Using Gauss-Jacobi quadrature rule to improve the accuracy of FEM for spatial fractional problems

2021 ◽  
Author(s):  
Zongze Yang ◽  
Jungang Wang ◽  
Zhanbin Yuan ◽  
Yufeng Nie
Keyword(s):  
Quadrature Rule

scholarly journals Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 923 ◽  
Author(s):  
Abdul Ghafoor ◽  
Sirajul Haq ◽  
Manzoor Hussain ◽  
Poom Kumam ◽  
Muhammad Asif Jan
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Quadrature Rule ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Haar Wavelets ◽  
Test Problems ◽  
Algebraic Equations ◽  
Diffusion Wave ◽  
Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

scholarly journals Quadrature Integration Techniques for Random Hyperbolic PDE Problems

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 160
Author(s):  
Rafael Company ◽  
Vera N. Egorova ◽  
Lucas Jódar
Keyword(s):  
Numerical Methods ◽  
Laplace Transform ◽  
Quadrature Rule ◽  
Inverse Laplace Transform ◽  
Process Time ◽  
Efficient Computation ◽  
Hyperbolic Pde ◽  
Numerical Convergence ◽  
Laplace Transform Technique ◽  
Integration Techniques

In this paper, we consider random hyperbolic partial differential equation (PDE) problems following the mean square approach and Laplace transform technique. Randomness requires not only the computation of the approximating stochastic processes, but also its statistical moments. Hence, appropriate numerical methods should allow for the efficient computation of the expectation and variance. Here, we analyse different numerical methods around the inverse Laplace transform and its evaluation by using several integration techniques, including midpoint quadrature rule, Gauss–Laguerre quadrature and its extensions, and the Talbot algorithm. Simulations, numerical convergence, and computational process time with experiments are shown.

Reliability analysis and state transfer scheduling optimization of degrading load-sharing system equipped with warm standby components

2021 ◽  
pp. 1748006X2110017
Author(s):  
Sheng-Jia Ruan ◽  
Yan-Hui Lin
Keyword(s):  
Reliability Analysis ◽  
System Reliability ◽  
High Reliability ◽  
Quadrature Rule ◽  
Load Sharing ◽  
Measurement Information ◽  
Model Parameters ◽  
Scheduling Optimization ◽  
State Transfer ◽  
Warm Standby

Standby redundancy can meet system safety requirements in industries with high reliability standards. To evaluate reliability of standby systems, failure dependency among components has to be considered especially when systems have load-sharing characteristics. In this paper, a reliability analysis and state transfer scheduling optimization framework is proposed for the load-sharing 1-out-of- N: G system equipped with M warm standby components and subject to continuous degradation process. First, the system reliability function considering multiple dependent components is derived in a recursive way. Then, a Monte Carlo method is developed and the closed Newton-Cotes quadrature rule is invoked for the system reliability quantification. Besides, likelihood functions are constructed based on the measurement information to estimate the model parameters of both active and standby components, whose degradation paths are modeled by the step-wise drifted Wiener processes. Finally, the system state transfer scheduling is optimized by the genetic algorithm to maximize the system reliability at mission time. The proposed methodology and its effectiveness are illustrated through a case study referring to a simplified aircraft hydraulic system.

scholarly journals On Appel-Type Quadrature Rules

2002 ◽  
Vol 9 (3) ◽  
pp. 405-412
Author(s):  
C. Belingeri ◽  
B. Germano
Keyword(s):  
Remainder Term ◽  
Quadrature Rule ◽  
Bernoulli Numbers ◽  
Quadrature Rules ◽  
Rule Based ◽  
Numerical Example ◽  
The Given ◽  
Derivatives Of

Abstract The Radon technique is applied in order to recover a quadrature rule based on Appel polynomials and the so called Appel numbers. The relevant formula generalizes both the Euler-MacLaurin quadrature rule and a similar rule using Euler (instead of Bernoulli) numbers and even (instead of odd) derivatives of the given function at the endpoints of the considered interval. In the general case, the remainder term is expressed in terms of Appel numbers, and all derivatives appear. A numerical example is also included.

scholarly journals Explicit Dynamic Finite Element Method for Failure with Smooth Fracture Energy Dissipations

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Jeong-Hoon Song ◽  
Thomas Menouillard ◽  
Alireza Tabarraei
Keyword(s):  
Finite Element ◽  
Fracture Energy ◽  
Computational Cost ◽  
Quadrature Rule ◽  
Dynamic Failure ◽  
Structural Dynamic ◽  
Integration Schemes ◽  
Hourglass Control ◽  
Quadrilateral Finite Element ◽  
Explicit Dynamic

A numerical method for dynamic failure analysis through the phantom node method is further developed. A distinct feature of this method is the use of the phantom nodes with a newly developed correction force scheme. Through this improved approach, fracture energy can be smoothly dissipated during dynamic failure processes without emanating noisy artifact stress waves. This method is implemented to the standard 4-node quadrilateral finite element; a single quadrature rule is employed with an hourglass control scheme in order to decrease computational cost and circumvent difficulties associated with the subdomain integration schemes for cracked elements. The effectiveness and robustness of this method are demonstrated with several numerical examples. In these examples, we showed the effectiveness of the described correction force scheme along with the applicability of this method to an interesting class of structural dynamic failure problems.

Fast and high-precision calculation of earth return mutual impedance between conductors over a multilayered soil

2018 ◽  
Vol 37 (3) ◽  
pp. 1214-1227 ◽  
Author(s):  
Junjie Ma
Keyword(s):  
Quadrature Rule ◽  
Content Type ◽  
Mutual Impedance ◽  
Numerical Tests ◽  
Sommerfeld Integral ◽  
The Earth ◽  
Finite Integral ◽  
Infinite Integral ◽  
Improper Integrals ◽  
Differential Matrix

Purpose Solutions for the earth return mutual impedance play an important role in analyzing couplings of multi-conductor systems. Generally, the mutual impedance is approximated by Pollaczek integrals. The purpose of this paper is devising fast algorithms for calculation of this kind of improper integrals and its applications. Design/methodology/approach According to singular points, the Pollaczek integral is divided into two parts: the finite integral and the infinite integral. The finite part is computed by combining an efficient Levin method, which is implemented with a Chebyshev differential matrix. By transforming the integration path, the tail integral is calculated with help of a transformed Clenshaw–Curtis quadrature rule. Findings Numerical tests show that this new method is robust to high oscillation and nearly singularities. Thus, it is suitable for evaluating Pollaczek integrals. Furthermore, compared with existing method, the presented algorithm gives high-order approaches for the earth return mutual impedance between conductors over a multilayered soil with wide ranges of parameters. Originality/value An efficient truncation strategy is proposed to accelerate numerical calculation of Pollaczek integral. Compared with existing algorithms, this method is easier to be applied to computation of similar improper integrals, such as Sommerfeld integral.

scholarly journals The number of lattice rules having given invariants

1992 ◽  
Vol 46 (3) ◽  
pp. 479-495 ◽  
Author(s):  
Stephen Joe ◽  
David C. Hunt
Keyword(s):  
Normal Form ◽  
Quadrature Rule ◽  
Unit Cube ◽  
Numerical Results ◽  
Smith Normal Form ◽  
Lattice Rules ◽  
Dimensional Unit ◽  
Lattice Rule ◽  
Positive Integers ◽  
Theoretical Results

A lattice rule is a quadrature rule used for the approximation of integrals over the s-dimensional unit cube. Every lattice rule may be characterised by an integer r called the rank of the rule and a set of r positive integers called the invariants. By exploiting the group-theoretic structure of lattice rules we determine the number of distinct lattice rules having given invariants. Some numerical results supporting the theoretical results are included. These numerical results are obtained by calculating the Smith normal form of certain integer matrices.

scholarly journals Mathematical Modelling of  RLC Circuit  Through Mixed Quadrature Rule

2018 ◽  
Author(s):  
Saumya jena ◽  
Damayanti Nayak
Keyword(s):  
Integral Equation ◽  
Mathematical Modelling ◽  
Electromotive Force ◽  
Quadrature Rule ◽  
Absolute Error ◽  
Definite Integral ◽  
Numerical Result ◽  
Rlc Circuit ◽  
Instantaneous Current

In this study, a mixed rule of degree of precision nine has been developed and implemented in the field of electrical sciences to obtain the instantaneous current in the RLC- circuit for particular value .The linearity has been performed with the Volterra’s integral equation of second kind with particular kernel . Then the definite integral has been evaluated through the mixed quadrature to obtain the numerical result which is very effective. A polynomial has been used to evaluate Volterra’s integral equation in the place of unknown functions. The accuracy of the proposed method has been tested taking different electromotive force in the circuit and absolute error has been estimated.

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