solving irregular strip packing problems by hybridising simulated annealing and linear programming abstract in this paper a hybrid algorithm to solve irregular strip packing problems is presented the metaheuristic simulated annealing is used to guide the search over the solution space while linear programming models are solved to generate neighbourhoods during the search process these linear programming models which are used to locally optimise the layouts derive from the application of compaction and separation algorithms computational tests were run using instances that are commonly used as benchmarks in the literature the best results published so far have been improved by this new hybrid packing algorithm

optimal quadrature formulas of euler maclaurin type in the present paper the problem of construction of optimal quadrature formulas in the sense of sard in the space l 2 m 0 1 is considered here the quadrature sum consists of values of the integrand at nodes and values of derivatives of the integrand at the end points of the integration interval the coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number n and for any m 2 s 1 using sobolev method which is based on discrete analogue of the differential operator d2m dx2m in particular for s m 2 optimality of the classical euler maclaurin quadrature formula is obtained starting from m 2 s new optimal quadrature formulas are obtained finally numerical results which confirm theoretical are presented

the euler maclaurin summation formula the sampling theorem and approximate integration over the real axis abstract the euler maclaurin summation formula is used to deduce the whittaker shannon sampling theorem for not necessarily band limited functions as well as to study numerical integration over the real axis concerning the latter error estimates are determined in case the function to be integrated is smooth but not necessarily analytic two characteristic examples are given

robust mixed integer linear programming models for the irregular strip packing problem abstract two dimensional irregular strip packing problems are cutting and packing problems where small pieces have to be cut from a larger object involving a non trivial handling of geometry increasingly sophisticated and complex heuristic approaches have been developed to address these problems but despite the apparently good quality of the solutions there is no guarantee of optimality therefore mixed integer linear programming mip models started to be developed however these models are heavily limited by the complexity of the geometry handling algorithms needed for the piece non overlapping constraints this led to pieces simplifications to specialize the developed mathematical models in this paper to overcome these limitations two robust mip models are proposed in the first model dtm the non overlapping constraints are stated based on direct trigonometry while in the second model nfp cm pieces are first decomposed into convex parts and then the non overlapping constraints are written based on nofit polygons of the convex parts both approaches are robust in terms of the type of geometries they can address considering any kind of non convex polygon with or without holes they are also simpler to implement than previous models this simplicity allowed to consider for the first time a variant of the models that deals with piece rotations computational experiments with benchmark instances show that nfp cm outperforms both dtm and the best exact model published in the literature new real world based instances with more complex geometries are proposed and used to verify the robustness of the new models

a 2 exchange heuristic for nesting problems abstract this paper describes a new heuristic for the nesting problem based on a 2 exchange neighbourhood generation strategy this mechanism guides the search through the solution space consisting of the sequences of pieces and relies on a low level placement heuristic to actually convert one sequence in a feasible layout the placement heuristic is based on a bottom left greedy procedure with the ability to fill holes in the middle of the layout at a later stage several variants of the 2 exchange nesting heuristic were implemented and tested with different initial solution ranking criteria different strategies for selecting the next solution and different neighbourhood sizes the computational experiments were based on data sets published in the literature in most of the cases the 2 exchange nesting algorithm generated better solutions than the best known solutions

discretization of nonlinear models by sinc collocation interpolation methods abstract this paper deals with the solution of initial boundary value problems for nonlinear evolution equations in one and two space dimensions the solution technique is based on collocation interpolation methods which use sinc functions the paper aims to provide a guideline towards a large number of nonlinear problems of interest in applied sciences

sinc nystro m method for numerical solution of a dominant system of cauchy singular integral equations given on a piecewise smooth contour a numerical method is proposed for obtaining an approximate solution to a dominant system of cauchy singular integral equations dscsies taken over a piecewise smooth contour l consisting of a finite number of arcs l j first the dscsies are reduced to a system of fredholm integral equations of the second kind which are then solved approximately by nystrom s method based on a sinc quadrature rule it is shown that the error of an n point approximation converges to zero at the rate o e cn 1 2 as n to infty provided that the coefficients of the dscsies are analytic in neighborhoods d j of each arc l j and satisfy a lipschitz condition in each d j

eventual periodicity of the forced oscillations for a korteweg de vries type equation on a bounded domain using a sinc collocation method abstract we demonstrate numerically the eventual time periodicity of solutions u t to the korteweg de vries type equation with periodic forcing at one end using the sinc collocation method this method approximates the space dimension of the solution with a cardinal expansion of sinc functions thus allowing the avoidance of a costly finite difference grid for a third order boundary value problem the first order time derivative is approximated with a weighted finite difference method the sinc collocation method was found to be more robust and more efficient than other numerical schemes when applied to this problem

a gaussian sinc collocation approach for a whipping cantilever with a follower shear force at its tip a spatial discretization scheme based on a set of gaussian sinc functions is proposed for the temporal projection of a set of partial differential equations pdes describing the non linear dynamics of an elastic plastic hardening softening ephs cantilever subjected to a follower shear force at its tip the dynamics so described correspond to planar whipping of a pipe conveying fluid ruptured near a right angled bend the constitutive ephs moment curvature relationship used here follows the earlier work of reid et al proceedings of the royal society of london series a 1998 454 997 1029 compared to the more classical lagrangian polynomial based collocation functions the gaussian sine functions have better localization properties moreover for a relatively large number of collocation points use of such functions does not lead to numerical overflow or underflow problems often associated with the use of higher order polynomial collocation functions the spatial discretization leads to a set of non linear ordinary differential equations odes in time which are in turn integrated via a fourth order adaptive runge kutta scheme some numerical results for a cantilever whipping in a plane are presented to further illustrate the present approach the method is a step forward towards the development of a mesh free non linear beam element suitable for dynamic analyses of pipe networks and pipe on pipe impact problems

approximation of the hilbert transform on the real line using hermite zeros the authors study the hilbert transform on the real line they introduce some polynomial approximations and some algorithms for its numerical evaluation error estimates in uniform norm are given

guided local search and its application to the traveling salesman problem abstract the traveling salesman problem tsp is one of the most famous problems in combinatorial optimization in this paper we are going to examine how the techniques of guided local search gls and fast local search fls can be applied to the problem gls sits on top of local search heuristics and has as a main aim to guide these procedures in exploring efficiently and effectively the vast search spaces of combinatorial optimization problems gls can be combined with the neighborhood reduction scheme of fls which significantly speeds up the operations of the algorithm the combination of gls and fls with tsp local search heuristics of different eiciency and effectiveness is studied in an effort to determine the dependence of gls on the underlying local search heuristic used comparisons are made with some of the best tsp heuristic algorithms and general optimization techniques which demonstrate the advantages of gls over alternative heuristic approaches suggested for the problem

the crane equation uuxx 2 the boundary value problem uuxx 2 appears in cranes theory of laminar convection from a point source we show that the solution is real only when x 2 on this interval denoting the constants of integration by a and s the general solution is av xs a where the crane function v is the parameter free function v exp erfinv 2 x 2 and erfinv z is the inverse of the error function v x is weakly singular at both endpoints its chebyshev polynomial coefficients an decrease proportionally to 1 n3 exponential convergence can be restored by writing v x n 0a2nt2n z x where the mapping is z arctanh x l2 arctanh x 2 2 another option is singular basis functions v 1x2 2 10 216log 1x2 2 has a maximum pointwise error that is less 1 2000 of the maximum of the crane function

optimal quadrature formulae on the real line in the present paper we treat quadrature formulae on the real line with equidistant nodes involving a sequence of not necessarily consecutive derivatives in theorem 3 1 we obtain the optimal quadrature formula for sobolev space of functions with an additional restriction on its fourier transforms the quadrature which has been proved to be optimal was introduced in 111 to the best of our knowledge the quadrature formulae considered in the present paper were studied in 7 with respect to the optimal degree of precision an explicit expression for the error of the optimal quadrature formula is given a comparison of the results with those for quadrature formulae without derivatives given in 2 under the same computational complexity is given other results concerning numerical integration on the real line can be found in s 6 131

barycentric formulae for cardinal sinc interpolants we present a barycentric representation of cardinal interpolants as well as a weighted barycentric formula for their efficient evaluation we also propose a rational cardinal function which in some cases agrees with the corresponding cardinal interpolant and in other cases is even more accurate in numerical examples we compare the relative accuracy of those various interpolants with one another and with a rational interpolant proposed in former work

the sinc galerkin method for fourth order differential equations the sinc galerkin method originally proposed by stenger is extended to handle fourth order ordinary differential equations the exponential convergence rate of the method o e kappa sqrt m is carefully developed and the special features of the discrete system are described spectral properties and conditioning of the associated matrices are given the appropriate choice of weight function in the galerkin inner product is discussed with primary emphasis given to choices that are best suited to fourth order partial differential equations numerical results are included to help illustrate the parameter selections made and confirm the efficiency and accuracy of the method

generalized collocation method for two dimensional reaction diffusion problems with homogeneous neumann boundary conditions nonlinear reaction diffusion problems are ubiquitous in science ranging from physics to engineering from economy to biology in particular in this paper two dimensional problems with generic dirichlet boundary conditions or homogeneous neumann boundary conditions are focused on to simulate this class of mathematical problems a quite simple generalized collocation method is proposed and developed moreover a test case and an application to the schnakenberg model are described in order to show the efficiency of the simulation method and the goodness of the numerical results

numerical solution of certain classes of transport equations in any dimension by shannon sampling a method is developed for computing solutions to some class of linear and nonlinear transport equations hyperbolic partial differential equations with smooth solutions in any dimension which exploits shannon sampling widely used in information theory and signal processing the method can be considered a spectral or a wavelet method strictly related to the existence of characteristics but allows in addition for some precise error estimates in the reconstruction of continuous profiles from discrete data non dissipativity and in some case parallelizability are other features of this approach monotonicity preserving cubic splines are used to handle nonuniform sampling several numerical examples in dimension one or two pertaining to single linear and nonlinear integro differential equations as well as to certain systems are given

sinc type approximations in h 1 norm with applications to boundary value problems abstract sinc functions are used for approximation on a general arc in the complex plane it is proved that for certain functions analytic in a domain containing the arc the error of an n term approximation converges to zero in the sobolev space h1 at the rate o e n 1 2 as n special attention is given to the important case of a b where a and b are finite real numbers an application is also considered of the approximate result to the numerical solution of two point boundary value problems it is shown that the classical sinc galerkin method with n basis functions has o e n 1 2 convergence rate in the h1 norm

hybridising tabu search with optimisation techniques for irregular stock cutting sequential meta heuristic implementations for the irregular stock cutting problem have highlighted a number of common problems the literature suggests a consensus that it is more efficient to allow configurations with overlapping pieces in the solution space and to penalise these in the evaluation function however depending on the severity of the penalty this relaxation results in a tendency to converge toward infeasible solutions or to seek out feasible solutions at the expense of overall quality a further problem is encountered in defining a neighbourhood search strategy that can deal with the infinite solution space inherent in the irregular stock cutting problem the implementation in this paper adopts a hybrid tabu search approach that incorporates two very different optimisation routines that utilise alternative neighbourhoods to address the described problems

a spectral order method for inverting sectorial laplace transforms laplace transforms which admit a holomorphic extension to some sector strictly containing the right half plane and exhibiting a potential behavior are considered a spectral order parallelizable method for their numerical inversion is proposed the method takes into account the available information about the errors arising in the evaluations several numerical illustrations are provided

the asymptotic chebyshev coefficients for functions with logarithmic endpoint singularities mappings and singular basis functions abstract when a function is singular at the ends of its expansion interval its chebyshev coefficients a n converge very poorly we analyze three numerical strategies for coping with such singularities of the form 1 x k log 1 x and in the process make some modest additions to the theory of chebyshev expansions the first two numerical methods are the convergence improving changes of coordinate x sin 2 y and x tanh ly 1 y 2 1 2 we derive the asymptotic chebyshev coefficients in the limit n for both mappings and for the original untransformed chebyshev series for the original function the asymptotic approximation for general k is augmented by the exact chebyshev coefficients for integer k numerical tests show that the sine mapping is excellent for k 1 increasing the rate of convergence to b n o 1 n 4 k 1 although the tanh transformation is guaranteed to be better for sufficiently large n we offer both theoretical and numerical evidence to explain why the sine mapping is usually better in practice sufficiently large n is usually huge instead of mapping one may use a third strategy supplementing the chebyshev polynomials with singular basis functions simple experiments show that this approach is also successful

adaptive fast and oblivious convolution in evolution equations with memory to approximate convolutions which occur in evolution equations with memory terms a variable step size algorithm is presented for which advancing n steps requires only o n log n operations and o log n active memory in place of o n 2 operations and o n memory for a direct implementation a basic feature of the fast algorithm is the reduction via contour integral representations to differential equations which are solved numerically with adaptive step sizes rather than the kernel itself its laplace transform is used in the algorithm the algorithm is illustrated on three examples a blowup example originating from a schrodinger equation with concentrated nonlinearity chemical reactions with inhibited diffusion and viscoelasticity with a fractional order constitutive law

global properties of pseudospectral methods abstract we apply polynomial interpolation methods both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations the derivative matrix for a general sequence of collocation points is explicitly constructed we explore the effect of several factors on the performance of these methods we show that global methods cannot be interpreted in terms of local methods for example the accuracy of the approximation differs when large gradients of the function occur near the center of the region or in the vicinity of the boundary this difference does not depend on the density of the collocation points near the boundaries hence intuition based on finite difference methods can lead to false results

multigrid sinc methods a galerkin method using whittaker cardinal or sinc functions as basis functions is described for the solution of boundary value problems when the solution is analytic in the interior of the domain the error of approximation using 2n 1 points is o e c n 1 2 even if derivatives of the solution are singular at the boundaries a multigrid method with overall complexity o nlogn is used to solve the discrete equations this paper contains a description of the multigrid sinc algorithm along with some preliminary numerical results for two point boundary value problems

compaction and separation algorithms for non convex polygons and their applications abstract given a two dimensional non overlapping layout of convex and non convex polygons compaction can be thought of as simulating the motion of the polygons resulting from applied forces by moving many polygons simultaneously compaction can improve the material utilization of even tightly packed layouts compaction is hard finding the tightest layout that a valid motion can reach is pspace hard and even compacting to a local optimum can require an exponential number of moves our first compaction algorithm uses a new velocity based optimization model based on existing physical simulation approaches the performance of this algorithm illustrates that these approaches cannot quickly compact tightly packed layouts we then present a new position based optimization model this model represents the forces as a linear objective function and it permits direct calculation via linear programming of new non overlapping polygon positions at a local minimum of the objective the new model yields a translational compaction algorithm that runs two orders of magnitude faster than physical simulation methods we also consider the problem of separating overlapping polygons using the least amount of motion prove optimal separation to be np complete and show that our position based model also yields an efficient algorithm for finding a locally optimal separation the compaction algorithm has improved cloth utilization of human generated layouts of trousers and other garments given a database of human generated markers and with an efficient technique for matching the separation algorithm can automatically generate layouts that approach human performance

quadrature formulae and functions of exponential type we obtain certain generalizations of the trapezoidal rule and the euler maclaurin formula that involve derivatives in the case of quadrature of functions of exponential type over infinite intervals we find conditions under which existence of the improper integral and convergence of the approximating series become equivalent

sinc collocation method with orthogonalization for singular poisson like problems this paper uses the sinc collocation method to solve singular poisson like problems a firstor higher order partial derivative of the exact solution is unbounded on the boundary a linear system is obtained which is the same as that obtained by using the sinc galerkin method with a smart choice of the stepsize and the number of the gridpoints the orthogonalization technique is successfully applied to solve the linear system obtained and a numerical approximation is obtained with an exponential accuracy o exp cn7 where n is a truncation parameter and c is a constant independent of n

a new spectral galerkin method for solving the two dimensional hyperbolic telegraph equation abstract telegraph equation is more suitable than ordinary diffusion equation in modeling reaction diffusion for several branches of sciences and engineering in this paper a new numerical technique is proposed for solving the second order two dimensional hyperbolic telegraph equation subject to initial and dirichlet boundary conditions firstly a time discrete scheme based on the finite difference method is obtained unconditional stability and convergence of this semi discrete scheme are established secondly a fully discrete scheme is obtained by the sinc galerkin method and the problem is converted into a sylvester matrix equation especially when a symmetric sinc galerkin method is used the resulting matrix equation is a discrete adi model problem then the alternating direction sinc galerkin adsg method is applied for solving this matrix equation also the exponential convergence rate of sinc galerkin method is proved finally some examples are given to illustrate the accuracy and efficiency of proposed method for solving such types of differential equations compared to some other well known methods

uniform norm minimization for two signed solutions in an earlier paper huestis 1986 i gave an iterative algorithm for efficient linear programming construction of the minimum uniform norm solution of a linear inverse problem in such a problem d data values ei are linear functionals of the unknown solution m r usually expressible as integrals over an appropriate domain v ei vgi r m r dr ei i 1 d the gi are known functions and ei allows for measurement error

symmetrization of the sinc galerkin method for boundary value problems on procede a la symetrisation de la methode de galerkin en modifiant la fonction poids pour une classe etendue de problemes aux valeurs limites

idb2 a fortran program for computing extremal bounds in gravity data interpretation it is well known that the interpretation of gravity anomaly data suffers from a fundamental nonuniqueness no matter how complete a gravity data set may be there are an unlimited number of subsurface density solutions compatible with it this generally remains true even when the class of mathematically acceptable solutions is limited by imposed constraints based on physical or geologic arguments the common practice constructing a single solution fitting or only approximately fitting the anomaly data is therefore of limited value

sampling series approximation of continuous weak sense stationary processes the shannon sampling theorem is proved for weak sense stationary stochastic processes using the theory of integral functions of exponential type estimates together with rates of convergence are given for the error which arises when a non bandlimited random signal function is approximated by its sampling series expansion the construction of certain generalized sampling series leads to a better behaviour of approximation

the double exponential sinc collocation method for singular sturm liouville problems sturm liouville problems are abundant in the numerical treatment of scientific and engineering problems in the present contribution we present an efficient and highly accurate method for computing eigenvalues of singular sturm liouville boundary value problems the proposed method uses the double exponential formula coupled with sinc collocation method this method produces a symmetric positive definite generalized eigenvalue system and has exponential convergence rate numerical examples are presented and comparisons with single exponential sinc collocation method clearly illustrate the advantage of using the double exponential formula

solution approaches to irregular nesting problems this paper reviews some of the approaches which have been adopted in the solution of problems involving the nesting of irregularly shaped pieces such problems arise in a wide variety of application areas and the purpose of this paper is to bring together problems and solution approaches from a number of diverse domains

convergence of the sinc method for a fourth order ordinary differential equation with an application we analytically show the exponential convergence of the sinc method applied to fourth order ordinary differential equations odes linear numerical examples are included to confirm our analytical results the solution obtained with the sinc method is compared with the one obtained with a second and a sixth order finite difference scheme and with the ode boundary value solver colnew in the case of the linear equations we then compute the steady state solution of the one dimensional cahn hilliard equation to illustrate our results

double exponential sinc collocation method for solving the energy eigenvalues of harmonic oscillators perturbed by a rational function we show that the double exponential sinc collocation method provides an efficient uniformly accurate solution to the one dimensional time independent schrodinger equation for a general class of rational potentials of the form v x p x q x the derived algorithm is based on the discretization of the hamiltonian of the schrodinger equation using sinc expansions this discretization results in a generalized eigenvalue problem the eigenvalues of which correspond to approximations of the energy values of the starting hamiltonian a systematic numerical study is conducted beginning with test potentials with known eigenvalues and moving to rational potentials of increasing degree

near optimality of the sinc approximation near optimality of the sinc approximation is established in a variety of spaces of functions analytic in a strip region about the real axis each space being characterized by the decay rate of their elements functions in the neighborhood of the infinity

summary of sinc numerical methods sinc approximation methods excel for problems whose solutions may have singularities or infinite domains or boundary layers this article summarizes results obtained to date on sinc numerical methods of computation sinc methods provide procedures for function approximation over bounded or unbounded regions encompassing interpolation approximation of derivatives approximate definite and indefinite integration solving initial value ordinary differential equation problems approximation and inversion of fourier and laplace transforms approximation of hilbert transforms and approximation of indefinite convolutions the approximate solution of partial differential equations and the approximate solution of integral equations methods for constructing conformal maps and methods for analytic continuation indeed sinc are ubiquitous for approximating every operation of calculus

gravity ideal bodies the interpretation of gravity anomaly data suffers from a fundamental nonuniqueness even when the solution set is bounded by physical or geologic constraints therefore constructing a single solution that fits or approximately fits the data is of limited value consequently much effort has been applied in recent years to developing inverse techniques for rigorous deduction of properties common to all possible solutions to this end parker developed the theory of an ideal body which characterizes the extremal solution with the smallest possible maximum density gravity ideal body analysis is an excellent reconaissance exploration tool because it is especially well suited for handling sparse data contaminated with noise for finding useful rigorous bounds on the infinite solution set and for predicting accurately what data need to be collected in order to tighten those bounds we present a practical three dimensional gravity ideal body computer code idb that can optimize a mesh with over 105 cells

fast neighborhood search for two and three dimensional nesting problems abstract in this paper we present a new heuristic solution method for two dimensional nesting problems it is based on a simple local search scheme in which the neighborhood is any horizontal or vertical translation of a given polygon from its current position to escape local minima we apply the meta heuristic method guided local search the strength of our solution method comes from a new algorithm which is capable of searching the neighborhood in polynomial time more precisely given a single polygon with m edges and a set of polygons with n edges the algorithm can find a translation with minimum overlap in time o mn log mn solutions for standard test instances are generated by an implementation and a comparison is done with recent results from the literature the solution method is very robust and most of the best solutions found are also the currently best results published our approach to the problem is very flexible regarding problem variations and special constraints and as an example we describe how it can handle materials with quality regions finally we generalize the algorithm for the fast neighborhood search and present a solution method for three dimensional nesting problems

an error analysis of the cauchy principal value integral evaluation by the imt scheme we provide an error analysis for the cauchy principal value integral evaluation by the imt method for analytic functions we get the discretization error e d in terms of a contour integral and we derive an asymptotic estimate for this error integral which shows that e d o e n 2 n 3 4 we give two numerical examples where the asymptotically estimated errors are compared with known actual values

the tanh transformation for the solution of singular integral equations using a tanh transformation a quadrature formula for the evaluation of singular integrals is obtained the formula has the same step length h as the formula for regular integrals derived by f stenger these quadrature formulae are valid for end point singularities of any order and their error exhibits an exponential decay when the number of integrations tends to infinity using these formulae the solution of singular integral equations does not depend on the order of the end point singularities furthermore the collocation points are given by a very simple equation and in the case of constant coefficients by a closed form formula

the sampling theorem dirichlet series and hankel transforms abstract some very surprising relations between fundamental theorems and formulas of signal analysis of analytic number theory and of applied analysis are presented it is shown that generalized forms of the classical whittaker kotelnikov shannon sampling theorem as well as of the brown butzer splettstoser approximate sampling expansion for non band limited signal functions can be deduced via the theory of dirichlet series with functional equations from a new summation formula for hankel transforms this counterpart to poisson s summation formula is shown to be essentially equivalent to the famous functional equation of riemann s zeta function to the modular relation of the theta function to the nielsen doetsch summation formula for bessel functions and to the partial fraction expansion of the periodic hilbert kernel

galerkin schemes and the sinc galerkin method for singular sturm liouville problems abstract galerkin schemes for the computation of the eigenvalues of both regular and singular sturm liouville problems are compared to the sine galerkin method contrasted are numerical results as well as qualitative features the schemes discussed include finite element spectral and collocation the equivalence of the spectral method with sine basis functions to the sinc collocation scheme provides this technique with many distinctive features highlighted among these features through the course of the computational comparisons is its ease of implementation its exponential accuracy in the presence of singularities and its all around versatility

numerical methods based on whittaker cardinal or sinc functions this paper summarizes the results known to date for using sine functions composed with other functions as bases for approximations in numerical analysis described in this paper are methods of interpolation and approximation of functions and their derivatives quadrature the approximate evaluation of transforms hilbert fourier laplace hankel and mellin and the approximate solution of differential and integral equations the methods have many advantages over classical methods which use polynomials as bases in addition all of the methods converge at an optimal rate if singularities on the boundary of approximation are ignored

the use of linear programming in the construction of extremal solutions to linear inverse problems while a finite collection of data does not specify a unique solution to a linear inverse problem it can allow bounds to be placed on certain nonlinear solution functionals using the dirichlet problem for the unit disc as an example this note demonstrates the use of linear programming in constructing extremal solutions associated with a variety of such bounds

solution of nonlinear problems in applied sciences by generalized collocation methods and mathematica abstract this paper deals with the developments of mathematical methods for the discretization of continuous models and the solution of nonlinear problems of interest in applied sciences the contents refer to developments of the differential quadrature method which leads to the so called generalized collocation methods the method is developed and applied to the solution of initial boundary value problems the computational problems are technically solved with mathematica

a sinc hunter quadrature rule for cauchy principal value integrals a sine function approach is used to derive a new hunter type quadrature rule for the evaluation of cauchy principal value integrals of cer tain analytic functions integration over a general arc in the complex plane is considered special treatment is given to integrals over the interval 1 1 it is shown that the quadrature error is of order 0 e where n is the number of nodes used and where c is a positive constant which is indepen dent of n an application of the rule to the approximate solution of cauchy singular integral equations is also discussed numerical examples are included to illustrate the performance of the rule

numerical indefinite integration of functions with singularities we derive an indefinite quadrature formula based on a theorem of ganelius for h p functions for p 1 over the interval 1 1 the main factor in the error of our indefinite quadrature formula is o e n q with 2n nodes and 1 p 1 1 the convergence rate of our formula is better than that of the stenger type formulas by a factor of 2 in the constant of the exponential we conjecture that our formula has the best possible value for that constant the results of numerical examples show that our indefinite quadrature formula is better than haber s indefinite quadrature formula for h p functions

a quadrature formula for entire functions of exponential type a quadrature formula over a semi infinite interval for entire functions of exponential type is established an alternative approach to a known expansion formula and an extension of it are also presented

optimality of the double exponential formula functional analysis approach in the light of the functional analysis theory we establish the optimality of the double exponential formula the argument consists of the following three ingredients 1 introduction of a number of spaces of functions analytic in a strip region about the real axis each space being characterized by the decay rate of their elements functions in the neighborhood of the infinity 2 proof of the near optimality of the trapezoidal formula in each space introduced in 1 by showing the near equality between an upper estimate for the error norm of the trapezoidal formula and a lower estimate for the minimum error norm of quadratures 3 nonexistence theorem for the spaces the characterizing decay rate of which is more rapid than the double exponential

an average sampling theorem for bandlimited stochastic processes in this correspondence we give an average sampling theorem for bandlimited stochastic processes which shows that many average sampling theorems have their counterparts for stochastic signals

bessel transforms and rational extrapolation a numerical method is developed which handles the bessel transform of functions having slow rates of decrease the method replaces the bessel transform by a related damped transform for which the sinc quadrature rule provides an efficient and accurate approximation it is then shown that the value of the original bessel transform can be obtained from the damped transform by extrapolation with the thiele algorithm

an adaptive algorithm for spectral computations on unbounded domains abstract we consider spectral methods for solving differential equations on unbounded domains in particular we consider a spectral rational function method and a method based on domain truncation combined with fourier series both methods contain a free parameter which determines the accuracy of the spectral method when solving evolution equations the optimal parameter may change with time resulting in a significant loss of accuracy if the parameter is kept fixed we develop an automatic algorithm which adapts the parameter at regular time levels to maintain optimum accuracy in numerical tests we found that the proposed method is robust and efficient