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1 – 7 of 7Yangin Fan and Emmanuel Guerre
The asymptotic bias and variance of a general class of local polynomial estimators of M-regression functions are studied over the whole compact support of the multivariate…
Abstract
The asymptotic bias and variance of a general class of local polynomial estimators of M-regression functions are studied over the whole compact support of the multivariate covariate under a minimal assumption on the support. The support assumption ensures that the vicinity of the boundary of the support will be visited by the multivariate covariate. The results show that like in the univariate case, multivariate local polynomial estimators have good bias and variance properties near the boundary. For the local polynomial regression estimator, we establish its asymptotic normality near the boundary and the usual optimal uniform convergence rate over the whole support. For local polynomial quantile regression, we establish a uniform linearization result which allows us to obtain similar results to the local polynomial regression. We demonstrate both theoretically and numerically that with our uniform results, the common practice of trimming local polynomial regression or quantile estimators to avoid “the boundary effect” is not needed.
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Zain ul Abdeen and Mujeeb ur Rehman
The purpose of this paper is to present a computational technique based on Newton–Cotes quadrature rule for solving fractional order differential equation.
Abstract
Purpose
The purpose of this paper is to present a computational technique based on Newton–Cotes quadrature rule for solving fractional order differential equation.
Design/methodology/approach
The numerical method reduces initial value problem into a system of algebraic equations. The method presented here is also applicable to non-linear differential equations. To deal with non-linear equations, a recursive sequence of approximations is developed using quasi-linearization technique.
Findings
The method is tested on several benchmark problems from the literature. Comparison shows the supremacy of proposed method in terms of robust accuracy and swift convergence. Method can work on several similar types of problems.
Originality/value
It has been demonstrated that many physical systems are modelled more accurately by fractional differential equations rather than classical differential equations. Therefore, it is vital to propose some efficient numerical method. The computational technique presented in this paper is based on Newton–Cotes quadrature rule and quasi-linearization. The key feature of the method is that it works efficiently for non-linear problems.
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Presents the scientific methodology from the enlarged cybernetical perspective that recognizes the anisotropy of time, the probabilistic character of natural laws, and the entry…
Abstract
Presents the scientific methodology from the enlarged cybernetical perspective that recognizes the anisotropy of time, the probabilistic character of natural laws, and the entry that the incomplete determinism in Nature opens to the occurrence of innovation, growth, organization, teleology communication, control, contest and freedom. The new tier to the methodological edifice that cybernetics provides stands on the earlier tiers, which go back to the Ionians (c. 500 BC). However, the new insights reveal flaws in the earlier tiers, and their removal strengthens the entire edifice. The new concepts of teleological activity and contest allow the clear demarcation of the military sciences as those whose subject matter is teleological activity involving contest. The paramount question “what ought to be done”, outside the empirical realm, is embraced by the scientific methodology. It also embraces the cognitive sciences that ask how the human mind is able to discover, and how the sequence of discoveries might converge to a true description of reality.
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Hongwei Li, Xiao Wang, Junmu Lin, Lei Wu and Tong Liu
This study aims to provide a solution of the power flow calculation for the low-voltage ditrect current power grid. The direct current (DC) power grid is becoming a reliable and…
Abstract
Purpose
This study aims to provide a solution of the power flow calculation for the low-voltage ditrect current power grid. The direct current (DC) power grid is becoming a reliable and economic alternative to millions of residential loads. The power flow (PF) in the DC network has some similarities with the alternative current case, but there are important differences that deserve to be further concerned. Moreover, the dispatchable distributed generators (DGs) in DC network can realize the flexible voltage control based on droop-control or virtual impedance-based methods. Thus, DC PF problems are still required to further study, such as hosting all load types and different DGs.
Design/methodology/approach
The DC power analysis was explored in this paper, and an improved Newton–Raphson based linear PF method has been proposed. Considering that constant impedance (CR), constant current (CI) and constant power (CP) (ZIP) loads can get close to the practical load level, ZIP load has been merged into the linear PF method. Moreover, DGs are much common and can be easily connected to the DC grid, so V nodes and the dispatchable DG units with droop control have been further taken into account in the proposed method.
Findings
The performance and advantages of the proposed method are investigated based on the results of the various test systems. The two existing linear models were used to compare with the proposed linear method. The numerical results demonstrate enough accuracy, strong robustness and high computational efficiency of the proposed linear method even in the heavily-loaded conditions and with 10 times the line resistances.
Originality/value
The conductance corresponding to each constant resistance load and the equivalent conductance for the dispatchable unit can be directly merged into the self-conductance (diagonal component) of the conductance matrix. The constant current loads and the injection powers from dispatchable DG units can be treated as the current sources in the proposed method. All of those make the PF model much clear and simple. It is capable of offering enough accuracy level, and it is suitable for applications in DC networks that require a large number of repeated PF calculations to optimize the energy flows under different scenarios.
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Farshid Mossaiby and Mehdi Ghaderian
The purpose of this paper is to extend the meshless local exponential basis functions (MLEBF) method to the case of nonlinear and linear, variable coefficient partial differential…
Abstract
Purpose
The purpose of this paper is to extend the meshless local exponential basis functions (MLEBF) method to the case of nonlinear and linear, variable coefficient partial differential equations (PDEs).
Design/methodology/approach
The original version of MLEBF method is limited to linear, constant coefficient PDEs. The reason is that exponential bases which satisfy the homogeneous operator can only be determined for this class of problems. To extend this method to the general case of linear PDEs, the variable coefficients along with all involved derivatives are first expanded. This expanded form is evaluated at the center of each cloud, and is assumed to be constant over the entire cloud. The solution procedure is followed as in the former version. Nonlinear problems are first converted to a succession of linear, variable coefficient PDEs using the Newton-Kantorovich scheme and are subsequently solved using the aforementioned approach until convergence is achieved.
Findings
The results obtained show good performance of the method as solution to a wide range of problems. The results are compared with the well-known methods in the literature such as the finite element method, high-order finite difference method or variants of the boundary element method.
Originality/value
The MLEBF method is a simple yet effective tool for analyzing various kinds of problems. It is easy to implement with high parallelization potential. The proposed method addresses the biggest limitation of the method, and extends it to linear, variable coefficient PDEs as well as nonlinear ones.
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Jalil Rashidinia and Zahra Mahmoodi
The purpose of this paper is to develop a numerical method based on quintic B‐spline to solve the linear and nonlinear Fredholm and Volterra integral equations.
Abstract
Purpose
The purpose of this paper is to develop a numerical method based on quintic B‐spline to solve the linear and nonlinear Fredholm and Volterra integral equations.
Design/methodology/approach
The solution is collocated by quintic B‐spline and then the integral equation is approximated by the Gauss‐Kronrod‐Legendre quadrature formula.
Findings
The arising system of linear or nonlinear algebraic equations can solve the linear combination coefficients appearing in the representation of the solution in spline basic functions.
Practical implications
The error analysis of proposed numerical method is studied theoretically. Numerical results are given to illustrate the efficiency of the proposed method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient.
Originality/value
The paper provides new method to solve the linear and nonlinear Fredholm and Volterra integral equations.
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