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Article
Publication date: 23 August 2011

Alper Korkmaz and İdris Dağ

This paper aims to present a numerical solution of non‐linear Burger's equation using differential quadrature method based on sinc functions.

Abstract

Purpose

This paper aims to present a numerical solution of non‐linear Burger's equation using differential quadrature method based on sinc functions.

Design/methodology/approach

Sinc Differential Quadrature Method is used for space discretization and four stage Runge‐Kutta algorithm is used for time discretization. A rate of convergency analysis is also performed for shock‐like solution. Numerical stability analysis is performed.

Findings

Sinc Differential Quadrature Method generates more accurate solutions of Burgers' equation when compared with the other methods.

Originality/value

This combination, Sinc Differential Quadrature and Runge‐Kutta of order four, has not been used to obtain numerical solutions of Burgers' equation.

Details

Engineering Computations, vol. 28 no. 6
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 4 June 2021

Junning Qin and Hongzhi Zhong

Various time integration methods and time finite element methods have been developed to obtain the responses of structural dynamic problems, but the accuracy and…

Abstract

Purpose

Various time integration methods and time finite element methods have been developed to obtain the responses of structural dynamic problems, but the accuracy and computational efficiency of them are sometimes not satisfactory. The purpose of this paper is to present a more accurate and efficient formulation on the basis of the weak form quadrature element method to solve linear structural dynamic problems.

Design/methodology/approach

A variational principle for linear structural dynamics, which is inspired by Noble's work, is proposed to develop the weak form temporal quadrature element formulation. With Lobatto quadrature rule and the differential quadrature analog, a system of linear equations is obtained to solve the responses at sampling time points simultaneously. Computation for multi-elements can be carried out by a time-marching technique, using the end point results of the last element as the initial conditions for the next.

Findings

The weak form temporal quadrature element formulation is conditionally stable. The relation between the normalized length of element and the suggested number of integration points in one element is given by a simple formula. Results show that the present formulation is much more accurate than other time integration methods and its dissipative property is also illustrated.

Originality/value

The weak form temporal quadrature element formulation provides a choice with high accuracy and efficiency for solution of linear structural dynamic problems.

Details

Engineering Computations, vol. 38 no. 10
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 29 July 2019

Umer Saeed and Muhammad Umair

The purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a…

Abstract

Purpose

The purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a semi-infinite domain.

Design/methodology/approach

The proposed method is the combination of the Legendre polynomials and differential quadrature method. The authors derived and constructed the new operational matrices for the fractional derivatives, which are used for the solutions of non-linear time and space fractional partial differential equations.

Findings

The fractional derivative of Lagrange polynomial is a big hurdle in classical DQM. To overcome this problem, the authors represent the Lagrange polynomial in terms of shifted Legendre polynomial. They construct a transformation matrix which transforms the Lagrange polynomial into shifted Legendre polynomial of arbitrary order. Then, they obtain the new weighting coefficients matrices for space fractional derivatives by shifted Legendre polynomials and use these in conversion of a non-linear fractional partial differential equation into a system of fractional ordinary differential equations. Convergence analysis for the proposed method is also discussed.

Originality/value

Many engineers can use the presented method for solving their time and space fractional non-linear partial differential equation models. To the best of the authors’ knowledge, the differential quadrature method has never been extended or implemented for non-linear time and space fractional partial differential equations.

Article
Publication date: 29 March 2013

Alper Korkmaz and İdris Dağ

The purpose of this paper is to simulate numerical solutions of nonlinear Burgers' equation with two well‐known problems in order to verify the accuracy of the cubic…

Abstract

Purpose

The purpose of this paper is to simulate numerical solutions of nonlinear Burgers' equation with two well‐known problems in order to verify the accuracy of the cubic B‐spline differential quadrature methods.

Design/methodology/approach

Cubic B‐spline differential quadrature methods have been used to discretize the Burgers' equation in space and the resultant ordinary equation system is integrated via Runge‐Kutta method of order four in time. Numerical results are compared with each other and some former results by calculating discrete root mean square and maximum error norms in each case. A matrix stability analysis is also performed by determining eigenvalues of the coefficient matrices numerically.

Findings

Numerical results show that differential quadrature methods based on cubic B‐splines generate acceptable solutions of nonlinear Burgers' equation. Constructing hybrid algorithms containing various basis to determine the weighting coefficients for higher order derivative approximations is also possible.

Originality/value

Nonlinear Burgers' equation is solved by cubic B‐spline differential quadrature methods.

Article
Publication date: 26 October 2012

Alper Korkmaz and İdris Dağ

Cubic B‐spline differential quadrature methods have been introduced. As test problems, two different solutions of advection‐diffusion equation are chosen. The first test…

Abstract

Purpose

Cubic B‐spline differential quadrature methods have been introduced. As test problems, two different solutions of advection‐diffusion equation are chosen. The first test problem, the transportion of an initial concentration, and the second one, the distribution of an initial pulse, are simulated. The purpose of this paper is to simulate the test problems.

Design/methodology/approach

The cubic B‐spline functions are chosen as test functions in order to construct the differential quadrature method. The error between the numerical solutions and analytical solutions are measured using various error norms.

Findings

The cubic B‐spline differential quadrature methods have produced acceptable solution for advection‐diffusion equation.

Originality/value

The advection‐diffusion equation has never been solved by any differential quadrature method based on cubic B‐splines.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 18 January 2019

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.

Details

Engineering Computations, vol. 36 no. 2
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 12 October 2018

Umer Saeed, Mujeeb ur Rehman and Qamar Din

The purpose of this paper is to propose a method for solving nonlinear fractional partial differential equations on the semi-infinite domain and to get better and more…

Abstract

Purpose

The purpose of this paper is to propose a method for solving nonlinear fractional partial differential equations on the semi-infinite domain and to get better and more accurate results.

Design/methodology/approach

The authors proposed a method by using the Chebyshev wavelets in conjunction with differential quadrature technique. The operational matrices for the method are derived, constructed and used for the solution of nonlinear fractional partial differential equations.

Findings

The operational matrices contain many zero entries, which lead to the high efficiency of the method and reasonable accuracy is achieved even with less number of grid points. The results are in good agreement with exact solutions and more accurate as compared to Haar wavelet method.

Originality/value

Many engineers can use the presented method for solving their nonlinear fractional models.

Article
Publication date: 6 April 2022

Sinan Maraş and Mustafa Yaman

This study aims to demonstrate the numerical application of differential quadrature (DQ) methods and show the experimental application of free vibration analysis of…

Abstract

Purpose

This study aims to demonstrate the numerical application of differential quadrature (DQ) methods and show the experimental application of free vibration analysis of fiber-metal laminated composite (FML) plates with various boundary conditions.

Design/methodology/approach

The FMLs are hybrid structures consisting of fiber-reinforced polymer matrix composites such as carbon, glass, aramid and different metal sheets, and are currently widely used in the automobile, aircraft and aerospace industries. Thus, free vibration analysis of these hybrid materials is necessary for the design process. The governing equations of motion are derived based on the classical plate theory. The DQ, generalized DQ (GDQ) and harmonic DQ (HDQ) differential quadrature methods have been used to solve the governing equations of an FML composite plate numerically. The accuracy and convergence of the numerical model have been verified by comparing the results available in the published literature with the results obtained from these methods. Moreover, an experimental procedure has been performed in order to compare the results against those of the numerical methods.

Findings

It is noteworthy that a high degree of similarity and accuracy was observed between the numerical results obtained by the DQ methods and the experimental results. Thus, the present study validates the applicability of the DQ methods for designing the FML composite plates.

Originality/value

In this study, the advantages of the DQ methods have been demonstrated differently from previous studies on the vibration analysis of the FML plates.

Details

Engineering Computations, vol. 39 no. 6
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 5 March 2018

Aswin V.S. and Ashish Awasthi

This paper aims to investigate the features of three vectorized iterative numerical schemes used to simulate the behavior of modified Burgers equation (MBE).

Abstract

Purpose

This paper aims to investigate the features of three vectorized iterative numerical schemes used to simulate the behavior of modified Burgers equation (MBE).

Design/methodology/approach

Two of the schemes comprise differential quadrature and finite difference methods, while the third scheme consists of only differential quadrature for the derivative approximations. Proposed schemes are simulated for well-posed problems of MBE having known the analytic solution. The computational complexity of the schemes is examined through monitoring the time taken to complete the simulation. The results are compared with the analytic solution with the help of discrete error norms. Also, the accuracy of the proposed schemes is compared with that of the existing schemes in the literature. Vectorized MATLAB programs of the schemes are used for all investigations.

Findings

It is observed that all the three schemes succeeded in producing a good replication of the exact solution. The results are closer to the analytical solution than the results in the literature. Among the three schemes, the scheme labeled as FDTDQS is found highly accurate and computationally cheaper using fewer grid points. From the vectorized MATLAB programs provided, it is evident that the implementation of the schemes is simple.

Originality/value

This study gives an idea about three numerical schemes for a highly nonlinear problem. This mathematical framework can be adopted to any one-dimensional partial differential equation as well, and the provided program will be helpful to generate more fast and accurate vectorized code in MATLAB.

Details

Engineering Computations, vol. 35 no. 1
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 26 August 2014

Anjali Verma, Ram Jiwari and Satish Kumar

The purpose of this paper is to propose a numerical scheme based on forward finite difference, quasi-linearisation process and polynomial differential quadrature method to…

Abstract

Purpose

The purpose of this paper is to propose a numerical scheme based on forward finite difference, quasi-linearisation process and polynomial differential quadrature method to find the numerical solutions of nonlinear Klein-Gordon equation with Dirichlet and Neumann boundary condition.

Design/methodology/approach

In first step, time derivative is discretised by forward difference method. Then, quasi-linearisation process is used to tackle the non-linearity in the equation. Finally, fully discretisation by differential quadrature method (DQM) leads to a system of linear equations which is solved by Gauss-elimination method.

Findings

The accuracy of the proposed method is demonstrated by several test examples. The numerical results are found to be in good agreement with the exact solutions and the numerical solutions exist in literature. The proposed scheme can be expended for multidimensional problems.

Originality/value

The main advantage of the present scheme is that the scheme gives very accurate and similar results to the exact solutions by choosing less number of grid points. Secondly, the scheme gives better accuracy than (Dehghan and Shokri, 2009; Pekmen and Tezer-Sezgin, 2012) by choosing less number of grid points and big time step length. Also, the scheme can be extended for multidimensional problems.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 24 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

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