A **differential** **quadrature** **method** is a numerical **method** for evaluating deriv- atives of sufficiently smooth function as proposed by [1]. The basic idea of diffe- rential **quadrature** came from Gauss **Quadrature** [2], 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** [1]. 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 [3].

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Abstract: In this paper, A multi-domain **differential** **quadrature** **method** is employed to solve a mode III crack problem. The domain of the problem is assumed to be irregular rather than it possesses discontinuities, (cracks). The entire domain is divided into several subdomains, according to the crack locations. A conformal mapping is applied to transform the irregular subdomains to regular ones. Then the **differential** **quadrature** **method** is employed to solve the problem over the transformed domains. Further, it’s focused on the crack regions by applying the localized version of **differential** **quadrature** **method**. The out of plane deflection is obtained at the immediate vicinity of the crack tips, such that the stress intensity factor can be calculated. The obtained results are compared with the previous analytical ones. Furthermore a parametric study is introduced to investigate the effects of elastic and geometric characteristics on the values of stress intensity factor.

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.

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.

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Using **differential** **quadrature** **method**, this study investigated pull-in instability of beam-type nano-switches under the effects of small-scale and intermolecular forces including the van der Waals and the Casimir forces. In these nano-switches, electrostatic forces were served as the driving force, and von-Karman type nonlinear strain was used to examine nonlinear geometric effects. To derive nonlinear governing equations as well as the related boundary conditions for the nano-beam, variation **method** was used. Besides, to study the influence of size effect, the nonlocal elasticity theory was employed and the resulting governing equations were solved using **differential** **quadrature** **method**. Finally, the pull-in parameters were studied using the nonlocal theory and the results were compared with the numerical results of the classical continuum theory as well as experimental results contained in the references. Results demonstrated that taking into consideration the von-Karman type nonlinear strain increases the beam stiffness and hence, the pull-in voltage. Besides, use of the small scale, compared with the classical theory of elasticity, yields results much closer to experimental results.

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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.

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A hybrid technique consists of **differential** **quadrature** **method** and Runge-Kutta fourth order **method** is employed to solve three reaction-diffusion problems. The validity of the proposed technique is proved by comparing the obtained results with the previous analytical ones. The technique

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.

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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.

The **differential** **quadrature** **method** as a powerful semi- analytical tool is employed to obtain the discretized forms governing and boundary equations of the conical shell. The general rule of DQM postulates that derivatives of any smooth function at a discrete point in the domain can be expressed as a weighted linear summation of all the functional values at all discrete sampling points. The key to DQM is to determine the weighting coefficients for discretization of a derivative of any order.

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On the basis of di ﬀ erential unitary space-time coding and diﬀerential orthogonal space-time coding, by using the star QAM **method**, two kinds of multiple amplitudes diﬀerential space-time coding schemes are presented in this paper; one is multiple amplitudes di ﬀ erential unitary space-time cod- ing; the other is multiple amplitudes di ﬀ erential orthogo- nal space-time coding. The two schemes can avoid the per- formance degradation of conventional DSTC scheme based on PSK modulation due to the decrease of minimum pro- duce distance in high spectrum e ﬃ ciency. The developed MDUSTC scheme can be applied to any number of antennas, and implement di ﬀ erent data rates, and low-complexity dif- ferential modulation due to the application of cyclic group codes. It has higher coding gain than existing diﬀerential unitary space-time coding. For the developed MDOSTC scheme, it has higher coding gain than existing di ﬀ erential orthogonal space-time coding schemes. Moreover, it has sim- pler decoder and can obtain higher code rate in the case of three or four transmit antennas. The simulation results in fading channel also show that our schemes have lower BER than the corresponding di ﬀ erential unitary space-time codes

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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

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performance is better compared to ACO-OFDM when min- imum mean square error (MMSE) detection is employed. The latter property is due to the inherent frequency diversity gain of SCFDE [20] and its low PAPR. Since the LED has limited linear range in its transfer characteristics, any values outside of that limited range will be clipped and distorted resulting in performance loss. We also propose in this paper two other schemes for generating real, positive signals with low PAPR for IM/DD optical DOW communications using SCFDE. The rest of the paper is organized as follows. In Section 2, we review the ACO-OFDM scheme. In Section 3, we present the proposed ACO-SCFDE. The two other newly proposed low PAPR schemes for optical communication using SCFDE which we call Repeat-and-Clipped Optical SCFDE (RCO-SCFDE) and Decomposed **Quadrature** Opti- cal SCFDE (DQO-SCFDE) are presented in Sections 4 and 5, respectively followed by an analysis of the PAPR issues for DOW in Section 4. Performance analyses are presented in Section 7 followed by the conclusion in Section 8.

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Turner presented a sensitivity evaluation analysis of the bending and torsional vibrations of V-shaped cantilevers [11]. Chang [12] studied the bending vibrations of a rectangular cantilever as a function of the cantilever angle. Later in his studies, Chang assessed the influence of a damper on rectangular cantilever vibration sensitivity [13]. Several discussions have been introduced as controlling and positioning problem for scanning probe devices, especially the dynamic mode AFM [14]. The necessity of real-time imaging in the nanoworld and the high sensitivity of nanoobjects make simulation-based works highly important. So, several modeling and simulation works have been developed. Nowadays, developing the V- shaped cantilever arrays and applications have been increasingly motivated because of emerged technologies and knowledge. As an example, in [15], for measuring concentrated masses or particles, a novel mass sensor incorporating V- shaped cross section cantilever was proposed by locating the analyte at predefined positions for both improving the mass detecting sensitivity and reducing the measuring deviation. In [16], the resonant frequency of flexural vibration for a V- shaped AFM cantilever has been investigated using the Timoshenko beam theory. It has been mentioned that the resonant frequency is sensitive to the width ratio and by increasing this ratio, the resonant frequency decreases, but critical contact stiffness increases, and finally, the variations of the height and breadth taper ratios and width ratio are affected on the sensitivity to the contact stiffness. Also, evaluation of optimum geometric parameters and optimum cantilever slope is considered as a significant purpose in order to obtain maximum flexural sensitivity by using genetic algorithm optimization **method** [17]. Adopting the parameters for the design of V-shape micro cantilever according to the sample contact stiffness, maximum flexural sensitivity can be obtained, so

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A **differential** frequency source is a critical component in fully- integrated radio-frequency transceivers. The design of high-frequency source entails trade-offs among several parameters including the phase noise, power consumption, frequency tuningrange, power consumption and chip area. These parameters are lumped into an equation to indicate the figure of merit of the voltage-controlled oscillator (VCO) for performance comparison. The frequency signal source can be formed with (i) a fundamental oscillator, (ii) a low frequency source in conjunction with frequency multipliers, and (iii) a high- frequency source with frequency dividers. The second approach is

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Abstract. In this work, we extended the work of [12] to approximate the solution of fractional order **differential** equations by an integral representation in the complex plane. The resultant integral is approximated to high order accuracy using **quadrature**. The accuracy of the **method** depends on the selection of optimal contour of integration. Several contour have been proposed in the literature for solving fractional **differential** equations. In the present work, we will investigate the applicability of the recently developed optimal contour in [16] for solving fractional **differential** equations. Various fractional order **differential** equations are approximated and the results are compared with other methods to demonstrate the efficiency and accuracy of the **method** for various optimal contour of integrations.

In this section, an error estimate of the applied **method** for the smooth solutions of fractional Logistic diﬀerential equation will be provided. For the sake of applying the theory of orthogonal polynomials we employ the variable transformations t = (1+x)/2, and let u(x) = y((1+x)/2) to rewrite (1), (2) as follows

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.

The accurate early identification of developmental delay in young children is important. The aim of this study was to highlight and propose a solution to problems associated with scoring a UK developmental screening tool known as the Schedule of Growing Skills II. Potential problems associated with the sensi- tivity of this screening tool were identified. As a possible solution to this problem, an alternative scoring **method** was developed to yield a developmental quotient. A pilot investigation of the new scoring **method** was conducted through comparisons with the Griffiths Mental Development Scales. Forty-three children aged 0 - 5 years were recruited and administered both developmental assessments. Results from both as- sessments were compared to examine validity. Both the new and published scoring methods showed good concurrent validity, however the new scoring **method** demonstrated better criterion-related validity in terms of higher sensitivity, comparable specificity, generally higher over-referrals, and lower un- der-referrals. The Schedule of Growing Skills II could be a valid, cost-effective way of screening for de- velopmental delay in young children using this new, more sensitive scoring **method**.

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One of the techniques to solve the Burgers equation is by using the Finite Difference **Method** (FDM) which is the simplest **method**. This **method** solved by replacing the values at certain grid points and approximates the derivatives by differences in these values. The partial derivatives in the PDE at each grid point are approximated from the neighborhood values.

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