scholarly journals A Coupled Pseudospectral-Differential Quadrature Method for a Class of Hyperbolic Telegraph Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Fangzong Wang ◽  
Yong Wang
Keyword(s):  
Differential Quadrature Method ◽  
Pseudospectral Method ◽  
Engineering Calculation ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Quadrature Methods ◽  
Telegraph Equations ◽  
Lagrange Interpolation Polynomials ◽  
Grid Points ◽  
Interpolation Polynomials

Pseudospectral methods and differential quadrature methods are two kinds of important meshless methods, both of which have been widely used in scientific and engineering calculation. The Lagrange interpolation polynomials are used as the trial function of the two methods, and the same distribution of grid points is used. This paper points out that the differential quadrature method is a special form of the pseudospectral method. On the basis of the above, a coupled pseudospectral-differential quadrature method (PSDQM) is proposed to solve a class of hyperbolic telegraph equations. Theoretical analysis and numerical tests show that the new method has spectral precision convergence in spatial domain and has A-stability in time domain. And it is suitable for solving multidimensional telegraph equations.

scholarly journals Differential Quadrature Solution of Hyperbolic Telegraph Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
B. Pekmen ◽  
M. Tezer-Sezgin
Keyword(s):  
Numerical Solution ◽  
Good Accuracy ◽  
Differential Quadrature Method ◽  
Telegraph Equation ◽  
Differential Quadrature ◽  
Time Interval ◽  
Quadrature Method ◽  
Time Level ◽  
Time Direction ◽  
Grid Points

Differential quadrature method (DQM) is proposed for the numerical solution of one- and two-space dimensional hyperbolic telegraph equation subject to appropriate initial and boundary conditions. Both polynomial-based differential quadrature (PDQ) and Fourier-based differential quadrature (FDQ) are used in space directions while PDQ is made use of in time direction. Numerical solution is obtained by using Gauss-Chebyshev-Lobatto grid points in space intervals and equally spaced and/or GCL grid points for the time interval. DQM in time direction gives the solution directly at a required time level or steady state without the need of iteration. DQM also has the advantage of giving quite good accuracy with considerably small number of discretization points both in space and time direction.

Harmonic Differential Quadrature Method for Surface Location Error Prediction and Machining Parameter Optimization in Milling

2015 ◽  
Vol 137 (2) ◽  
Author(s):  
Ye Ding ◽  
XiaoJian Zhang ◽  
Han Ding
Keyword(s):  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Machining Parameters ◽  
Quadrature Method ◽  
Location Error ◽  
Sampling Grid ◽  
Surface Location Error ◽  
Harmonic Differential Quadrature Method ◽  
Surface Location ◽  
Grid Points

This paper presents a semi-analytical numerical method for surface location error (SLE) prediction in milling processes, governed by a time-periodic delay-differential equation (DDE) in state-space form. The time period is discretized as a set of sampling grid points. By using the harmonic differential quadrature method (DQM), the first-order derivative in the DDE is approximated by the linear sums of the state values at all the sampling grid points. On this basis, the DDE is discretized as a set of algebraic equations. A dynamic map can then be constructed to simultaneously determine the stability and the steady-state SLE of the milling process. To obtain optimal machining parameters, an optimization model based on the milling dynamics is formulated and an interior point penalty function method is employed to solve the problem. Experimentally validated examples are utilized to verify the accuracy and efficiency of the proposed approach.

A numerical study of two-dimensional coupled systems and higher order partial differential equations

2019 ◽  
Vol 12 (05) ◽  
pp. 1950071
Author(s):  
R. Rohila ◽  
R. C. Mittal
Keyword(s):  
Differential Equations ◽  
Numerical Study ◽  
Differential Quadrature Method ◽  
Bernstein Polynomials ◽  
Differential Quadrature ◽  
Coupled Systems ◽  
Two Dimensional ◽  
Quadrature Method ◽  
Grid Points ◽  
The Stability

In this paper, a new approach and methodology is developed by incorporating differential quadrature technique with Bernstein polynomials. In differential quadrature method, approximations are done in a way that the derivatives of the function are replaced by a linear sum of functional values at the grid points of the given domain. In Bernstein differential quadrature method (BDQM), Bernstein polynomials are employed for spatial discretization so that a system of ordinary differential equations (ODE’s) is obtained which is solved by SSPRK-43 method. The stability of the method is also studied. The accuracy of the present method is checked by performing numerical experiments on two-dimensional coupled Burgers’ and Brusselator systems and fourth-order extended Fisher Kolmogorov (EFK) equation. Implementation of the method is very easy, efficient and capable of reducing the size of computational efforts.

Modeling of Transient Flow in Pipeline Systems Using Incremental Differential Quadrature Method

2008 ◽  
Author(s):  
M. R. Hashemi ◽  
M. J. Abedini
Keyword(s):  
Method Of Characteristics ◽  
Transient Flow ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Test Case ◽  
Quadrature Method ◽  
Governing Equations ◽  
Friction Term ◽  
Grid Points ◽  
And Performance

Our objective in this paper is to introduce the detailed implementation and performance of a new method so called Incremental Differential Quadrature Method (IDQM) for modeling transient flow in pipelines. The basics of IDQM, its structure, and the formulation of the method for numerical simulation of transient flow in pipelines are discussed. The results obtained from this numerical exercise are compared with the method of characteristics and experimental data. The test case studies show that the best structure for IDQM solution is cosine grid distribution. In terms of viscous damping, the inclusion of unsteady friction term in the governing equations can improve the results significantly. This study demonstrates that IDQM can be regarded as a new alternative for numerical solution of transient flow such as water hammer in pipelines. IDQM is highly implicit in nature which makes the method unconditionally stable. It gives accurate results using few grid points.

Generalized Differential Quadrature Method for Burgers Equation

2003 ◽  
Author(s):  
Mladen Mesˇtrovic´
Keyword(s):  
Burgers Equation ◽  
Differential Quadrature Method ◽  
Computational Effort ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Generalized Differential Quadrature Method ◽  
Grid Points ◽  
Generalized Differential ◽  
Weighting Coefficients

The generalized differential quadrature method as an accurate and efficient numerical method is developed for the Burgers equation. The numerical algorithm for this class of problem is presented. Differential quadrature approximation of needed derivatives is given by a weighted linear sum of the function values at grid points. Recurrence relationship is used for calculation of weighting coefficients. The calculated numerical results are compared with exact solutions to show the quality of the generalized differential quadrature solutions for each example. Numerical examples have shown accuracy of the GDQ method with relatively small computational effort.

A numerical scheme based on differential quadrature method for numerical simulation of nonlinear Klein-Gordon equation

2014 ◽  
Vol 24 (7) ◽  
pp. 1390-1404 ◽  
Author(s):  
Anjali Verma ◽  
Ram Jiwari ◽  
Satish Kumar
Keyword(s):  
Numerical Scheme ◽  
Gordon Equation ◽  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Content Type ◽  
Multidimensional Problems ◽  
Klein Gordon ◽  
Grid Points

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.

Stability Analysis of Milling Via the Differential Quadrature Method

2013 ◽  
Vol 135 (4) ◽  
Author(s):  
Ye Ding ◽  
LiMin Zhu ◽  
XiaoJian Zhang ◽  
Han Ding
Keyword(s):  
Stability Analysis ◽  
Analytical Solution ◽  
Free Vibration ◽  
Forced Vibration ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Algebraic Equations ◽  
Sampling Grid ◽  
Grid Points

This paper presents a time-domain semi-analytical method for stability analysis of milling in the framework of the differential quadrature method. The governing equation of milling processes taking into account the regenerative effect is formulated as a linear periodic delayed differential equation (DDE) in state space form. The tooth passing period is first separated as the free vibration duration and the forced vibration duration. As for the free vibration duration, the analytical solution is available. As for the forced vibration duration, this time interval is discretized by sampling grid points. Then, the differential quadrature method is employed to approximate the time derivative of the state function at a sampling grid point within the forced vibration duration by a weighted linear sum of the function values over the whole sampling grid points. The Lagrange polynomial based algorithm (LPBA) and trigonometric functions based algorithm (TFBA) are employed to obtain the weight coefficients. Thereafter, the DDE on the forced vibration duration is discretized as a series of algebraic equations. By combining the analytical solution of the free vibration duration and the algebraic equations of the forced vibration duration, Floquet transition matrix can be constructed to determine the milling stability according to Floquet theory. Simulation results and experimentally validated examples are utilized to demonstrate the effectiveness and accuracy of the proposed approach.

A modified differential quadrature procedure for numerical solution of moving load problem

2015 ◽  
Vol 230 (5) ◽  
pp. 715-731 ◽  
Author(s):  
SA Eftekhari
Keyword(s):  
Differential Equations ◽  
Differential Quadrature Method ◽  
Moving Load ◽  
Finite Difference Methods ◽  
Delta Function ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Dirac Delta ◽  
Singular Functions ◽  
Grid Points

The differential quadrature method is a powerful numerical method for the solution of partial differential equations that arise in various fields of engineering, mathematics, and physics. It is easy to use and also straightforward to implement. However, similar to the conventional point discretization methods like the collocation and finite difference methods, the differential quadrature method has some difficulty in solving differential equations involving singular functions like the Dirac-delta function. This is due to complexities introduced by the singular functions to the discretization process of the problem region. To overcome this difficulty, this paper presents a combined differential quadrature–integral quadrature procedure in which such singular functions are simply handled. The mixed scheme can be easily applied to the problems in which the location of the singular point coincides with one of the differential quadrature grid points. However, for problems in which such condition is not fulfilled (i.e. for the case of arbitrary arranged grid points), especially for moving load class of problems, the coupled approach may fail to produce accurate solutions. To solve this drawback, we also introduce two simple approximations and show that they can yield accurate results. The reliability and applicability of the proposed method are demonstrated herein through the solution of some illustrative problems, including the moving load problems of Euler–Bernoulli and Timoshenko beams. The results generated by the proposed method are compared with analytical and numerical results available in the literature and excellent agreement is achieved.

A local adaptive differential quadrature method for solving multi-dimensional heat transport equation at the microscale

2012 ◽  
Vol 227 (1) ◽  
pp. 33-41 ◽  
Author(s):  
Jiun-Yu Wu ◽  
Hui-Ching Wang ◽  
Ming-I Char ◽  
Bo-Chen Tai
Keyword(s):  
Transport Equation ◽  
Heat Transport ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Heat Transport Equation ◽  
Grid Points ◽  
Boundary Treatments ◽  
The Given ◽  
Accurate Numerical Solution

In this article, we employed the local adaptive differential quadrature method to solve multi-dimensional heat transport equation at the microscale. The local adaptive differential quadrature method was employed to tackle the boundary conditions by using both localized interpolation basis functions and exterior grid points. We found that accurate numerical solution can be obtained by using a small number of gird points for boundary treatments. The governing equation of heat transport was employed to describe the thermal behavior of microstructures, which was of vital importance in microtechnology applications. Four examples showed the effectiveness and accuracy of our algorism in providing excellent estimates of unknown temperature from the given data. We found that the proposed scheme is applicable to the heat transport equation at the microscale.

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