on stable numerical differentiation based on a regularized volterra equation two different approaches for numeri cal differentiation are considered the first approach consists of solving a regularized volterra equation while the second approach is based on solving a disretized version of the regularized volterra equation numerical experiments show that these methods are efficient and compete fa vorably with the variational regularization method for stable calculating the derivatives of noisy functions

numerical solution of systems of second order boundary value problems using continuous genetic algorithm in this paper continuous genetic algorithm is introduced as an efficient solver for systems of second order boundary value problems where smooth solution curves are used throughout the evolution of the algorithm to obtain the required nodal values of the unknown variables the solution methodology is based on representing each derivative in the system of differential equations by its finite difference approximation after that the overall residue for all nodes in the given system of differential equations is formulated the solution to the system of differential equations is then converted into the problem of minimizing the overall residue or maximizing the fitness function based on the nodal values generated from the genetic operators three numerical test problems including linear and nonlinear systems were analyzed to illustrate the procedure and confirm the performance of the proposed method in addition to that a convergence and sensitivity analysis to genetic operators and control parameters of the algorithm has been carried out the numerical results show that the proposed algorithm is a robust and accurate procedure for solving systems of second order boundary value problems furthermore the obtained accuracy for the solutions using cga is much better than the results obtained using some modern methods

explicit inverse of a generalized vandermonde matrix in this paper the author gives an explicit closed form expression for the nxn inverse matrix v g k n 1 of the generalized nxn vandermonde matrix v g k n by using the elementary symmetric functions symbolic and numerical results are presented

a note on the stirling matrix of the second kind in this article a new algorithm for generating the elements of the n 1 by n 1 stirling matrix of the second kind is presented based on this algorithm a computational maple procedure is written and a sample output of this procedure is also given for some values of n

existence results and iterative method for solving a nonlinear biharmonic equation of kirchhoff type abstract in this paper for investigating a boundary value problem for a nonlinear biharmonic equation of kirchhoff type we propose an efficient approach by the reduction of the problem to an operator equation for the nonlinear part of the equation the result is that we have established the existence and uniqueness of a solution under some easily verified conditions also we propose an iterative method for finding the solution some examples demonstrate the applicability of the theoretical results and the efficiency of the iterative method

dsc time domain solution of maxwell s equations a new computational algorithm the discrete singular convolution dsc is introduced for solving scattering and guided wave problems described by time domain maxwell s equations the dsc algorithm is utilized for the spatial discretization and the fourth order runge kutta scheme is used for the time advancing staggered meshes are used for electromagnetic fields four standard test problems a hollow air filled waveguide a dielectric slab loaded rectangular waveguide a shield microstrip line and a dielectric square are employed to illustrate the usefulness to test the accuracy and to explore the limitation of the dsc algorithm results are compared with those of finite difference scaling function multi resolution time domain and finite element based high frequency structure simulator numerical experiments indicate that the present algorithm is a promising approach for achieving high accuracy in electromagnetic wave computations

staggered time integrators for wave equations we consider variations of the adams bashforth backward differentiation and runge kutta families of time integrators to solve systems of linear wave equations on uniform time staggered grids these methods are found to have smaller local truncation errors and to allow larger stable time steps than traditional nonstaggered versions of equivalent orders we investigate the accuracy and stability of these methods analytically experimentally and through the use of a novel root portrait technique

numerical treatment of two dimensional interfaces for acoustic and elastic waves we present a numerical method to take into account 2d arbitrary shaped interfaces in classical finite difference schemes on a uniform cartesian grid this work extends the explicit simplified interface method esim previously proposed in 1d j comput phys 168 2001 227 248 the physical problem under study concerns the linear hyperbolic systems of acoustics and elastodynamics with stationary interfaces our method maintains near the interfaces properties of the schemes in homogeneous medium such as the order of accuracy and the stability limit moreover it enforces the numerical solution to satisfy the exact interface conditions lastly it provides subcell geometrical features of the interface inside the meshing the esim can be coupled automatically with a wide class of numerical schemes lax wendroff flux limiter schemes etc for a negligible additional computational cost throughout the paper we focus on the challenging case of an interface between a fluid and an elastic solid in numerical experiments we provide comparisons between numerical solutions and original analytic solutions showing the efficiency of the method

on the numerical integration of the schrodinger equation a finite difference method for solving the schrodinger eigenvalue equation is generalised in order to treat a larger number of potentials results are shown for asymmetrical two well oscillators

high rate self synchronizing codes self synchronization under the presence of additive noise can be achieved by allocating a certain number of bits of each codeword as markers for synchronization difference systems of sets are combinatorial designs which specify the positions of synchronization markers in codewords in such a way that the resulting error tolerant self synchronizing codes may be realized as cosets of linear codes ideally difference systems of sets should sacrifice as few bits as possible for a given code length alphabet size and error tolerance capability however it seems difficult to attain optimality with respect to known bounds when the noise level is relatively low in fact the majority of known optimal difference systems of sets are for exceptionally noisy channels requiring a substantial amount of bits for synchronization to address this problem we present constructions for difference systems of sets that allow for higher information rates while sacrificing optimality to only a small extent our constructions utilize optimal difference systems of sets as ingredients and when applied carefully generate asymptotically optimal ones with higher information rates we also give direct constructions for optimal difference systems of sets with high information rates and error tolerance that generate binary and ternary self synchronizing codes

reconstruction of thermally symmetrized quantum autocorrelation functions from imaginary time data in this paper i propose a technique for recovering quantum dynamical information from imaginary time data via the resolution of a one dimensional hamburger moment problem it is shown that the quantum autocorrelation functions are uniquely determined by and can be reconstructed from their sequence of derivatives at origin a general class of reconstruction algorithms is then identified according to theorem 3 the technique is advocated as especially effective for a certain class of quantum problems in continuum space for which only a few moments are necessary for such problems it is argued that the derivatives at origin can be evaluated by monte carlo simulations via estimators of finite variances in the limit of an infinite number of path variables finally a maximum entropy inversion algorithm for the hamburger moment problem is utilized to compute the quantum rate of reaction for a one dimensional symmetric eckart barrier

a variational method for numerical differentiation a new method is given for effecting numerical differentiation by means of an optimization procedure the method is shown to be effective for the differentiation of noisy functions and stability and convergence results for a steepest descent implementation are proved

closed form expressions for the finite approximations of first and higher derivatives based on taylor series numerical differentiation formulas based on interpolating polynomials operators and lozenge diagrams can be simplified to one of the finite difference approximations based on taylor series in this paper we have presented closed form expressions of these approximations of arbitrary order for first and higher derivatives a comparison of the three types of approximations is given with an ideal digital differentiator by comparing their frequency responses the comparison reveals that the central difference approximations can be used as digital differentiators because they do not introduce any phase distortion and their amplitude response is closer to that of an ideal differentiator it is also observed that central difference approximations are in fact the same as maximally flat digital differentiators in the appendix a computer program written in mathematica is presented which can give the approximation of any order to the derivative of a function at a certain mesh point

higher order and higher floating point precision numerical approximations of finite strain elasticity moduli two real domain numerical approximation methods for accurate computation of finite strain elasticity moduli are developed and their accuracy and computational efficiency investigated with reference to hyperelastic constitutive models with known analytical solutions the methods are higher order and higher floating point precision numerical approximation the latter being novel in this context a general formula for higher order approximation finite difference schemes is derived and a new procedure is proposed to implement increased floating point precision the accuracy of the approximated elasticity moduli is investigated numerically using higher order approximations in standard double precision and increased quadruple precision it is found that as the order of the approximation increases the elasticity moduli tend towards the analytical solution using higher floating point precision the approximated elasticity moduli for all orders of approximation are found to be more accurate than the standard double precision evaluation of the analytical moduli application of the techniques to a finite element problem shows that the numerically approximated methods obtain convergence equivalent to the analytical method but require greater computational effort it is concluded that numerical approximation of elasticity moduli is a powerful and effective means of implementing advanced constitutive models in the finite element method without prior derivation of difficult analytical solutions

high order fdtd methods via derivative matching for maxwell s equations with material interfaces this paper introduces a series of novel hierarchical implicit derivative matching methods to restore the accuracy of high order finite difference time domain fdtd schemes of computational electromagnetics cem with material interfaces in one 1d and two spatial dimensions 2d by making use of fictitious points systematic approaches are proposed to locally enforce the physical jump conditions at material interfaces in a preprocessing stage to arbitrarily high orders of accuracy in principle while often limited by numerical instability orders up to 16 and 12 are achieved respectively in 1d and 2d detailed stability analyses are presented for the present approach to examine the upper limit in constructing embedded fdtd methods as natural generalizations of the high order fdtd schemes the proposed derivative matching methods automatically reduce to the standard fdtd schemes when the material interfaces are absent an interesting feature of the present approach is that it encompasses a variety of schemes of different orders in a single code another feature of the present approach is that it can be robustly implemented with other high accuracy time domain approaches such as the multiresolution time domain method and the local spectral time domain method to cope with material interfaces numerical experiments on both 1d and 2d problems are carried out to test the convergence examine the stability access the efficiency and explore the limitation of the proposed methods it is found that operating at their best capacity the proposed high order schemes could be over 2000 times more efficient than their fourth order versions in 2d in conclusion the present work indicates that the proposed hierarchical derivative matching methods might lead to practical high order schemes for numerical solution of time domain maxwell s equations with material interfaces

numerical differentiation from a viewpoint of regularization theory in this paper we discuss the classical ill posed problem of numerical differentiation assuming that the smoothness of the function to be differentiated is unknown using recent results on adaptive regularization of general ill posed problems we propose new rules for the choice of the stepsize in the finite difference methods and for the regularization parameter choice in numerical differentiation regularized by the iterated tikhonov method these methods are shown to be effective for the differentiation of noisy functions and the order optimal convergence results for them are proved

solvability of dynamic antiplane frictional contact problems for viscoelastic cylinders abstract we study a mathematical model which describes the antiplane shear deformations of a cylinder in frictional contact with a rigid foundation the process is dynamic the material behavior is described with a linearly viscoelastic constitutive law and friction is modeled with a general subdifferential boundary condition we derive a variational formulation of the model which is in a form of an evolutionary hemivariational inequality for the displacement field then we prove the existence of a weak solution to the model the proof is based on an abstract result for second order evolutionary inclusions in banach spaces also we prove that under additional assumptions the weak solution to the model is unique we complete our results with concrete examples of friction laws for which our results are valid

mathematical proof of closed form expressions for finite difference approximations based on taylor series taylor series based finite difference approximations of derivatives of a function have already been presented in closed forms with explicit formulas for their coefficients however those formulas were not derived mathematically and were based on observation of numerical results in this paper we provide a mathematical proof of those formulas by deriving them mathematically from the taylor series

extrapolation methods and derivatives of limits of sequences let s m be an infinite sequence whose limit or antilimit s can be approximated very efficiently by applying a suitable extrapolation method e 0 to s m assume that the s m and hence also s are differentiable functions of some parameter d d s being the limit or antilimit of d d s m and that we need to approximate d d s a direct way of achieving this would be by applying again a suitable extrapolation method e 1 to the sequence d d s m and this approach has often been used efficiently in various problems of practical importance unfortunately as has been observed at least in some important cases when d d s m and s m have essentially different asymptotic behaviors as m the approximations to d d s produced by this approach despite the fact that they are good do not converge as quickly as those obtained for s and this is puzzling in this paper we first give a rigorous mathematical explanation of this phenomenon for the cases in which e 0 is the richardson extrapolation process and e 1 is a generalization of it thus showing that the phenomenon has very little to do with numerics following that we propose a procedure that amounts to first applying the extrapolation method e 0 to s m and then differentiating the resulting approximations to s and we provide a thorough convergence and stability analysis in conjunction with the richardson extrapolation process it follows from this analysis that the new procedure for d d s has practically the same convergence properties as e 0 for s we show that a very efficient way of implementing the new procedure is by actually differentiating the recursion relations satisfied by the extrapolation method used and we derive the necessary algorithm for the richardson extrapolation process we demonstrate the effectiveness of the new approach with numerical examples that also support the theory we discuss the application of this approach to numerical integration in the presence of endpoint singularities we also discuss briefly its application in conjunction with other extrapolation methods

presentation error analysis and numerical experiments on a group of 1 step ahead numerical differentiation formulas in order to achieve higher computational precision in approximating the first order derivative of the target point the 1 step ahead numerical differentiation formulas are presented these formulas greatly remedy some intrinsic weaknesses of the backward numerical differentiation formulas and overcome the limitation of the central numerical differentiation formulas in addition a group of formulas are proposed to obtain the optimal step length moreover the error analysis of the 1 step ahead numerical differentiation formulas and the backward numerical differentiation formulas is further investigated numerical studies show that the proposed optimal step length formulas are effective and the performance of the 1 step ahead numerical differentiation formulas is much better than that of the backward numerical differentiation formulas in the first order derivative approximation

multidimensional optimization of finite difference schemes for computational aeroacoustics because of the long propagation distances computational aeroacoustics schemes must propagate the waves at the correct wave speeds and lower the isotropy error as much as possible the spatial differencing schemes are most frequently analyzed and optimized for one dimensional test cases therefore in multidimensional problems such optimized schemes may not have isotropic behavior in this work optimized finite difference schemes for multidimensional computational aeroacoustics are derived which are designed to have improved isotropy compared to existing schemes the derivation is performed based on both taylor series expansion and fourier analysis various explicit centered finite difference schemes and the associated boundary stencils have been derived and analyzed the isotropy corrector factor a parameter of the schemes can be determined by minimizing the integrated error between the phase or group velocities on different spatial directions the order of accuracy of the optimized schemes is the same as that of the classical schemes the advantage being in reducing the isotropy error the present schemes are restricted to equally spaced cartesian grids so the generalized curvilinear transformation method and cartesian grid methods are good candidates the optimized schemes are tested by solving various multidimensional problems of aeroacoustics

a unified approach to whiteman s and ding helleseth s generalized cyclotomy over residue class rings the theory of cyclotomy dates back to gauss and has a number of applications in combinatorics coding theory and cryptography cyclotomy over a residue class ring bbzv can be divided into classical cyclotomy or generalized cyclotomy depending on v prime or composite in this paper we introduce a generalized cyclotomy of order d over bbzp1e1p2e2 pnen which includes whiteman s and ding helleseth s generalized cyclotomy as special cases here p1 p2 pn are pairwise distinct odd primes satisfying d pi 1 for all 1 i n and e1 e2 en are positive integers we derive some basic properties of the corresponding cyclotomic numbers and obtain a general formula to compute them via classical cyclotomic numbers as applications we completely solve an open problem and a conjecture on whiteman s generalized cyclotomy of order four over bbzp1p2 besides we also construct an infinite series of near optimal codebooks over bbzp1p2 as well as some infinite series of asymptotically optimal difference systems of sets over bbzp1e1p2e2 pnen

a family of numerov type exponentially fitted predictor corrector methods for the numerical integration of the radial schro dinger equation a family of predictor corrector exponential numerov type methods is developed for the numerical integration of the one dimensional schrodinger equation the formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions the new methods are very simple and integrate more exponential functions than both the well known fourth order numerov type exponentially fitted methods and the sixth algebraic order runge kutta type methods numerical results also indicate that the new methods are much more accurate than the other exponentially fitted methods mentioned above

a decomposed immersed interface method for variable coefficient elliptic equations with non smooth and discontinuous solutions a second order accurate finite difference method is presented for solving two dimensional variable coefficient elliptic equations on cartesian grids in which the coefficients the source term the solution and its derivatives may be nonsmooth or discontinuous across an interface a correction term is introduced to the standard central difference stencil so that the numerical discretization is well defined across the interface we also propose a new method to approximate the correction term as part of the iterative procedure the method is easy to implement since the correction term only needs to be added to the right hand side of the system therefore the coefficient matrix remains symmetric and diagonally dominant allowing for most standard solvers to be used numerical examples show good agreements with exact solutions and the order of accuracy is comparable with other immersed interface methods

a hierarchical optical flow estimation algorithm based on the interlevel motion smoothness constraint abstract an optical flow estimation algorithm is proposed based on the multiresolution representation of the image i e the image pyramid since the image pyramid consists of several low pass filtered versions of the image sequence the motion vector at each pyramid level is also the low pass filtered versions of the motion vector at the finest level thus the interlevel motion smoothness constraint is introduced in the motion estimation by defining the operators to establish an appropriate relationship between two adjacent pyramid levels the employment of the image pyramid allows us to estimate the large flow vectors and to adopt easily the multigrid technique to improve the convergence speed of the proposed algorithm the simulation results reveal that the proposed algorithm provides more accurate motion estimation compared to the methods using the smoothness constraints such as horn and schunck s algorithm in addition the theoretical analysis of the convergence behavior of the proposed algorithm is also provided in this paper

finding a control parameter in one dimensional parabolic equations an inverse problem concerning diffusion equation with source control parameter is considered several finite difference schemes are presented for identifying the control parameter which produces at each time a desired temperature at a given point in a spatial domain these schemes are based on the second order 3 point forward time centred space ftcs explicit formula and the fourth order 5 point ftcs explicit scheme and the second order 3 point backward time centred space btcs implicit technique and the fourth order 3 3 crandall s implicit formula the 5 point ftcs scheme has a bounded range of stability but its fourth order accuracy is significant the 3 point btcs method uses less central processor cpu time than the 3 3 crandall s technique but it is only second order accurate the basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach developed from the 1974 work of warming and hyett this allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes the results of a numerical experiment are presented and accuracy and the cpu time needed for the parabolic inverse problem are discussed

modified backward differentiation methods of the adams type based on exponential interpolation abstract for the numerical solution of first order differential equations multistep methods of the adams type are established which integrate exactly the functions exp kx x j j 0 1 n where k is a parameter these methods have stepsize dependent coefficients which also depend on k within each integration segment one can choose for k the value which minimizes in a certain sense the local truncation error in this way the new method leads to highly accurate results in comparison with the classical methods of the adams type especially when the solution of the differential equation shows an exponential behaviour numerical examples are provided to illustrate this

the immersed interface multigrid methods for interface problems new multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic pdes that involve interfaces and discontinuities for elliptic interface problems the multigrid solver developed in this paper works while some other multigrid solvers do not for linear parabolic equations we have developed the second order maximum principle preserving finite difference scheme in this paper we use the crank nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives numerical examples are also presented

normalization of temporal distance parameters in pediatric gait several techniques for the normalization of temporal distance parameters in pediatric gait are given the resulting normalized data can be used to compare or discriminate individuals or groups of individuals without the effect of variables such as age and height the normalization with respect to a reference data set or with respect to the data set itself can be accommodated three novel techniques for normalization of gait data have been given offset decorrelation and detrending for normalization of stride length with respect to height all three are superior to the commonly accepted technique of dividing by the height the offset technique has obscure units and will have a residual correlation the decorrelation technique also has obscure units but will have zero correlation provided it is being normalized using a linear model or piecewise linear model fitted to the data in a least squares sense the detrending technique will also result in a zero correlation if piecewise linear or polynomial models fitted to the data in a least squares sense are used the detrending technique is the most useful of the three techniques proposed as it will also generate the same units as the original data set and can be easily scaled so that its magnitude is also comparable with the original data both the decorrelation and detrending techniques can be used simultaneously to normalize data with respect to two or more variables

new finite difference formulas for numerical differentiation conventional numerical differentiation formulas based on interpolating polynomials operators and lozenge diagrams can be simplified to one of the finite difference approximations based on taylor series and closed form expressions of these finite difference formulas have already been presented in this paper we present new finite difference formulas which are more accurate than the available ones especially for the oscillating functions having frequency components near the nyquist frequency closed form expressions of the new formulas are given for arbitrary order a comparison of the previously available three types of approximations is given with the presented formulas a computer program written in mathematica based on new formulas is given in the appendix for numerical differentiation of a function at a specified mesh point

on a connection between symmetric polynomials generalized stirling numbers and the newton general divided difference interpolation polynomial this paper gives a connection between symmetric polynomials generalized stirling numbers and the newton general divided difference interpolation polynomial the generalized stirling numbers of the first and second kind denoted s n k a and s n k a respectively are given symbolically for the case n 5 as an illustrative example by using maple programming maple v r3 programming reference manual these numbers depend on n k and n distinct values a 1 a 2 a n the ordinary stirling numbers s n k and s n k may be obtained as special cases by taking a i i 1 i 1 1 n

on the numerical integration of the schrodinger equation in the finite difference schemes abstract formulae for the integration of the schrodinger equation for bound states based on the scheme of central differences are generated from the taylor expansion with the help of formal pade approximants these methods are studied in matrix form and a limiting formula the best for a given discretization is obtained

general explicit difference formulas for numerical differentiation through introducing the generalized vandermonde determinant the linear algebraic system of a kind of vandermonde equations is solved analytically by use of the basic properties of this determinant and then we present general explicit finite difference formulas with arbitrary order accuracy for approximating first and higher derivatives which are applicable to unequally or equally spaced data comparing with other finite difference formulas the new explicit difference formulas have some important advantages basic computer algorithms for the new formulas are given and numerical results show that the new explicit difference formulas are quite effective for estimating first and higher derivatives of equally and unequally spaced data

lagrange interpolation for continuous piecewise smooth functions this note is devoted to lagrange interpolation for continuous piecewise smooth functions a new family of interpolatory functions with explicit approximation error bounds is obtained we apply the theory to the classical lagrange interpolation

a remainder formula of numerical differentiation for the generalized lagrange interpolation through exploiting the generalized lagrange interpolation for continuous piecewise smooth functions a remainder term of numerical differentiation is given also the error estimates for the numerical differentiation formula are presented

constructions of difference systems of sets from finite projective geometry difference systems of sets dsss are combinatorial structures introduced by levenshtein in connection with code synchronization in this paper some recursive constructions of dsss obtained from finite projective geometry are presented as a consequence new infinite families of optimal dsss are obtained

constructions of difference systems of sets and disjoint difference families difference systems of sets dsss are combinatorial structures that are a generalization of cyclic difference sets and arise in connection with code synchronization in this correspondence we give some constructions of dss from cyclic designs and get some infinite classes of optimal difference systems of sets

simplified analytical expressions for numerical differentiation via cycle index based on the lagrange interpolation to the function f x 0 for arbitrarily chosen x 0 and logarithmic differentiation we give a simple approach to analytical expressions for numerical differentiation using cycle index a detailed analysis for the remainder is also included

parallel techniques for a boundary value problem with non classic boundary conditions new parallel techniques are developed for obtaining approximate solutions to an initial boundary value problem for the three dimensional parabolic partial differential equation pde with non classic boundary conditions while sharing some common features with the one dimensional models the solution of three dimensional equations are substantially more difficult thus some considerations are taken to be able to extend some ideas of one dimensional case using a suitable transformation the solution of this problem is equivalent to the solution of two other problems the former which is a one dimensional non classic boundary value problem giving the value of m through using an unconditionally stable explicit scheme using this result the second problem will be changed to a classical three dimensional diffusion equation with neumann s boundary conditions which will be solved numerically by using the method of lines semi discretization approach the space derivatives in the pde are approximated by finite difference replacements the solution of the resulting system of first order linear ordinary differential equations satisfies a recurrence relation which involves a matrix exponential function the accuracy in time is controlled by choosing several subdiagonal pade approximants to replace this matrix exponential term numerical techniques are developed to compute the required solution using a splitting method leading to algorithms for sequential and parallel implementation the algorithms are tested on a model problem and the resulting numerical experiments are also presented the central processor unit times needed are also reported and are compared

stability and numerical dispersion analysis of a fourth order accurate fdtd method in order to obtain high order accuracy a fourth order accurate finite difference time domain fdtd method is presented by kyu pyung hwang unlike conventional fdtd methods a staggered backward differentiation scheme instead of the leapfrog scheme is used to approximate the temporal partial differential operator however the high order of its characteristic equation derived by the von neumann method makes the analysis of its numerical dispersion and stability very difficult in automatic control theory there are two effective methods for the stability analysis i e the routh hurwitz test and the jury test the combination of the von neumann method with each of the two can strictly derive the stability condition which only makes use of the coefficients of its characteristic equation without numerically solving it the method of analysis in this paper is also applicable in the stability and numerical analysis of other high order accurate fdtd methods

taylor series based finite difference approximations of higher degree derivatives a new type of taylor series based finite difference approximations of higher degree derivatives of a function are presented in closed forms with their coefficients given by explicit formulas for arbitrary orders characteristics and accuracies of presented approximations and already presented central difference higher degree approximations are investigated by performing example numerical differentiations it is shown that the presented approximations are more accurate than the central difference approximations especially for odd degrees however for even degrees central difference approximations become attractive as the coefficients of the presented approximations of even degrees do not correspond to equispaced input samples

uniformly best wavenumber approximations by spatial central difference operators we construct accurate central difference stencils for problems involving high frequency waves or multi frequency solutions over long time intervals with a relatively coarse spatial mesh and with an easily obtained bound on the dispersion error this is done by demonstrating that the problem of constructing central difference stencils that have minimal dispersion error in the infinity norm can be recast into a problem of approximating a continuous function from a finite dimensional subspace with a basis forming a chebyshev set in this new formulation characterising and numerically obtaining optimised schemes can be done using established theory

numerical solution of one dimensional parabolic inverse problem this paper is concerned with numerical techniques for the solution of a one dimensional parabolic inverse problem with a control parameter the discrete approximation of the problem is based on the finite difference schemes these techniques are presented for identifying the control parameter which produces at each time a desired temperature at a given point in a spatial domain the main idea behind the finite difference methods for obtaining the solution of a given partial differential equation is to approximate the derivatives appearing in the equation by a set of values of the function at a selected number of points the most usual way to generate these approximations is through the use of taylor series the methods developed here are based on the modified equivalent partial differential equation as described by warming and hyett j comput phys 14 2 1974 159 this approach allows the simple determination of the theoretical order of accuracy thus allowing methods to be compared with one another also from the truncation error of the modified equivalent equation it is possible to eliminate the dominant error terms associated with the finite difference equations that contain free parameters weights thus leading to more accurate methods the results of a numerical experiment are presented and accuracy and the central processor cpu time needed for the parabolic inverse problem are discussed

a class of generalized mixed variational hemivariational inequalities i existence and uniqueness results abstract we investigate a generalized lagrange multiplier system in a banach space called a mixed variational hemivariational inequality mvhvi for short which contains a hemivariational inequality and a variational inequality first we employ the minty technique and a monotonicity argument to establish an equivalence theorem which provides three different equivalent formulations of the inequality problem without compactness for one of operators in the problem a general existence theorem for mvhvi is proved by using the fan knaster kuratowski mazurkiewicz principle combined with methods of nonsmooth analysis furthermore we demonstrate several crucial properties of the solution set to mvhvi which include boundedness convexity weak closedness and continuity finally a uniqueness result with respect to the first component of the solution for the inequality problem is proved by using the ladyzhenskaya babuska brezzi lbb condition all results are obtained in a general functional framework in reflexive banach spaces

comparison of taylor finite difference and window finite difference and their application in fdtd the finite difference time domain fdtd method is an important tool in numerical electromagnetic simulation there are many ways to construct a finite difference approximation such as the taylor series expansion theorem the filtering theory etc this paper aims to provide the comparison between the taylor finite difference tfd scheme based on the taylor series expansion theorem and the window finite difference wfd scheme based on the filtering theory their properties have been examined in detail separately in addition the formula of the generalized finite difference gfd scheme is presented which can include both the tfd scheme and the wfd scheme furthermore their application in the numerical solution of maxwell s equations is presented the formulas for the stability criterion and the numerical dispersion relation are derived and analyzed in order to evaluate their performance more accurately a new definition of error is presented upon it the effect of several factors including the grid resolution the courant number and the aspect ratio of the cell on the performance of the numerical dispersion is examined

derivative kernels numerics and applications a generalized framework for numerical differentiation nd is proposed for constructing a finite impulse response fir filter in closed form the framework regulates the frequency response of nd filters for arbitrary derivative order and cutoff frequency selected parameters relying on interpolating power polynomials and maximally flat design techniques compared with the state of the art solutions such as gaussian kernels the proposed nd filter is sharply localized in the fourier domain with ripple free artifacts here we construct 2d maxflat kernels for image directional differentiation to calculate image differentials for arbitrary derivative order cutoff level and steering angle the resulted kernel library renders a new solution capable of delivering discrete approximation of gradients hessian and higher order tensors in numerous applications we tested the utility of this library on three different imaging applications with main focus on the unsharp masking the reported results highlight the high efficiency of the 2d maxflat kernel and its versatility with respect to robustness and parameter control accuracy

a finite difference method for the numerical solution of the schro dinger equation a new approach which is based on a new property of phase lag for computing eigenvalues of schrodinger equations with potentials is developed in this paper we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wave functions tend to zero for x ii the general case of the morse potential and of the woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient compared to previously derived methods

a wavenumber based extrapolation and interpolation method for use in conjunction with high order finite difference schemes the errors incurred in using extrapolation and interpolation in large scale computations are analyzed and quantified in the wavenumber space if a large extrapolation stencil is used the errors in the low wavenumbers can be significantly reduced however the errors in the high wavenumbers are at the same time greatly increased the opposite is true if the stencil size is reduced based on the wavenumber analysis an optimized extrapolation and interpolation method is proposed the optimization is carried out over a selected band of wavenumbers it is known that extrapolation often leads to numerical instability the instability is the result of large error amplification in the high wavenumber range to reduce the tendency to trigger numerical instability it is proposed that an extra constraint be imposed on the optimized extrapolation method the added constraint aims to reduce error amplification over the high wavenumbers numerical examples are provided to illustrate that accurate and stable numerical results can be obtained in large scale simulation using a high order finite difference scheme and the proposed optimized extrapolation method when the same problems are recomputed using the familiar high order polynomials extrapolation method in the lagrange form in one case the numerical results are plagued by large errors and ultimately instability in another problem it is found that the use of the lagrange polynomials extrapolation method would lead immediately to numerical instability

chaotic motion of a parametrically forced simple pendulum abstract this paper describes the behavior of a simple pendulum with support in a steady and rather slow circular motion in a vertical plane the governing equation of this model is nonlinear and its motion is about the direction of gravity in the equation of motion there are three parameters which are related to the radius and angular velocity of the circular motion the length of the pendulum and the damping constant for some combinations of these parameters this model is found to have chaotic motion by numerical integration this paper not only draws the time series and poincare map to show the characteristics of regular and chaotic motions but also explores the feasibility of using the amplitude probability distribution by numerical method to diagnose whether the motion is chaotic the effect of geometry and damping constant on the motion of the system is investigated furthermore whether the poincare map and amplitude probability distribution depend sensitively on the initial conditions is also studied

on the computational complexity of the empirical mode decomposition algorithm abstract it has been claimed that the empirical mode decomposition emd and its improved version the ensemble emd eemd are computation intensive in this study we will prove that the time complexity of the emd eemd which has never been analyzed before is actually equivalent to that of the fourier transform numerical examples are presented to verify that emd eemd is in fact a computationally efficient method

a mixed hemivariational variational problem and applications abstract we consider an abstract system with lagrange multipliers which consists of a hemivariational inequality and a variational inequality the hemivariational inequality is governed by a hemicontinuous generalized monotone possibly nonlinear operator as well as by a bilinear form this bilinear form also governs the variational inequality we are looking for a pair solution in a subset of a product of two real reflexive banach spaces in order to illustrate the applicability of the abstract results two examples in terms of pdes are delivered each example is related to the weak solvability of a boundary value problem arising in contact mechanics

fourier analysis of numerical algorithms for the maxwell equations the fourier method is used to analyze the dispersive dissipative and isotropy errors of various spatial and time discretizations applied to the maxwell equations on multi dimensional grids both cartesian grids and non cartesian grids based on hexagons and tetradecahedra are studied and compared the numerical errors are quantitatively determined in terms of phase speed wavenumber propagation direction gridspacings and cfl number the study shows that centered schemes are more efficient and accurate than upwind schemes and the non cartesian grids yield superior isotropy than the cartesian ones for the centered schemes the staggered grids produce less errors than the unstaggered ones a new unstaggered algorithm which has all the best properties is introduced using an optimization technique to determine the nodal weights the new algorithm provides the highest accuracy among all the schemes discussed the study also demonstrates that a proper choice of time discretization can reduce the overall numerical errors due to the spatial discretization

summation by parts operators and high order quadrature summation by parts sbp operators are finite difference operators that mimic integration by parts the sbp operator definition includes a weight matrix that is used formally for discrete integration however the accuracy of the weight matrix as a quadrature rule is not explicitly part of the sbp definition we show that sbp weight matrices are related to trapezoid rules with end corrections whose accuracy matches the corresponding difference operator at internal nodes for diagonal weight matrices the accuracy of sbp quadrature extends to curvilinear domains provided the jacobian is approximated with the same sbp operator used for the quadrature this quadrature has significant implications for sbp based discretizations in particular the diagonal norm accurately approximates the l 2 norm for functions and multi dimensional sbp discretizations accurately approximate the divergence theorem

a stabilized lagrange multiplier method for the finite element approximation of contact problems in elastostatics in this work we consider a stabilized lagrange or kuhn tucker multiplier method in order to approximate the unilateral contact model in linear elastostatics the particularity of the method is that no discrete inf sup condition is needed in the convergence analysis we propose three approximations of the contact conditions well adapted to this method and we study the convergence of the discrete solutions several numerical examples in two and three space dimensions illustrate the theoretical results and show the capabilities of the method

finite element analysis of dynamic response of wrinkling membranes a new iterative scheme for finite element analysis of wrinkling membranes originally devised for static analysis is extended for analyzing dynamic response of wrinkling membranes the scheme is found to be successfully implemented with an explicit total lagrangian finite element code based upon the central difference method the finite element implementation of the scheme is straightforward and only minor modifications are needed for existing membrane finite element codes in light of application for contact impact discussed are a couple of existing contact algorithms which are well suited for treating contact impact of wrinkling membranes with rigid bodies the validity of the scheme is demonstrated via a numerical simulation of an inflating automotive airbag made of orthotropic membranes under impulse pressure loading

a recursive algorithm for optimizing differentiation in this paper a recursive algorithm will be introduced to improve the differentiation method proposed by hasan et al 2012 their algorithm is based on the undetermined coefficient method and uses the taylor series expansion and vandermonde matrix inverse to calculate weighting coefficients with a complexity of o n 4 our method reduces time complexity from o n 4 to o n 3 moreover we obtain a substantial optimality when the required degree and the order of accuracy increase finally the implementation of the proposed method follows and experimentations prove the validity of the algorithm and compare previous and new methods in terms of execution time besides the combination of our optimization and parallel processing provides valuable results in real world applications particularly qrs detection in ecg signal processing has been considered and we have shown that how features of our method could be used in order to increase accuracy and speed for this application

an optimal a priori error estimate for nonlinear multibody contact problems nonconforming domain decomposition methods provide a powerful tool for the numerical approximation of partial differential equations for the discretization of a nonlinear multibody contact problem we use linear mortar finite elements based on dual lagrange multipliers under some regularity assumptions on the solution an optimal convergence order of h 0 5 nu 0 nu leq 0 5 can be established in two dimensions 2d and three dimensions 3d compared with a standard linear saddle point formulation two additional terms which provide a measure for the nonconformity and the nonlinearity of the approach have to be taken in account numerical examples illustrating the performance of the nonconforming method and confirming our theoretical result are presented

taylor type 1 step ahead numerical differentiation rule for first order derivative approximation and znn discretization in order to achieve higher computational precision in approximating the first order derivative and discretize more effectively the continuous time zhang neural network znn a taylor type numerical differentiation rule is proposed and investigated in this paper this rule not only greatly remedies some intrinsic weaknesses of the backward and central numerical differentiation rules but also overcomes the limitation of the lagrange type numerical differentiation rules in znn discretization in addition a formula is proposed to obtain the optimal step length of the taylor type numerical differentiation rule moreover based on the proposed numerical differentiation rule the stability convergence and residual error of the taylor type discrete time znn dtznn are analyzed numerical experimental results further substantiate the efficacy and advantages of the proposed taylor type numerical differentiation rule for first order derivative approximation and znn discretization a formula is proposed to approximate the first order derivative an optimal step length rule for the proposed formula is investigated a taylor type znn model is derived for time varying matrix inversion

efficient algorithms for problems with friction in this paper a nonconforming discretization method for the frictional contact between two bodies subjected to antiplane shear deformation is considered the method is based on a mixed variational formulation where dual basis functions are used for the discretization of the lagrange multiplier under some regularity assumptions on the solution an optimal a priori error estimate is obtained to solve the discrete nonlinear problem an inexact primal dual active set strategy is introduced finally numerical examples confirming the theoretical result for the a priori error estimate and illustrating the performance of the algorithm are presented the results can easily be generalized to the case of coulomb friction for this case a numerical example is given

identifying a control function in two dimensional parabolic inverse problems numerical solution of the parabolic partial differential equations with an unknown parameter play a very important role in engineering applications this study presents numerical schemes for solving two two dimensional parabolic inverse problems i e for the determination of some unknown functions in the parabolic partial differential equations these schemes are developed for determining the parameter p t which satisfy u t u x x u y y p t email protected in rx 0 t u x y 0 f x y x y r 0 1 2 it is assumed that u is known on the boundary of r and subject to the integral overspecification over a portion of the spatial domain 0 email protected 0 1u x y t dxdy e t 0

an algorithm for the finite difference approximation of derivatives with arbitrary degree and order of accuracy in this paper we introduce an algorithm and a computer code for numerical differentiation of discrete functions the algorithm presented is suitable for calculating derivatives of any degree with any arbitrary order of accuracy over all the known function sampling points the algorithm introduced avoids the labour of preliminary differencing and is in fact more convenient than using the tabulated finite difference formulas in particular when the derivatives are required with high approximation accuracy moreover the given matlab computer code can be implemented to solve boundary value ordinary and partial differential equations with high numerical accuracy the numerical technique is based on the undetermined coefficient method in conjunction with taylor s expansion to avoid the difficulty of solving a system of linear equations an explicit closed form equation for the weighting coefficients is derived in terms of the elementary symmetric functions this is done by using an explicit closed formula for the vandermonde matrix inverse moreover the code is designed to give a unified approximation order throughout the given domain a numerical differentiation example is used to investigate the validity and feasibility of the algorithm and the code it is found that the method and the code work properly for any degree of derivative and any order of accuracy

representation for the lagrangian numerical differentiation formula involving elementary symmetric functions by using elementary symmetric functions this paper presents an explicit representation for the lagrangian numerical differentiation formula as well as the error estimate for local approximation and we also point out that the numerical differentiation formula constructed by li j p li general explicit difference formulas for numerical differentiation j comput appl math 183 2005 29 52 is actually a special case of the lagrangian numerical differentiation formula to approximate the values of the derivatives at the nodes

an analytical and numerical method for solving linear and nonlinear vibration problems a direct analytical and numerical method independent of the existing classical methods for solving linear and nonlinear vibration problems is given with the introduction of a piecewise constant argument nt n a new numerical method which produces sufficiently accurate results with good convergence is introduced development of the formulae for numerical calculations is based on the original governing differential equations

the explicit jump immersed interface method finite difference methods for pdes with piecewise smooth solutions many boundary value problems bvps or initial bvps have nonsmooth solutions with jumps along lower dimensional interfaces the explicit jump immersed interface method ejiim was developed following li s fast iterative immersed interface method fiiim recognizing that the foundation for the efficient solution of many such problems is a good solver for elliptic bvps ejiim generalizes the class of problems for which fiiim is applicable it handles interfaces between constant and variable coefficients and extends the immersed interface method iim to bvps on irregular domains with neumann and dirichlet boundary conditions proofs of second order convergence for a one dimensional 1d problem with piecewise constant coefficients and for two dimensional 2d problems with singular sources are given other problems are reduced to the singular sources case with additional equations determining the source strengths the advantages of ejiim are high quality of solutions even on coarse grids and easy adaptation to many problems with complicated geometries while still maintaining the efficiency of the fiiim

atomft solving odes and daes using taylor series abstract taylor series methods compute a solution to an initial value problem in ordinary differential equations by expanding each component of the solution in a long taylor series the series terms are generated recursively using the techniques of automatic differentiation the atomft system includes a translator to transform statements of the system of odes into a fortran 77 object program that is compiled linked with the atomft runtime library and run to solve the problem we review the use of the atomft system for nonstiff and stiff odes the propagation of global errors and applications to differential algebraic equations arising from certain control problems to boundary value problems to numerical quadrature and to delay problems

a finite difference domain decomposition algorithm for numerical solution of the heat equation a domain decomposition algorithm for numerically solving the heat equation in one and two space dimensions is presented in this procedure interface values between subdomains are found by an explicit finite difference formula once these values are calculated interior values are determined by backward differencing in time a natural extension of this method allows for the use of different time steps in different subdomains maximum norm error estimates for these procedures are derived which demonstrate that the error incurred at the interfaces is higher order in the discretization parameters

three new unconditionally stable fdtd methods with high order accuracy three novel finite difference time domain fdtd methods based on the split step ss scheme with high order accuracy are presented which are proven to be unconditionally stable in the first novel method symmetric operator and uniform splitting are adopted simultaneously to split the matrix derived from the classical maxwell s equations into six sub matrices accordingly the time step is divided into six sub steps the second and third proposed methods are obtained by adjusting the sequence of the sub matrices deduced in the first method so all the novel methods presented in the paper have similar formulations of which the numerical dispersion errors and the anisotropic errors are lower than the alternating direction implicit finite difference time domain adi fdtd method the initial ss fdtd method and the modified ss fdtd method based on the strang splitting scheme specifically for the second method corresponding to a certain cell per wavelength cpw there is a courant number value making the numerical anisotropic error to be zero while in the third novel method corresponding to a certain courant number value there exists a cpw making the numerical anisotropic error to be zero in order to demonstrate the high order accuracy and efficiency of the proposed methods numerical results are presented