# differential quadrature method (DQ)

## Top PDF differential quadrature method (DQ): ### Numerical Solution of the Coupled Viscous Burgers’ Equation Using Differential Quadrature Method Based on Fourier Expansion Basis

A differential quadrature method is a numerical method for evaluating deriv- atives of sufficiently smooth function as proposed by . The basic idea of diffe- rential quadrature came from Gauss Quadrature , which is a useful numerical integration method. Gauss Quadrature is characterized by approximating a defi- nite integral with weighting sum of integrand value of a group at Gauss point. Extending it to find the derivatives of various orders of sufficiently smooth func- tion gives rise to DQ . In other words the derivatives of smooth function are approximated with weighted sum of the function values at a group of so called nodes. Differential quadrature can be formulated through either approximation theory or solving a system of linear equations. However, the rapid development over the recent years on problems involving nonlinear, discontinuity, multiple scales, singularity and irregularity are challenges in the field of computational science and engineering of the various numerical solutions. DQ method has dis- tinguished themselves because of their high accuracy, straight forward imple- mentation and generality in variety of problems . ### The effect of small scale and intermolecular forces on the nonlinear Pull-in instability behavior of nano-switches using differential quadrature method

problems in engineering and physical sciences. The numerical methods for the solution of initial or boundary-value problems, in general, seek to transform, either through a differential or an integral formulation into an analogous set of first-order or algebraic equations in terms of the discrete values of the field variable at some specified discrete points of the solution domain. The differential quadrature method is a numerical solution technique has been successfully employed in a variety of problems in engineering and physical sciences. The method has been projected by its proponents as a potential alternative to the conventional numerical solution techniques such as the finite difference and finite element methods. In order to solve the equation using DQM, first, the equation of motion and governing boundary conditions are made dimensionless. To do this, the following dimensionless parameters are defined: ### Calculation of finite difference method and differential quadrature method for burgers equation

Another technique which is discussed in this study is Differential Quadrature Method (DQM). As stated by (C. Shu, 2000), DQM is an extension of FDM for the highest order of finite difference scheme. This method represents by sum up all the derivatives of the function at any grid points, and then the equation transforms to a system of ordinary differential equations (ODEs) or a set of algebraic equations (R.C. Mittal and Ram Jiwari, 2009). The system of ordinary differential equations is then solved by numerical methods such as the implicit Runge-Kutta method that will be discussed in order to get the solutions in this study. ### Analysis Of Reaction Diffusion Problems Using Differential Quadrature Method

Abstract-- In this paper, a hybrid technique of differential quadrature method and Runge-Kutta fourth order method is employed to analyze reaction-diffusion problems. The obtained results are compared with the available analytical ones. Further, a parametric study is introduced to investigate the influence of reaction and diffusion characteristics on behavior of the obtained results. ### Parallel calculation of differential quadrature method for the burgers-huxley equation

Another numerical discretization technique that will be discussed in this study is the Differential Quadrature method (DQM). DQM is an extension of FDM for the highest order of finite difference scheme (C. Shu, 2000). As stated by R.C. Mittal and Ram Jiwari (2009), this method linearly sum up all the derivatives of a function at any location of the function values at a finite number of grid points, then the equation can be transformed into a set of ordinary differential equations (ODEs) or a set of algebraic equations. The set of ordinary differential equations or algebraic equations is then treated by standard numerical methods such as the implicit Runge-Kutta (RK) method that will be discussed in this study in order to obtain the solutions. ### The effect of small scale on the vibrational response of nano-column based on differential quadrature method

ABSTRACT: The present paper deals with the dynamic behavior of nano-column subjected to follower force using the nonlocal elasticity theory. The nonlocal elasticity theory is used to analyze the mechanical behavior of nanoscale materials. The used method of solution is the Differential Quadrature Method (DQM). It is shown that the nonlocal effect plays an important role in the vibrational behavior of nano-columns. The results can provide useful guidance for the study and design of the next generation of nanodevices and could be useful in biomedical and bioengineering applications as well as in other fields related with the nanotechnology. ### Analysis of Cracked Plates Using Localized Multi-Domain Differential Quadrature Method

More recently, the differential quadrature method (DQM), is introduced for solving several engineering problems, such that in thermodynamics, aerodynamics, structural and fracture mechanics. The method possesses the capability to achieve accurate results with a minimal computational effort [10-13], The classical version of DQM can’t deal with discontinuous or irregular domains. So, a new version of DQM, (termed by multi-domain differential quadrature technique), is developed for solving discontinuity problems. The philosophy of multi-domain DQM inherits the merits of the flexibility of finite element method and at the same time retains the high accuracy of the DQM. The main advantages of multi-domain DQM can be summarized as follows: multi-domain DQM is able to deal with problems with either geometric or material discontinuity, therefore it is recommended for solving crack problems. Also it can deal with problems with doubly or multiply connected regions. Further, it is able to treat problems with inconsistent boundary conditions [14-18]. ### Numerical solution of fractional order Riccati differential equation by differential quadrature method based on Chebyshev polynomials

We apply the Chebyshev polynomial-based diﬀerential quadrature method to the solution of a fractional-order Riccati diﬀerential equation. The fractional derivative is described in the Caputo sense. We derive and utilize explicit expressions of weighting coeﬃcients for approximation of fractional derivatives to reduce a Riccati diﬀerential equation to a system of algebraic equations. We present numerical examples to verify the eﬃciency and accuracy of the proposed method. The results reveal that the method is accurate and easy to implement. ### NUMERICAL SOLUTION OF NON-CONSERVATIVE LINEAR TRANSPORT PROBLEMS

interval leads linear systems of equations with three-banded coefficient matrix. Solving this system, we determine the weighting coefficients of the derivative approximations at nodes. Substituting the space derivative approximations obtained by the differential quadrature method into the AD equation, we construct a system of ordinary differential equation of order one in time variable. Then, we integrate this system with respect to time variable by implicit third-fourth order Rosenbrock method due to its strong stability properties. In order to see the validity of the suggested algorithm, we solve some IBVPs for the AD equation. The simulation plots of the numerical results show that the results are in a good agreement with the analytical results. The plots indicating the maximum errors agree that the errors decrease as time goes due to accuracy, validity and stability of the method and the natures of the models. The implementation of the suggested algorithm can be extended to the other problems for the nonlinear equations and systems. ### Nonlinear Bending Analysis of Sector Graphene Sheet Embedded in Elastic Matrix Based on Nonlocal Continuum Mechanics

The nonlinear bending behavior of sector graphene sheets is studied subjected to uniform transverse loads resting on a Winkler-Pasternak elastic foundation using the nonlocal elasticity theory. Considering the nonlocal differential constitutive relations of Eringen theory based on first order shear deformation theory and using the von-Karman strain field, the equilibrium partial differential equations are derived. The nonlinear partial differential equations system is solved using the differential quadrature method (DQM) and a new semi analytical polynomial method (SAPM). By using the DQM or SAPM, the partial differential equations are converted to nonlinear algebraic equations, then the Newton–Raphson iterative scheme is applied to solve the resulting nonlinear algebraic equations system. The obtained results from DQM and SAPM are compared. It is observed that the SAPM results are so close to DQM. The SAPM’s formulations are considerably simpler than the DQM. Different boundary conditions including clamped, simply supported and free edges are considered. The obtained results are validated with available researches, then the small scale effects is investigated on the results due to various conditions such as outer radius to thickness ratio, boundary conditions, linear to nonlinear analysis, nonlocal to local analysis ratio, angle of the sector and stiffness value of elastic foundation. ### Three-dimensional Magneto-thermo-elastic Analysis of Functionally Graded Truncated Conical Shells

steady state thermal and mechanical stresses in a hollow thick sphere made of functionally graded material. The analytical solution of heat conduction equation and the Navier equation were presented using the direct method. Paliwal and sinha  considered large deflection static analysis of shallow spherical shells on Winkler foundation, applying Bergler's and Modified Bergler's methods. Jane and Wu  studied thermo-elasticity problem in the curvilinear circular conical coordinate system. The hybrid Laplace transformation and finite difference were developed to obtain the solution of two dimensional axisymmetric coupled thermo-elastic equations. Chandrashekhara and bhimaraddi  presented the thermal stress analysis of doubly curved shallow shells using shear flexible finite element method. The basic equations were the extensions of Sanders shell theory to include shear deformation and thermal strains. Obata et al.  carried out thermal stresses analysis of a thick hollow cylinder, under two- dimensional temperature distribution. Xing and Liu  studied the magneto-thermo-elastic stresses in a conducting rectangular plate subjected to an arbitrary variation of magnetic field using differential quadrature method. Higuchi et al.  investigated the magneto- thermo-elastic stress fields induced by a transient magnetic field in an infinite conducting plate and numerically solved the corresponding electromagnetic, thermal and elastic equations. Lee et al.  considered three-dimensional axisymmetric coupled magneto- thermo-elasticity problems for laminated circular conical shells subjected to magneto-thermo-elastic loads, using Laplace transform and finite difference methods. Bodaghi and Shakeri  carried out an analytical investigation on free vibration and transient response of functionally graded piezoelectric cylindrical panels subjected to impulsive loads. The present work, investigates the three-dimensional problem of an FG truncated conical shell made of non-ferromagnetic metal such as aluminum permeated by a primary uniform magnetic field and subjected to internal pressure and rapid temperature change at the inner surface. The corresponding governing equations in three dimensions are extracted and the differential quadrature approach is applied to discretize the governing equations, boundary conditions and heat conduction equations. Different values of the in-homogeneity constant and inner-wall temperature are used to demonstrate their important roles on the distribution of displacement, stresses, temperature and induced magnetic fields. Results obtained by the present method are validated through comparison with results of the finite element method. ### Evaluating Displacements on a Circular Cylindrical Shell with the Use of Polynomial Quadrature Method

A polynomial differential quadrature method (PDQ) is proposed for the solution of complex sets of differential equations. In the PDQ method displacements are discretized as matrices whose indexes correspond to spatial coordinates; differential operators are replaced by special matrices embedding contour conditions. So partial differential equations lead naturally to a linear algebraic system easily solvable using the computational apparatus. In order to display the performance of the method, an important problem in the analysis of reactor technology is considered – the evaluation of the displacements on a circular cylindrical shell. ### EFFECTS OF VARYING IN-PLANE FORCES ON VIBRATION OF ORTHOTROPIC RECTANGULAR PLATES RESTING ON PASTERNAK FOUNDATION

uniform in-plane loadings. The first variation from the uniform loading is one, which varies linearly (Fauconneau and Marangoni , Gorman  and Naguleswaran ); the other variations may be parabolic (Bert and Devarakonda ; Hu et al. ) or harmonic variations (Benoy ). Recently, Kang and Leissa  and Leissa and Kang  have analyzed the buckling and vibration of isotropic rectangular plates using Frobénius method i.e. exact solutions are obtained when two opposite edges are simply supported and these are subjected to linearly varying in-plane loading while the other two are either clamped (i.e. SS-C-SS-C plate) or free (i.e. SS-F-SS-F plate), respectively. Very recently, Wang et al.  employed differential quadrature method to obtain the numerical results for the buckling and vibration of isotropic SS- C-SS-C rectangular plate subjected to linearly varying in-plane stresses along the simply supported edges on the basis of classical plate theory. In the wide literature devoted to buckling and vibrational behaviour of plates, there is a lack of analysis on the buckling and vibration of orthotropic rectangular plates subjected to non-uniform in-plane loads. ### Nonlinear Buckling of Circular Nano Plates on Elastic Foundation

The following article investigates nonlinear symmetric buckling of moderately thick circular Nano plates with an orthotropic property under uniform radial compressive in-plane mechanical load. Taking into account Eringen nonlocal elasticity theory, principle of virtual work, first order shear deformation plate theory (FSDT) and nonlinear Von-Karman strains, the governing equations are obtained based on displacements. The differential quadrature method (DQM) as a numerical procedure is applied for solving the equations. In this analysis, for solving the stability equations, adjacent equilibrium methodis employed. In nonlinear buckling analyses and for obtaining the buckling load, generally the available nonlinear terms of the stability equation are neglected. However, in this study, for getting the most accurate data, nonlinear terms are considered and the non-dimensional buckling load is compared with the condition of considering or neglecting that of terms and the effect of that of terms are also studied . The accuracy of the present results is validated by comparing the solutions with available studies. The effects of nonlocal parameter, thickness, radiusand elastic foundation are investigated on non-dimensional buckling loads. The results of analyses based on local and non-local theories are compared. From the results, it can be seen that the effect of nonlocal parameter on simply support condition is less than clamped condition. It can be observed that with increasing the radius of the plate, the difference between local and non-local analyses,increases. ### Solving Singularly Perturbed Differential-Difference Equations using Special Finite Difference Method

The highest order derivative is multiplied by a perturbation parameter is known as the singular perturbation problem. A differential equation in which the highest derivative is multiplied by a small positive parameter and containing at least one shift term(delay or advance) is known as singularly perturbed differential- difference equation. Prasad and Reddy  developed differential quadrature method for finding the numerical solution of boundary-value problems for a singularly perturbed differential-difference equation of mixed type. In the current reviews led by Kadalbajoo and Sharma ,, Kadalbajoo and Ramesh  and Kadalbajoo and Kumar ,, the terms negative or left shift and positive or right shift have been utilized for delay and advance respectively. The differential-difference equation plays an essential role in the mathematical modeling of various practical phenomena in the biosciences and control theory. Any system including a feedback control will quite include often time delays. These arise because a finite time is required to sense information and after that respond to it. For a point by point examination on differential-difference equation, one may refer to Bellen and Zennaro , Driver , Bellman and Cooke  books and high level monographs. ### GDQEM Analysis for Free Vibration of V-shaped Atomic Force Microscope Cantilevers

V-shaped and triangular cantilevers are widely employed in atomic force microscope (AFM) imaging techniques due to their stability. For the design of vibration control systems of AFM cantilevers which utilize patched piezo actuators, obtaining an accurate system model is indispensable prior to acquiring the information related to natural modes. A general differential quadrature element method (GDQEM) analysis based on layer-wise displacement beam theory was performed to obtain the natural frequencies of V-shaped AFM cantilevers with piezoelectric actuators. A finite element analysis was applied to validate the accuracy of numerical results. Finally, a parametric investigation of the sensitivity of natural frequencies with respect to beam geometry was performed. Simulations show that presented approach is considerably accurate and does not need a lot computational costs. Based on the governing equations, general differential quadrature method (GDQM) and GDQM could be applied for uniform and stepped plates, respectively. Thus, presented approach covers the V-shaped and triangular cantilevers perfectly and could be utilized to derive the dynamic response of such systems with a little substitution. ### Novel Techniques of Single-CarrierFrequency-Domain Equalization for Optical Wireless Communications

Single-carrier modulation using frequency domain equalization is a promising alternative to OFDM for highly dispersive channels in broadband wireless communications [14, 15]. In both approaches, a cyclic prefix (CP) is appended to each block for eliminating the interblock interference and converting, with respect to the useful part of the transmitted block, the linear convolution with the channel to circular. This allows low-complexity fast-Fourier transform-(FFT-) based receiver implementations. In recent years, SCFDE has become a powerful and an attractive link access method for the next-generation broadband wireless networks [16–18]. Because it is essentially a single-carrier system, SCFDE does not have some of the inherent problems of OFDM such as high PAPR. As a result, it has recently been receiving remarkable attention and has been adopted in the uplink of the Third Generation Partnership Project (3GPP) Long- Term Evolution (LTE)  system. ### Free Vibration Analysis of a Rotating Non Uniform Blade with Multiple Open Cracks Using DQEM

DQEM analysis of free transverse vibration of a rotating non-uniform Timoshenko blade with multiple open cracks was presented. Comparison of the proposed method with the exact solutions available in the literature revealed the excellent accuracy of this method. Also, effects of location, depth, and numbers of cracks as well as the angular velocity ### A Low Power Push-Push Differential VCO Using Current-Reuse Circuit Design Technique

VCOs with similar frequency band. The differential VCOs [16, 17] use a standard cross-coupled oscillator topology, and they cannot be used to generate frequency sources with higher frequency than the cutoff-frequency of active transistors. The QVCO  uses two differential VCOs to form a quadrature VCO and can be used as a differential VCO. Its upper operation frequency is also limited by the cut-off frequency of transistors. The proposed VCO has potential for generating higher frequency source than the reference VCOs. ### A differential QPSK modem using the TMS320C6711 DSK

We report on a student project implementing a D-QPSK mo- dem on a TI C6711 DSK. The modem incorporates func- tionalities such as QPSK symbol generation, differential en- coding and decoding, transmit- and receive filtering, quadra- ture modulation, timing synchronisation, and bit error detec- tion. Both transmitter and receiver have to be operated con- currently and in real time. The DSK’s on-board digital to analogue and analogue to digital converters were employed to interface the transmitted and received signals of the mo- dem. Based on a set of specifications, the various solutions implemented by the students had to be compatible and be able to synchronise with each other.