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Article
Publication date: 17 November 2021

Mehdi Dehghan, Baharak Hooshyarfarzin and Mostafa Abbaszadeh

This study aims to use the polynomial approximation method based on the Pascal polynomial basis for obtaining the numerical solutions of partial differential equations. Moreover…

Abstract

Purpose

This study aims to use the polynomial approximation method based on the Pascal polynomial basis for obtaining the numerical solutions of partial differential equations. Moreover, this method does not require establishing grids in the computational domain.

Design/methodology/approach

In this study, the authors present a meshfree method based on Pascal polynomial expansion for the numerical solution of the Sobolev equation. In general, Sobolev-type equations have several applications in physics and mechanical engineering.

Findings

The authors use the Crank-Nicolson scheme to discrete the time variable and the Pascal polynomial-based (PPB) method for discretizing the spatial variables. But it is clear that increasing the value of the final time or number of time steps, will bear a lot of costs during numerical simulations. An important purpose of this paper is to reduce the execution time for applying the PPB method. To reach this aim, the proper orthogonal decomposition technique has been combined with the PPB method.

Originality/value

The developed procedure is tested on various examples of one-dimensional, two-dimensional and three-dimensional versions of the governed equation on the rectangular and irregular domains to check its accuracy and validity.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 32 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

Open Access
Article
Publication date: 8 March 2021

Md Abu Hanif Sarkar

The purpose of this paper is to find a doubly nonlinear parabolic equation of fast diffusion in a bounded domain.

Abstract

Purpose

The purpose of this paper is to find a doubly nonlinear parabolic equation of fast diffusion in a bounded domain.

Design/methodology/approach

For positive and bounded initial data, the authors study the initial zero-boundary value problem.

Findings

The findings of this study showed the complete extinction of a continuous weak solution at a finite time.

Originality/value

The extinction time is studied earlier but for the Laplacian case. The authors presented the finite extinction time for the case of p-Laplacian.

Details

Arab Journal of Mathematical Sciences, vol. 28 no. 1
Type: Research Article
ISSN: 1319-5166

Keywords

Open Access
Article
Publication date: 18 August 2021

Omar Benslimane, Ahmed Aberqi and Jaouad Bennouna

In the present paper, the authors will discuss the solvability of a class of nonlinear anisotropic elliptic problems (P), with the presence of a lower-order term and a…

Abstract

Purpose

In the present paper, the authors will discuss the solvability of a class of nonlinear anisotropic elliptic problems (P), with the presence of a lower-order term and a non-polynomial growth which does not satisfy any sign condition which is described by an N-uplet of N-functions satisfying the Δ2-condition, within the fulfilling of anisotropic Sobolev-Orlicz space. In addition, the resulting analysis requires the development of some new aspects of the theory in this field. The source term is merely integrable.

Design/methodology/approach

An approximation procedure and some priori estimates are used to solve the problem.

Findings

The authors prove the existence of entropy solutions to unilateral problem in the framework of anisotropic Sobolev-Orlicz space with bounded domain. The resulting analysis requires the development of some new aspects of the theory in this field.

Originality/value

To the best of the authors’ knowledge, this is the first paper that investigates the existence of entropy solutions to unilateral problem in the framework of anisotropic Sobolev-Orlicz space with bounded domain.

Details

Arab Journal of Mathematical Sciences, vol. 29 no. 1
Type: Research Article
ISSN: 1319-5166

Keywords

Article
Publication date: 16 June 2010

Alexander G. Bagdoev, Anna V. Vardanyan, Sedrak V. Vardanyan and Ashot N. Martirosyan

The purpose of this paper is to investigate the problem of fracture of construction by solution of several mixed unsteady boundary value problems of elasticity, determination of…

Abstract

Purpose

The purpose of this paper is to investigate the problem of fracture of construction by solution of several mixed unsteady boundary value problems of elasticity, determination of stress intensity factors and concentration of stresses near edges of cracks and by numerical calculations of them obtained by explicit formulae.

Design/methodology/approach

The main methods of solution are integral transformations of Laplace and Fourier, method of Winner‐Hopf system solution by avoiding the singularities of coefficients of their matrices and factorization of them using numerical solution of the same order system of Fredholm integral equations. The solution for stresses is obtained in originals by effective Smirnov‐Sobolev form. The obtained integrals for stress intensity coefficients are calculated for considered cases of plane and anti‐plane problems of cracks, and for more complex space problem of crack are carried out all mentioned analytical investigations, including derivation of stresses distributions formulae near crack edge.

Findings

These analytic and numerical methods based on dynamic elasticity approximation on account of singularities near cracks edge allow precise calculation of the possibility and character of fracture of media under any loading of rather complex type.

Originality/value

Results can be useful for investigation of constructions responsibility. The developed mathematical methods are original and modern, using all actual effective methods of investigation of solutions of linear system of equations with three and four independent variables for complex initial, boundary value problems.

Details

Multidiscipline Modeling in Materials and Structures, vol. 6 no. 1
Type: Research Article
ISSN: 1573-6105

Keywords

Article
Publication date: 2 November 2015

Guo-Dong Zhang and Yinnian He

The purpose of this paper is to consider the numerical implementation of the Euler semi-implicit scheme for three-dimensional non-stationary magnetohydrodynamics (MHD) equations

Abstract

Purpose

The purpose of this paper is to consider the numerical implementation of the Euler semi-implicit scheme for three-dimensional non-stationary magnetohydrodynamics (MHD) equations. The Euler semi-implicit scheme is used for time discretization and (P 1b , P 1, P 1) finite element for velocity, pressure and magnet is used for the spatial discretization.

Design/methodology/approach

Several numerical experiments are provided to show this scheme is unconditional stability and unconditional L2−H2 convergence with the L2−H2 optimal error rates for solving the non-stationary MHD flows.

Findings

In this paper, the authors mainly focus on the numerical investigation of the Euler semi-implicit scheme for MHD flows. First, the unconditional stability and the L2−H2 unconditional convergence with optimal L2−H2 error rates of this scheme are validated through our numerical tests. Some interesting phenomenons are presented.

Originality/value

The Euler semi-implicit scheme is used to simulate a practical physics model problem to investigate the interaction of fluid and induced magnetic field. Some interesting phenomenons are presented.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 January 2006

Andrey B. Andreev and Todor D. Todorov

To study and to analyze a second order finite‐element boundary‐flux approximation using isoparametric numerical integration.

Abstract

Purpose

To study and to analyze a second order finite‐element boundary‐flux approximation using isoparametric numerical integration.

Design/methodology/approach

The numerical finite‐element integration is the main method used in this research. Since a domain with curved boundary is considered we apply an isoparametric approach. The lumped flux formulation is another method of approach in this paper.

Findings

This research study presents a careful analysis of the combined effect of the numerical integration and isoparametric FEM on the boundary‐flux error. Some L2‐norm estimates are proved for the approximate solutions of the problem under consideration.

Research limitations/implications

The authors offer a general study within the framework of the boundary‐flux approximation theory, which completes the results of published works in this scientific field of research.

Practical implications

A useful application is to employ appropriate quadrature formulae without violating the precision of the boundary‐flux FEM. The lumped mass approximation is also an important practical approach to the problem in question.

Originality/value

The paper presents an entire investigation in FE boundary‐flux approximation theory, in particular, elements of arbitrary degree and domains with curved boundaries. The work is addressed to the possible related fields of interest of postgraduate students and specialists in fluid mechanics and numerical analysis.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 16 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 19 June 2007

Todor D. Todorov

To obtain error estimates for 3D consistent boundary‐flux approximations.

Abstract

Purpose

To obtain error estimates for 3D consistent boundary‐flux approximations.

Design/methodology/approach

Isoparametric approach is used for constructing finite‐element approximations.

Findings

This research study presents a convergence analysis of 3D boundary‐flux approximations. Error estimates are proved for the approximate solutions of the problem under consideration.

Research limitations/implications

General results for a consistent boundary‐flux problem are obtained for all 3D domains with Lipschitz‐continuous boundary. This investigation will be continued studying combined effect of curved boundaries and isoparametric numerical integration. An optimal refined strategy with respect to algorithmic aspects for solving 3D boundary‐flux problem also will be considered.

Practical implications

The obtained results enable engineers to calculate the flux across the curved boundaries using finite element method (FEM).

Originality/value

The paper presents an isoparametric finite‐element method for a 3D consistent boundary‐flux problem in domains with complex geometry. The work is addressed to the possible‐related fields of interest of postgraduate students and specialists in fluid mechanics and numerical analysis.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 17 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 June 2005

Bogdan Cranganu‐Cretu, Joerg Ostrowski and Zoran Andjelic

To provide first insight onto the application of hierarchical matrices and adaptive cross approximation (ACA) techniques for electromagnetic scattering problems.

Abstract

Purpose

To provide first insight onto the application of hierarchical matrices and adaptive cross approximation (ACA) techniques for electromagnetic scattering problems.

Design/methodology/approach

The shielding effectiveness of metallic casings with apertures is analyzed via an electric field integral equation. To reduce the storage needs and the complexity of matrix equation solution, a technique combining the use of hierarchical matrices (H‐matrix) in conjunction with the ACA technique is used.

Findings

Provides first results for compression of a matrix resulting from a Helmholtz problem by means of hierarchical matrices and ACA techniques. Gives insight into the importance of obtaining a “cheap” preconditioner.

Research limitations/implications

The technique resides on the smotheness of kernel functions – which is no longer valid for big wave numbers.

Practical implications

Gives means of solving problems of big dimensions in terms of number of unknowns – without the need to tailor the approach for the specific kernel function. The original integration functions used to fill the full matrix can be used here.

Originality/value

The paper represents one of the first attempts to use the above‐mentioned techniques for the high frequency domain.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 24 no. 2
Type: Research Article
ISSN: 0332-1649

Keywords

Open Access
Article
Publication date: 14 March 2023

Svetlin Georgiev, Aissa Boukarou, Keltoum Bouhali and Khaled Zennir

This paper is devoted to the generalized Kadomtsev–Petviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global…

Abstract

Purpose

This paper is devoted to the generalized Kadomtsev–Petviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results.

Design/methodology/approach

This paper is devoted to the generalized Kadomtsev–Petviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results.

Findings

This paper is devoted to the generalized Kadomtsev–Petviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results.

Originality/value

This article is devoted to the generalized Kadomtsev–Petviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results.

Details

Arab Journal of Mathematical Sciences, vol. ahead-of-print no. ahead-of-print
Type: Research Article
ISSN: 1319-5166

Keywords

Article
Publication date: 1 May 2000

K. Park, D.Y. Yang and Y.S. Kang

The present work is concerned with three‐dimensional finite element analysis of the hollow section extrusion process using a porthole die. The effects of related design parameters…

Abstract

The present work is concerned with three‐dimensional finite element analysis of the hollow section extrusion process using a porthole die. The effects of related design parameters are discussed through the finite element simulation for extrusion of a triply‐connected rectangular tubular section. For economic computation, mismatching refinement, an efficient domain decomposition method with different mesh density for each subdomain is implemented. In order to obtain the uniform flow at the outlet, design parameters such as the hole size and the hole position are investigated and compared through the numerical analysis. Comparing the velocity distribution with that of the original design, it is concluded that the design modification enables more uniform flow characteristics. The analysis results are then successfully reflected on the industrial porthole die design.

Details

Engineering Computations, vol. 17 no. 3
Type: Research Article
ISSN: 0264-4401

Keywords

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