solution of poisson and laplace equations by quadrilateral quadrature element the quadrature element method qem is a new concept in numerical methods which was introduced only recently a quadrilateral quadrature element is developed in the paper and applied to the solution of two dimensional potential problems governed by poisson or laplace equations the results of three examples are in good agreement with the available solutions it is shown that the present quadrilateral quadrature element is very efficient and effective most significantly it can be employed to solve differential equations of physical problems with irregular geometry and complex boundary conditions

a levinson type algorithm for modeling fast sampled data the standard discrete time autoregressive model is poorly suited for modeling series obtained by sampling continuous time processes at fairly rapid rates large computational errors can occur when the levinson algorithm is used to estimate the parameters of this model because the toeplitz covariance matrix is ill suited for inversion an alternative model is developed based on an incremental difference operator rather than the shift operator it is shown that as the sampling period goes to zero unlike the standard autoregressive parameters the coefficients of this model converge to certain parameters that depend directly on the statistics of the continuous time process a levinson type algorithm for efficiently estimating the parameters of this model is derived numerical examples are given to show that when the sampling interval is small this algorithm is considerably less sensitivity to arithmetic roundoff errors than the levinson algorithm

lattice filters for rls estimation of a delta operator based model a lattice filter algorithm is derived for the recursive least squares rls estimation of the parameters of the input output model of a plant based on the delta operator formulation the lattice derivation presented allows a numerically efficient implementation particularly for large orders the order recursive property of the lattices is ideally suited for order determination techniques and applications where the order of the plant may change during operation an example concerning parameter identification for a fourth order plant with measurement noise is given

design of a multichannel two dimensional delta domain lattice filter for noise removal in this paper a multichannel form of the two dimensional 2 d delta domain lattice filter has been designed motivated by the success of 2 d delta domain lattice filters delta domain concepts are generalized to the multichannel case by a good choice of the transformation matrix that relates the q domain and delta domain filters the channels can be handled independently which makes the algorithm simple and attractive however it is capable of handling multichannel problems even if the noise is correlated between channels resulting in extremely low snr values the proposed algorithm provides the ability of handling multichannel image frames and colored images in the so called delta domain that is known to be always better than the traditional q domain in handling ill conditioned problems on line image frames carrying gaussian and multiplicative noise simultaneously on different channels as well as color images are taken into consideration and successful results are obtained in terms of noise removal

an eight node curvilinear differential quadrature formulation for reissner mindlin plates a first endeavour to exploit the differential quadrature dq method as a simple accurate and efficient numerical technique for the bending of quadrilateral reissner mindlin plates with curvilinear boundaries is made the curvilinear quadrilateral is mapped onto a square domain by using the geometric coordinate transformation this geometric mapping technique is employed to transform the governing differential equations and boundary conditions of the problem from the physical domain onto the computational domain the dq procedures are then applied to discretise the transformed set of differential equations and boundary conditions into a set of linear algebraic equations the numerical solutions to the problem are obtained by solving this set of linear algebraic equations examples illustrating the accuracy and convergence of the dq method for reissner mindlin plates with curvilinear boundaries are presented the great development potential of the dq method as an alternative to other approximate techniques for solving problems related to engineering sciences and structural mechanics is exploited in the present study

on the free vibration analysis of circular plates with stepped thickness over a concentric region by the differential quadrature element method it can be seen the dqem retains all advantages of the differential quadrature method extends the applicability range of the dq method and is more convenient for treating various boundary conditions

stability and accuracy of differential quadrature method in solving dynamic problems in this paper the differential quadrature method is used to solve dynamic problems governed by second order ordinary differential equations in time the legendre radau chebyshev chebyshev gauss lobatto and uniformly spaced sampling grid points are considered besides two approaches using the conventional and modified differential quadrature rules to impose the initial conditions are also investigated the stability and accuracy properties are studied by evaluating the spectral radii and truncation errors of the resultant numerical amplification matrices it is found that higher order accurate solutions can be obtained at the end of a time step if the gauss and radau sampling grid points are used however the conventional approach to impose the initial conditions in general only gives conditionally stable time step integration algorithms unconditionally stable algorithms can be obtained if the modified differential quadrature rule is used unfortunately the commonly used chebyshev gauss lobatto sampling grid points would not generate unconditionally stable algorithms

a new approach to the differential quadrature method for fourth order equations a generalized and more complete methodology for treating boundary conditions in the differential quadrature method dqm is presented this improved approach eliminates the deficiencies of the type grid arrangement which represents an approximation by applying the boundary conditions exactly two kinds of basis functions chebyshev and lagrange are used for concept demonstration it is found that the new approach cures most deficiencies of the current dqm 1997 by john wiley sons ltd

a comparison of differential quadrature methods for the solution of partial differential equations abstract discrete ordinate methods for the solution of partial differential equations are examined and compared a novel approach for the discrete representation of differential operators based on split range polynomial expansions is introduced the utility of the method is demonstrated for the case of differentiation of functions involving steep gradients the solution of burgers equation is presented to illustrate the effectiveness of the technique for the solution of nonlinear partial differential equations exhibiting nearly discontinuous solutions comparisons are made with other differential quadrature methods

a restricted class of multiobjective linear fractional programming problems abstract in this paper a restricted class of multiobjective linear fractional programming problems in the sense that the denominators are identical is investigated a numerical example is presented to illustrate the solution procedure

modelling and identification of non linear deterministic systems in the delta domain this paper provides a formulation for using the delta operator in the modelling of non linear systems it is shown that a unique representation of a deterministic non linear auto regressive with exogenous input narx model can be obtained for polynomial basis functions using the delta operator and expressions are derived to convert between the shift and delta domain a delta narx model is applied to the identification of a test problem a van der pol oscillator a comparison is made with the standard shift operator non linear model and it is demonstrated that the delta domain approach improves the numerical properties of structure detection leads to a parsimonious description and provides a model that is closely linked to the continuous time non linear system in terms of both parameters and structure

application of generalized differential quadrature rule in blasius and onsager equations the generalized differential quadrature rule gdqr proposed recently by the authors is applied here to third order non linear differential equations of the blasius type and to sixth order linear onsager differential equations high 3rd order differential equations in fluid mechanics are dealt with without using point techniques the half space domain is simplified in a practical way accurate results are obtained for both kinds of problems the wide applicability of the gdqr in high order differential equations is manifested further through this work copyright 2001 john wiley sons ltd

static analysis of structures by the quadrature element method qem abstract the differential quadrature method dqm is an alternative discrete approach to solving directly the governing equations of engineering and mathematical physics since the dqm does not require the derivation of the weak forms of the governing equations like the fem it can greatly reduce formulation efforts in higher order approximation the dqm has been applied in the past to the analyses of various single structural components such as bars beams membranes and plates all of the component analyses yielded good to excellent results however previous studies were limited to simple geometries and simple boundary conditions due to the limitation of using global basis functions in the present study a domain decomposition technique for the dqm is proposed to analyse truss and frame structures where the whole structural domain is represented by a collection of simple element subdomains connected together at specific nodal points this method is named the quadrature element method qem

delta levinson and schur type rls algorithms for adaptive signal processing in this paper we develop delta operator based levinson and schur type on line rls algorithms such algorithms have the potential of improved numerical behavior for ill conditioned input data these new algorithms are obtained by a unified transformation on the existing q operator based ones we first show that the conventional lattice structure can be naturally derived when the backward delta operator is used with this operator levinson and schur algorithms for the stationary stochastic model in q domain can easily be transformed into the delta domain then same transformation will be applied to the q domain on line levinson and schur type rls algorithms to obtain the delta domain counterparts their normalized versions as well as a systolic array architecture implementing the new delta schur rls algorithm are proposed extension to the equal length multichannel case is also given computer simulations show the expected numerical advantages of the delta based algorithms for fast sampled data in real time over the q domain ones under finite precision implementation

a delta least squares lattice algorithm for fast sampling most shift operator based adaptive algorithms exhibit poor numerical behavior when the input discrete time process is obtained from a continuous time process by fast sampling this includes the shift operator based least squares lattice algorithm we develop a delta least squares lattice algorithm this algorithm has a low computational complexity compared with the delta levinson rls algorithm and shows better numerical properties compared with the shift least squares lattice algorithm under fast sampling computer simulations show that the new algorithm also outperforms an existing delta least squares lattice algorithm

a delta mywe algorithm for parameter estimation of noisy ar processes we develop a delta operator based modified yule walker equation algorithm mywe for parameter estimation of a noisy autoregressive ar process the methodology in developing this new algorithm is similar to the previous works on pure ar processes computer simulation results are given to show the improvement of performance in estimating ar parameters in white noise over the q operator mywe algorithm

static analysis of frame structures by the differential quadrature element method in this paper a new version of differential quadrature method dqm is proposed and then extended to analyse frame structures the new method called the differential quadrature element method dqem retains all advantages of the earlier version of the differential quadrature method and overcomes some critical shortcomings existing in the original dqm the proposed method is however different from the quadrature element method qem proposed earlier by striz et al the methodology is explained in details herein via a differential quadrature beam element and some numerical examples are given to show the efficiency of the present method 1997 by john wiley sons ltd

brief parameter identification with derivative shift operator parametrization a way of parametrizing linear systems using an operator which combines the two commonly used operators the shift q and the delta d operators is proposed it is shown that the proposed parametrization improves the numerical property of parameter identification process by using the proposed operator the condition number of the information matrix in the identification algorithm is significantly lower than that using the q or d operator it is also shown that accurate and fast converging parameter identification can be achieved by using the new operator at fast sampling rates especially for systems with relatively wide bandwidth the numerical superiority of the new parametrization is demonstrated by simulation results

a structural view of asymptotic convergence speed of adaptive iir filtering algorithms i infinite precision implementation for pt i see ibid vol 41 no 4 p 1493 1517 1993 finite precision fp implementation is the ultimately inevitable reality of all adaptive filters including adaptive infinite impulse response iir filters this paper continues to examine the asymptotic convergence speed of adaptive iir filters of various structures and algorithms including the simple constant gain type and the newton type but under fp implementation a stochastic differential equation sde approach is used in the analysis such an approach not only greatly simplifies the fp analysis which is traditionally very involved algebraically but it also provides valuable information about the first order as well as the second order moments that the latter are not available using the ordinary differential equation ode approach the asymptotic convergence speed as well as the convergent values of the pertinent moments of fp errors are examined in terms of unknown system pole zero locations the adverse effects of lightly damped low frequency ldlf poles resulting from fast sampling on the local transient and convergent behavior of various structures and algorithms are analyzed and compared the new results agree with the existing ones when reduced to the finite impulse response fir case in particular the explosive behavior of pertinent error variances of newton type iir algorithms when the forgetting factor spl lambda 1 is also concluded computer simulation verifies the predicted theoretical results

on properties of information matrices of delta operator based adaptive signal processing algorithms in this paper we study the properties of information matrices of the difference operator or the so called delta spl delta operator based algorithms for adaptive signal processing we show that the conditioning of a transformed information matrix in the spl delta domain is always better than that of the original information matrix in the conventional q domain for ill conditioned problems the results obtained in this paper give an explanation of the advantages of using delta operator algorithms for adaptive signal processing that have been developed the analysis in this paper also helps to clarify the problem about the effect of spl delta in the delta operator algorithms and justify the use of sampling interval as spl delta in most cases when the ill conditioning is caused by fast sampling of continuous time systems

j lossless and extended j lossless factorizations approach for domain h control the paper addresses problems of numerical conditioning of the discrete time h x control design based on j lossless factorizations of the chain scattering representation of a plant it is demonstrated that for a sufficiently small sampling period forward shift operator techniques may become ill conditioned and numerical robustness and reliability of design procedures can be significantly improved by utilizing a so called i operator form of the source problem state space models of all i domain controllers are given the solutions are obtained by considering two coupled i domain algebraic riccati equations that can be solved by applying a generalized eigenproblem formulation numerical conditioning of these solutions can effectively be evaluated by using a suitable relative condition number an extended j lossless factorization for generalized plants with zeros on the stability boundary circle is also given a so called zero compensator technique adapted to the i domain models is utilized to cancel suc

on optimal selection of interior points for applying discretized boundary conditions in dq vibration analysis of beams and plates the direct implementation of derivative boundary conditions at edge points was demonstrated to be simple flexible and computationally efficient in the vibration analysis of beams and plates by using the differential quadrature dq method in the approach the discretized governing equations at certain interior points need to be replaced by the discretized boundary conditions naturally one may have questions as to whether the replacement points can be selected arbitrarily and how to choose the optimal replacement points this paper tries to answer these questions and its focus is to investigate systematically the effect of the replacement location on the accuracy of the numerical solution from the error distribution analysis of derivative approximation it is found that the optimal position for the replacement is the interior point just adjacent to the boundary this conclusion is confirmed by the numerical experiments of free vibration analysis for beams and plates with various boundary conditions

a roundoff error analysis of the lms adaptive algorithm the steady state output error of the least mean square lms adaptive algorithm due to the finite precision arithmetic of a digital processor is analyzed it is found to consist of three terms 1 the error due to the input data quantization 2 the error due to the rounding of the arithmetic operations in calculating the filter s output and 3 the error due to the deviation of the filter s coefficients from the values they take when infinite precision arithmetic is used the last term is of paricular interest because its mean squared value is inversely proportional to the adaptation step size both fixed and floating point arithmetics are examined and the expressions for the final mean square error are found to be similar the relation between the quantization error and the error that occurs when adaptation possibly ceases due to quantization is also investigated

vibrations of non uniform rings studied by means of the differential quadrature method abstract the vibrational behavior of ring like structures is of interest in several technological situations from automotive and aircraft tires to electric motors and transformers this study deals with the solution of the governing sixth order differential equation by using the differential quadrature method the fundamental frequency of in plane inextensional vibration is thus determined and compared with the value predicted by the optimized rayleigh ritz method

the generalized differential quadrature rule for initial value differential equations abstract the generalized differential quadrature rule gdqr proposed recently by the authors is applied here to solve initial value differential equations of the 2nd to 4th order differential quadrature expressions are derived based on the gdqr for these equations the hermite interpolation functions are used as trial functions to obtain the explicit weighting coefficients for an easy and efficient implementation of the gdqr the numerical solutions for example problems demonstrate that the gdqr has high efficiency and accuracy a detailed discussion on the present method is presented by comparing with other existing methods the present method can be extended to other types of differential equation systems

conditioning of lms algorithms with fast sampling the lms algorithm is very commonly used in signal processing its convergence properties depend primarily on the step size chosen and the condition number of an information matrix associated with the system in most applications today the lms uses a regression vector based on the shift operator including the ubiquitous tapped delay line we demonstrate that generically when fast sampling is employed these regression vectors lead to poorly conditioned lms by comparison delta operator based regression vectors lend with rapid sampling to improved condition numbers hence to better performance

application of differential quadrature to static analysis of structural components the numerical technique of differential quadrature for the solution of linear and non linear partial differential equations first introduced by bellman and his associates is applied to the equations governing the deflection and buckling behaviour of one and two dimensional structural components separate transformations are used for higher order derivatives as suggested by mingle thus extending the method to treat fourth order equations and to include multiple boundary conditions in the respective co ordinate directions results are obtained for various boundary and loading conditions and are compared with existing exact and numerical solutions by other methods the application of differential quadrature to this class of problems is seen to lead to accurate results with relatively small computational effort

the generalized differential quadrature rule for fourth order differential equations the generalized differential quadrature rule gdqr proposed here is aimed at solving high order differential equations the improved approach is completely exempted from the use of the existing point technique by applying multiple conditions in a rigorous manner the gdqr is used here to static and dynamic analyses of bernoulli euler beams and classical rectangular plates numerical error analysis caused by the method itself is carried out in the beam analysis independent variables for the plate are first defined the explicit weighting coefficients are derived for a fourth order differential equation with two conditions at two different points it is quite evident that the gdqr expressions and weighting coefficients for two dimensional problems are not a direct application of those for one dimensional problems the gdqr are implemented through a number of examples good results are obtained in this work copyright 2001 john wiley sons ltd

maximum likelihood estimation of delta domain model parameters from noisy output signals fast sampling is desirable to describe signal transmission through wide bandwidth systems the delta operator provides an ideal discrete time modeling description for such fast sampled systems however the estimation of delta domain model parameters is usually biased by directly applying the delta transformations to a sampled signal corrupted by additive measurement noise this problem is solved here by expectation maximization where the delta transformations of the true signal are estimated and then used to obtain the model parameters the method is demonstrated on a numerical example to improve on the accuracy of using a shift operator approach when the sample rate is fast