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Article
Publication date: 2 September 2019

Farshid Mirzaee and Sahar Alipour

The purpose of this paper is to develop a new method based on operational matrices of two-dimensional delta functions for solving two-dimensional nonlinear quadratic integral

Abstract

Purpose

The purpose of this paper is to develop a new method based on operational matrices of two-dimensional delta functions for solving two-dimensional nonlinear quadratic integral equations (2D-QIEs) of fractional order, numerically.

Design/methodology/approach

For this aim, two-dimensional delta functions are introduced, and their properties are expressed. Then, the fractional operational matrix of integration based on two-dimensional delta functions is calculated for the first time.

Findings

By applying the operational matrices, the main problem would be transformed into a nonlinear system of algebraic equations which can be solved by using Newton's iterative method. Also, a few results related to error estimate and convergence analysis of the proposed method are investigated.

Originality/value

Two numerical examples are presented to show the validity and applicability of the suggested approach. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.

Details

Multidiscipline Modeling in Materials and Structures, vol. 15 no. 6
Type: Research Article
ISSN: 1573-6105

Keywords

Article
Publication date: 1 June 1994

Michael M. Grigor’ev

The paper gives the description of boundary element method(BEM) with subdomains for the solution ofconvection—diffusion equations with variable coefficients and Burgers’equations

Abstract

The paper gives the description of boundary element method (BEM) with subdomains for the solution of convection—diffusion equations with variable coefficients and Burgers’ equations. At first, the whole domain is discretized into K subdomains, in which linearization of equations by representing convective velocity by the sum of constant and variable parts is carried out. Then using fundamental solutions for convection—diffusion linear equations for each subdomain the boundary integral equation (in which the part of the convective term with the constant convective velocity is not included into the pseudo‐body force) is formulated. Only part of the convective term with the variable velocity, which is, as a rule, more than one order less than convective velocity constant part contribution, is left as the pseudo‐source. On the one hand, this does not disturb the numerical BEM—algorithm stability and, on the other hand, this leads to significant improvement in the accuracy of solution. The global matrix, similar to the case of finite element method, has block band structure whereas its width depends only on the numeration order of nodes and subdomains. It is noted, that in comparison with the direct boundary element method the number of global matrix non‐zero elements is not proportional to the square of the number of nodes, but only to the total number of nodal points. This allows us to use the BEM for the solution of problems with very fine space discretization. The proposed BEM with subdomains technique has been used for the numerical solution of one‐dimensional linear steady‐state convective—diffusion problem with variable coefficients and one‐dimensional non‐linear Burgers’ equation for which exact analytical solutions are available. It made it possible to find out the BEM correctness according to both time and space. High precision of the numerical method is noted. The good point of the BEM is the high iteration convergence, which is disturbed neither by high Reynolds numbers nor by the presence of negative velocity zones.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 4 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 June 2000

A. Savini

Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic community…

1128

Abstract

Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic community. Observes that computer package implementation theory contributes to clarification. Discusses the areas covered by some of the papers ‐ such as artificial intelligence using fuzzy logic. Includes applications such as permanent magnets and looks at eddy current problems. States the finite element method is currently the most popular method used for field computation. Closes by pointing out the amalgam of topics.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 19 no. 2
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 1 December 2005

Alessandro Corsini, Franco Rispoli and Andrea Santoriello

An original finite element scheme for advection‐diffusion‐reaction problems is presented. The new method, called spotted Petrov‐Galerkin (SPG), is a quadratic Petrov‐Galerkin (PG…

Abstract

Purpose

An original finite element scheme for advection‐diffusion‐reaction problems is presented. The new method, called spotted Petrov‐Galerkin (SPG), is a quadratic Petrov‐Galerkin (PG) formulation developed for the solution of equations where either reaction (associated to zero‐order derivatives of the unknown) and/or advection (proportional to first‐order derivatives) dominates on diffusion (associated to second‐order derivatives). The addressed issues are turbulence and advective‐reactive features in modelling turbomachinery flows.

Design/methodology/approach

The present work addresses the definition of a new PG stabilization scheme for the reactive flow limit, formulated on a quadratic finite element space of approximation. We advocate the use of a higher order stabilized formulation that guarantees the best compromise between solution stability and accuracy. The formulation is first presented for linear scalar one‐dimensional advective‐diffusive‐reactive problems and then extended to quadrangular Q2 elements.

Findings

The proposed advective‐diffusive‐reactive PG formulation improves the solution accuracy with respect to a standard streamline driven stabilization schemes, e.g. the streamline upwind or Galerkin, in that it properly accounts for the boundary layer region flow phenomena in presence of non‐equilibrium effects.

Research limitations/implications

The numerical method here proposed has been designed for second‐order quadrangular finite‐elements. In particular, the Reynolds‐Averaged Navier‐Stokes equations with a non‐linear turbulence closure have been modelled using the stable mixed element pair Q2‐Q1.

Originality/value

This paper investigated the predicting capabilities of a finite element method stabilized formulation developed for the purpose of solving advection‐reaction‐diffusion problems. The new method, called SPG, demonstrates its suitability in solving the typical equations of turbulence eddy viscosity models.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 15 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 June 2000

K. Wiak

Discusses the 27 papers in ISEF 1999 Proceedings on the subject of electromagnetisms. States the groups of papers cover such subjects within the discipline as: induction machines;…

Abstract

Discusses the 27 papers in ISEF 1999 Proceedings on the subject of electromagnetisms. States the groups of papers cover such subjects within the discipline as: induction machines; reluctance motors; PM motors; transformers and reactors; and special problems and applications. Debates all of these in great detail and itemizes each with greater in‐depth discussion of the various technical applications and areas. Concludes that the recommendations made should be adhered to.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 19 no. 2
Type: Research Article
ISSN: 0332-1649

Keywords

Book part
Publication date: 23 June 2016

Liangjun Su and Yonghui Zhang

In this paper, we study a partially linear dynamic panel data model with fixed effects, where either exogenous or endogenous variables or both enter the linear part, and the…

Abstract

In this paper, we study a partially linear dynamic panel data model with fixed effects, where either exogenous or endogenous variables or both enter the linear part, and the lagged-dependent variable together with some other exogenous variables enter the nonparametric part. Two types of estimation methods are proposed for the first-differenced model. One is composed of a semiparametric GMM estimator for the finite-dimensional parameter θ and a local polynomial estimator for the infinite-dimensional parameter m based on the empirical solutions to Fredholm integral equations of the second kind, and the other is a sieve IV estimate of the parametric and nonparametric components jointly. We study the asymptotic properties for these two types of estimates when the number of individuals N tends to and the time period T is fixed. We also propose a specification test for the linearity of the nonparametric component based on a weighted square distance between the parametric estimate under the linear restriction and the semiparametric estimate under the alternative. Monte Carlo simulations suggest that the proposed estimators and tests perform well in finite samples. We apply the model to study the relationship between intellectual property right (IPR) protection and economic growth, and find that IPR has a non-linear positive effect on the economic growth rate.

Article
Publication date: 1 September 2003

W. Florez, H. Power and F. Chejne

This paper presents a boundary element method (BEM) based on a subdomain approach for the solution of non‐Newtonian fluid flow problems which include thermal effects and viscous…

Abstract

This paper presents a boundary element method (BEM) based on a subdomain approach for the solution of non‐Newtonian fluid flow problems which include thermal effects and viscous dissipation. The volume integral arising from non‐linear terms is converted into equivalent boundary integrals by the multi‐domain dual reciprocity method (MD‐DRM) in each subdomain. Augmented thin plate splines interpolation functions are used for the approximation of field variables. The iterative numerical formulation is achieved by viewing the material as divided into small elements and on each of them the integral representation formulae for the velocity and temperature are applied and discretised using linear boundary elements. The final system of non‐linear algebraic equations is solved by a modified Newton's method. The numerical examples include non‐Newtonian problems with viscous dissipation, temperature‐dependent viscosity and natural convection due to bouyancy forces.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 13 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 January 1987

T.K. Hellen and W.S. Blackburn

A review is made of methods for calculating parameters characterizing crack tip behaviour in non‐linear materials. Convenient methods of calculating J‐integral type quantities are…

Abstract

A review is made of methods for calculating parameters characterizing crack tip behaviour in non‐linear materials. Convenient methods of calculating J‐integral type quantities are reviewed, classified broadly into two groups, as domain integrals and virtual crack extension techniques. In addition to considerations of how such quantities may be calculated by finite elements, assessment methods of conducting the actual incremental analyses are described.

Details

Engineering Computations, vol. 4 no. 1
Type: Research Article
ISSN: 0264-4401

Article
Publication date: 1 September 2003

J.P. Hernandez, T.A. Osswald and D.A. Weiss

In this paper, a novel boundary element formulation for the deformation of a viscous 2D‐planar cylindrical geometry, immersed in a different viscous fluid and moving towards a…

Abstract

In this paper, a novel boundary element formulation for the deformation of a viscous 2D‐planar cylindrical geometry, immersed in a different viscous fluid and moving towards a rigid wall, is proposed for moderate Reynolds number, considering surface tension effects. The boundary integral formulation for Stokes flow inside and outside the geometry is represented in terms of a combined distribution of a single‐layer and a double‐layer potential of Green functions over the geometry surface. Additionally, non‐linear terms describing effects absent in pure Stokes flow, such as the time derivative of the velocity and inertia, are included. These effects lead to the appearance of domain integrals. Traditional dual reciprocity is applied in order to approximate these domain integrals by a series of particular solutions which are then transformed into boundary integrals. Augmented thin‐plate splines, i.e. r2log(r), plus three additional linear terms from a Pascal triangle expansion were chosen for the dual reciprocity approximation. In order to avoid the discretization of the rigid wall, and using the fact that the velocity on the wall must vanish due to the no‐slip condition, the fundamental solution was modified with a combination of image singularities including an image Stokeslet, a potential dipole and a Stokes‐doublet.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 13 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 25 May 2012

Mehdi Dehghan and Masoud Hajarian

Solving the non‐linear equation f(x)=0 has nice applications in various branches of physics and engineering. Sometimes the applications of the numerical methods to solve non‐linear

275

Abstract

Purpose

Solving the non‐linear equation f(x)=0 has nice applications in various branches of physics and engineering. Sometimes the applications of the numerical methods to solve non‐linear equations depending on the second derivatives are restricted in physics and engineering. The purpose of this paper is to propose two new modified Newton's method for solving non‐linear equations. Convergence results show that the order of convergence of the proposed iterative methods for a simple root is four. The iterative methods are free from second derivative and can be used for solving non‐linear equations without computing the second derivative. Finally, several numerical examples are given to illustrate that proposed iterative algorithms are effective.

Design/methodology/approach

In this paper, first the authors introduce two new approximations for the definite integral arising from Newton's theorem. Then by considering these approximations, two new iterative methods are provided with fourth‐order convergence which can be used for solving non‐linear equations without computing second derivatives.

Findings

In this paper, the authors propose two new iterative methods without second derivatives for solving the non‐linear equation f(x)=0. From numerical results, it is observed that the new methods are comparable with various iterative methods. Also numerical results corroborate the theoretical analysis.

Originality/value

The best property of these schemes is that they are second derivative free. Also from numerical results, it is observed that the new methods are comparable with various iterative methods. The numerical results corroborate the theoretical analysis.

Details

Engineering Computations, vol. 29 no. 4
Type: Research Article
ISSN: 0264-4401

Keywords

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