified into five groups: Fancy Light, Fancy, Fancy Intense, Fancy Deep and Fancy Dark. The feasibility of the colour grading scheme was verified using the colour difference formula DE2000.

The Social, Economics, And Finance Analysis On Profit And Loss Sharing Of Islamic Partnership (Case Study of Tobacco Bussiness in Jember Regency, Indonesia)

The purpose of this study are: (1) to investigate the implementation of partnerships in the VOK tobacco business in an Islamic perspective, whether include mukhabarah, muzara’ah, or musaqah contract; (2) to explore the partnership implementation of VOK tobacco business on social, economic, and finance. This study use the snowball method with in depth interview. This research captured that there are two contracts of Islamic partnership on tobacco business, namely mukhabarah and muzara'ah. The social impact is the availability of jobs for people who do not own land/rice fields, thus helping the government to reduce unemployment and create close kinship in the community. The manifestation of the economic impact is that people who do not own land can still earn income for live. This study shows that there is a difference formula in calculating the profit sharing between the net farm income (NFI) and local culture formula. The local cultural formula does not use the real cost to count the net profit which is considered injustice. It is important to note that injustice is something that is against Islamic belief. Profit-sharing ratio between landowner and land tiller is 50%: 50% of net income.

New repetitive motion planning scheme with cube end-effector planning precision for redundant robotic manipulators

Repetitive motion planning (RMP) is important in operating redundant robotic manipulators. In this paper, a new RMP scheme that is based on the pseudoinverse formulation is proposed for redundant robotic manipulators. Such a scheme is derived from the discretization of an existing RMP scheme by utilizing the difference formula. Then, theoretical analysis and results are presented to show the characteristic of the proposed RMP scheme. That is, this scheme possesses the characteristic of cube pattern in the end-effector planning precision. The proposed RMP scheme is further extended and studied for redundant robotic manipulators under joint constraint. Based on a four-link robotic manipulator, simulation results substantiate the effectiveness and superiority of the proposed RMP scheme and its extended one.

Integral Representation of Polynomial Xn (x;a,b)

In the present paper we have obtained fine difference formula, contour integral representation, real integral representation, infinite single integral representation, finite single integral representation, finite double integral representation, finite double integral representation of polynomial ࢔ࢄ .(࢈ ,ࢇ ;࢞) Keywords: Finite difference, single integral representation, contour integral representation, simple generating relation, double integral representation

A New Interpolation Approach and Corresponding Instance-Based Learning

Starting from finding approximate value of a function, introduces the measure of approximation-degree between two numerical values, proposes the concepts of “strict approximation” and “strict approximation region”, then, derives the corresponding one-dimensional interpolation methods and formulas, and then presents a calculation model called “sum-times-difference formula” for high-dimensional interpolation, thus develops a new interpolation approach, that is, ADB interpolation. ADB interpolation is applied to the interpolation of actual functions with satisfactory results. Viewed from principle and effect, the interpolation approach is of novel idea, and has the advantages of simple calculation, stable accuracy, facilitating parallel processing, very suiting for high-dimensional interpolation, and easy to be extended to the interpolation of vector valued functions. Applying the approach to instance-based learning, a new instance-based learning method, learning using ADB interpolation, is obtained. The learning method is of unique technique, which has also the advantages of definite mathematical basis, implicit distance weights, avoiding misclassification, high efficiency, and wide range of applications, as well as being interpretable, etc. In principle, this method is a kind of learning by analogy, which and the deep learning that belongs to inductive learning can complement each other, and for some problems, the two can even have an effect of “different approaches but equal results” in big data and cloud computing environment. Thus, the learning using ADB interpolation can also be regarded as a kind of “wide learning” that is dual to deep learning.

A New Interpolation Approach and Corresponding Instance-Based Learning

Starting from finding approximate value of a function, introduces the measure of approximation-degree between two numerical values, proposes the concepts of “strict approximation” and “strict approximation region”, then, derives the corresponding one-dimensional interpolation methods and formulas, and then presents a calculation model called “sum-times-difference formula” for high-dimensional interpolation, thus develops a new interpolation approach, that is, ADB interpolation. ADB interpolation is applied to the interpolation of actual functions with satisfactory results. Viewed from principle and effect, the interpolation approach is of novel idea, and has the advantages of simple calculation, stable accuracy, facilitating parallel processing, very suiting for high-dimensional interpolation, and easy to be extended to the interpolation of vector valued functions. Applying the approach to instance-based learning, a new instance-based learning method, learning using ADB interpolation, is obtained. The learning method is of unique technique, which has also the advantages of definite mathematical basis, implicit distance weights, avoiding misclassification, high efficiency, and wide range of applications, as well as being interpretable, etc. In principle, this method is a kind of learning by analogy, which and the deep learning that belongs to inductive learning can complement each other, and for some problems, the two can even have an effect of “different approaches but equal results” in big data and cloud computing environment. Thus, the learning using ADB interpolation can also be regarded as a kind of “wide learning” that is dual to deep learning.