tensor product generalized adi methods for separable elliptic problems we consider solving separable second order linear elliptic partial differential equations if an elliptic problem is separable then for certain discretizations the matrices involved in the corresponding discrete problem can be expressed in terms of tensor products of lower order matrices in the most general case the discrete problem can be written in the form a 1 otimes b 2 b 1 otimes a 2 c f we present a new tensor product generalized alternating direction implicit tpgadi iterative method for solving such discrete problems we prove convergence and establish computational efficiency the tpgadi method is applied to the hermite bicubic collocation equations we conclude that the tpgadi method is an effective tool for solving the discrete elliptic problems arising from a large class of elliptic problems

stability of discrete breathers abstract we approach the problem of linear stability of discrete breathers in a rigorous way for networks of not necessarily identical oscillators with general type of coupling in the case of hamiltonian systems using symplectic signature theory we give general conditions on the spectrum of the monodromy map in the uncoupled limit such that the discrete breathers are l 2 linearly stable for weak enough coupling we consider also dissipative networks of oscillators in this case we prove that the discrete breathers are not only stable but also attract a neighbourhood of initial data for any choice of l p topology some examples are considered amongst which an instance of roto breather

a collection of 2d elliptic problems for testing adaptive grid refinement algorithms adaptive grid refinement is a critical component of the improvements that have recently been made in algorithms for the numerical solution of partial differential equations pdes the development of new algorithms and computer codes for the solution of pdes usually involves the use of proof of concept test problems 2d elliptic problems are often used as the first test bed for new algorithms and codes this paper contains a set of twelve parametrized 2d elliptic test problems for adaptive grid refinement algorithms and codes the problems exhibit a variety of types of singularities near singularities and other difficulties

comparing numerical methods for helmholtz equation model problem in this article we implement a relatively new numerical technique adomian s decomposition method for solving the linear helmholtz partial differential equations the method in applied mathematics can be an effective procedure to obtain for the analytic and approximate solutions a new approach to a linear or nonlinear problems is particularly valuable as a tool for scientists and applied mathematicians because it provides immediate and visible symbolic terms of analytic solution as well as its numerical approximate solution to both linear and nonlinear problems without linearization solving frontier problems of physics the decomposition method kluwer academic publishers boston 1994 j math anal appl 35 1988 501 it does also not require discretization and consequently massive computation in this scheme the solution is performed in the form of a convergent power series with easily computable components this paper will present a numerical comparison with the adomian decomposition and a conventional finite difference method the numerical results demonstrate that the new method is quite accurate and readily implemented

the sinc method in multiple space dimensions model problems the sinc galerkin method is applied to the canonical forms of second order partial differential equations in multiple space dimensions for time dependent problems the scheme is fully galerkin i e the domain of the basis elements includes time hence the approximate solution for each equation is specified by solving the associated linear system notation is developed which facilitates description of both the linear systems and the algorithms used to solve them alternative algorithms are offered dependent on a scalar versus vector computing environment numerical results for the test problems presented sustain the exponential convergence rate of the method even in the presence of singularities

on grid refinement at point singularities for h p methods the h p method for solving partial differential equations uses discretizations where the grid size h and the polynomial degree p are varied to obtain the most efficiency in the computing of approximate solutions it has been known for many years that increasing p can dramatically improve efficiency and that the current common practice of keeping p 1 is a particularly poor choice for problems with singularities good efficiency requires that the grid be refined near the points with singularities so the h p method also requires a grid refinement scheme we compare two such schemes here which we call e refinement proposed by babuska and colleagues recently and q refinement proposed by rice in the 1960 s both of these schemes have been shown to have asymptotically optimal order of convergence the actual behaviours of these two schemes are quite different in practice and the mechanisms for choosing good values of h p and the grid refinement parameters are also quite different this paper presents the results of a systematic experimental study of these two methods and discusses the difficulty of choosing highly efficient h p and grid refinement parameters our study confirms that these two refinements are of approximately equal efficiency given such good choices the data suggest that the q refinement is more efficient but the advantage is not dramatic given the uncertainties inherent in choosing numerical methods for practical applications we conclude that it is substantially simpler to make good choices for the q refinement than for the e refinement

sinc galerkin method for solving biharmonic problems there are many techniques available to numerically solve the biharmonic equation in this paper we show that the sinc galerkin method is a very effective tool in numerically solving this equation hermite interpolation is used to treat the nonhomogeneous boundary conditions our method is tested on examples and comparisons with other methods are made it is shown that the sinc galerkin method yields good results even when singularities occur at the boundaries

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

the calculation of singular coefficients abstract we consider in this paper the calculation of the singular coefficients associated with the solution of an elliptic partial differential equation near a singular point a re entrant corner or a crack tip etc these are the coefficients in the relevant singular expansion of the solution near the point of singularity they often have physical relevance and it is of interest to be able to calculate them accurately we consider the problem in the context of the global element method this is a variable order finite element method designed to be capable of producing highly accurate solutions for singular problems even in the neighbourhood of the singularity if the values of the singular coefficients are needed these must be extracted from the computed solution we show in this paper that a suitably defined least squares fitting procedure allows the calculation of values for the leading singular coefficients which are as accurate as the underlying solution

weighted a search unifying view and application the a algorithm is a well known heuristic best first search method several performance accelerated extensions of the exact a approach are known interesting examples are approximate algorithms where the heuristic function used is inflated by a weight often referred to as weighted a these methods guarantee a bounded suboptimality as a technical contribution this paper presents the previous results related to weighted a from authors like pohl pearl kim likhachev and others in a more condensed and unifying form with this unified view a novel general bound on suboptimality of the result is derived in the case of avoiding any reopening of expanded states for e 0 this bound is 1 e n 2 where n is an upper bound on an optimal solution length binary decision diagrams bdds are well known to ai e g from set based exploration of sparse memory and symbolic manipulation of state spaces the problem of exact or approximate bdd minimization is introduced as a possible new challenge for heuristic search like many classical ai domains this problem is motivated by real world applications several variants of weighted a search are applied to problems of bdd minimization and the more classical domains like blocksworld and sliding tile puzzles for bdd minimization the comparison of the evaluated methods also includes previous heuristic and simulation based methods such as rudell s hill climbing based sifting algorithm simulated annealing and evolutionary algorithms a discussion of the results obtained in the different problem domains gives our experiences with weighted a which is of value for the ai practitioner

spline collocation differentiation matrices spline collocation differentiation matrices arising when solving pdes with periodic boundary conditions have block circulant structure in this paper we use this fact to present an eigenvalue analysis for a large class of such spline collocation differentiation matrices we use this to discuss the stability of collocation algorithms for parabolic and hyperbolic problems and fast collocation algorithms for elliptic pdes

the complex wavenumber eigenvalues of laplace s tidal equations for oceans bounded by meridians the complex wavenumber eigenvalues of laplace s tidal equations are determined for an ocean of constant depth bounded by meridians a galerkin method is used to expand the tide height and velocities in series of associated legendre functions a homogeneous system of equations results from the continuity and momentum equations the frequency and depth are fixed so that the meridional wavenumbers are the eigenvalues this gives rise to a generalized eigenvalue problem that must be solved numerically by iteration the eigenvalues are not integers and represent inertia gravity wave solutions at the specified tidal forcing frequency that can be excited by the presence of meridional boundaries those complex eigenvalues represent solutions that decay away from meridional boundaries the eigenvalue spectrum is investigated for the semi diurnal fortnightly and monthly tides one complex wavenumber for the semi diurnal tide explains the amphidromic systems within 20 of a north south coastline the fortnightly and monthly tides have only real wavenumber eigenvalues the basin scale deviation of these tides from equilibrium is attributed to low wavenumber divergent inertia gravity waves

a chebyshev rational chebyshev spectral method for the helmholtz equation in a sector on the surface of a sphere defeating corner singularities when the boundaries of a domain meet at an angle the solutions to an elliptic partial differential equation will usually be singular at the corner using the example of the helmholtz equation on the surface of a sphere in a domain bounded by meridians we show how corner singularities can be defeated by mapping the corner to infinity by applying a chebyshev series in longitude and a rational chebyshev series in the mercator coordinate y arctanh cos colatitude we obtain an exponential rate of convergence despite the corner singularities

combinations of collocation and finite element methods for poisson s equation in this paper we provide a framework of combinations of collocation method cm with the finite element method fem the key idea is to link the galerkin method to the least squares method which is then approximated by integration approximation and led to the cm the new important uniformly v 0 h elliptic inequality is proved interestingly the integration approximation plays a role only in satisfying the uniformly v 0 h elliptic inequality for the combinations of the finite element and collocation methods fem cm the optimal convergence rates can be achieved the advantage of the cm is to formulate easily linear algebraic equations where the associated matrices are positive definite but nonsymmetric we may also solve the algebraic equations of fem and the collocation equations directly by the least squares method thus to greatly improve numerical stability numerical experiments are also carried for poisson s problem to support the analysis note that the analysis in this paper is distinct from the existing literature and it covers a large class of the cm using various admissible functions such as the radial basis functions the sinc functions etc

remarks on the spectral properties of tight binding and kronig penney models with substitution sequences we comment on some recent investigations on the electronic properties of models associated with the thue morse chain point out that their conclusions contradict rigorously proven theorems and indicate some of the sources of these misinterpretations we briefly review and explain the current status of mathematical results in this field and discuss some conjectures and open problems

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

a fast 3d poisson solver of arbitrary order accuracy we present a direct solver for the poisson and laplace equations in a 3d rectangular box the method is based on the application of the discrete fourier transform accompanied by a subtraction technique which allows reducing the errors associated with the gibbs phenomenon and achieving any prescribed rate of convergence the algorithm requireso n3logn operations wherenis the number of grid points in each direction we show that our approach allows accurate treatment of singular cases which arise when the boundary function is discontinuous or incompatible with the differential equation

localised modes on localised equilibria abstract we show that spatially localised equilibria for networks of dynamical systems can have localised eigenmodes on them and study their spectral properties and onset thresholds the theory is illustrated with numerics for a class of one dimensional one degree of freedom hamiltonian oscillator chains with nearest neighbour coupling which have a discrete kink in the uncoupled limit we discuss implications of localised eigenmodes on spatially localised equilibria on many aspects of localised dynamics of networks

traps and snares in eigenvalue calculations with application to pseudospectral computations of ocean tides in a basin bounded by meridians we make several observations about eigenvalue problems using as examples laplace s tidal equations and the differential equation satisfied by the associated legendre functions whatever the discretization only some of the eigenvalues of then dimensional matrix eigenvalue problem will be good approximations to those of the differential equation usuallythen 2 eigenvalues of smallest magnitude for the tidal problem however the good eigenvalues are scattered so our first point is it is important to plot the drift of eigenvalues with changes in resolution we suggest plotting the difference between a low resolution eigenvalue and the nearest high resolution eigenvalue divided by the magnitude of the eigenvalue or the intermodal separation whichever is smaller second as a final safeguard it is important to look at the chebyshev coefficients of the mode we show a numerically computed anti kelvin wave which has little eigenvalue drift but is completely spurious as is obvious from its spectral series third inverting the roles of parameters can drastically modify the spectrum legendre s equation may have either an infinite number of discrete modes or only a handful depending on which parameter is the eigenvalue fourth when the modes are singular but decay to zero at the endpoints as is true of tides a tanh mapping can retrieve the usual exponential accuracy of spectral methods fifth the pseudospectral method is more reliable than deriving a banded galerkin matrix by means of recurrence relations the pseudospectral code is simple to check whereas it is easy to make an untestable mistake with the intricate algebra required for the galerkin method sixth we offer a brief cautionary tale about overlooked modes all these cautions are applicable toallforms of spatial discretization including finite difference and finite element methods however we limit our illustrations to spectral schemes where these difficulties are most easily resolved with a bit of care the pseudospectral method is a very robust and efficient method for solving differential eigenproblems even with singular eigenmodes

the performance of the collocation and galerkin methods with hermite bi cubics this paper presents a study of the performance of the collocation and galerkin methods using hermite bi cubic basis functions it is a sequel to the studies of houstis et al 6 and weiser et al

matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions abstract bialecki b and g fairweather matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions journal of computational and applied mathematics 46 1993 369 386 in recent years several matrix decomposition algorithms have been developed for the efficient solution of the linear algebraic systems arising when finite difference finite element galerkin and spectral methods are applied to separable elliptic boundary value problems in a rectangle the success of these methods depends on knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two point boundary value problems in one space dimension the primary purpose of this paper is to provide an overview of matrix decomposition algorithms and show how they can be expressed in terms of a unifying framework particular emphasis is placed on algorithms formulated recently by the authors for solving the linear systems arising in orthogonal spline collocation that is spline collocation at gauss points all of the methods discussed in this paper are modular and possess a great deal of natural parallelism

breathers in nonlinear lattices existence linear stability and quantization abstract it has been proved by the principle of anti integrable limit renamed here anti continuous limit and the implicit function theorem that under rather weak hypotheses there exist spatially localized and time periodic solutions breathers in arrays of nonlinear coupled classical oscillators provided the coupling is not too strong the models can be translationally invariant or not and be of any dimension there are also time periodic solutions corresponding to arbitrary distributions on the lattice of such solutions multibreathers the whole set of these solutions can be labeled by a discrete coding sequence the condition that the frequency of the breather or multibreather solution has no harmonics in the phonon spectrum can be discarded for a large subset of spatially extended time periodic solutions called phonobreather which correspond to special codes which can be also spatially chaotic and where no oscillator is at rest these results extend almost without change to models with coupled rotors the existence of rotobreathers which are exact solutions consisting of a single rotor rotating while the other rotors oscillate is proved there are also multirotobreather solutions and phono rotobreathers etc it is also shown that the phase of the single breathers constituting a multibreather or multirotobreather solution can be submitted to a phase torsion these new dynamical structures carry a nonzero energy flow in models with two dimensions and more the phase torsion of multibreather solutions can generate new dynamical solutions with vortices in the energy flow the dynamics of a quantum electron coupled to a classical lattice is also studied within the same approach from the anti continuous limit the existence of two kinds of polarobreathers which are spatially exponentially localized dynamical solutions is proved for a small enough electronic transfer integral the polarobreathers of the first kind include the standard static polaron the lattice configuration is static time independent when the electron is in an excited state i e the electronic oscillator oscillates periodically they can be spatially chaotic a polarobreather of the second kind corresponds to a localized periodic lattice oscillation associated with a localized quasiperiodic electronic oscillation the most simple of these solutions are those for which both the lattice oscillation and the electron wave function are mostly concentrated close to the same single site the standard floquet analysis of the linear stability of the breather and multibreather solutions is shown to be related with the band spectrum property of the newton operator involved in the implicit function theorem the krein theory of bifurcations is reinterpreted from this point of view the linear stability of the single breather and rotobreather solutions is then proved at least at small enough coupling finally it is shown that in the quantum version of translationally invariant classical models where there cannot exist in principle any strictly localized excitation there are narrow bands of excitations consisting of bound states of many bosons which are the quantum counterpart of the classical breathers the bandwidth of these excitations goes exponentially to zero with the number of bound phonons this theory provides simultaneously existence theorems and a new powerful and accurate numerical method for calculating practically any of the solutions which has been formally predicted to exist these numerical applications are currently written elsewhere

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

block variables for deterministic aperiodic sequences we use the concept of block variables to obtain a measure of order disorder for some one dimensional deterministic aperiodic sequences for the thue morse sequence the rudin shapiro sequence and the period doubling sequence it is possible to obtain analytical expressions in the limit of infinite sequences for the fibonacci sequence we present some analytical results which can be supported by numerical arguments it turns out that the block variables show a wide range of different behaviour some of them indicating that some of the considered sequences are more random than other however the method does not give any definite answer to the question of which sequence is more disordered than the other and in this sense the results obtained are negative we compare this with some other ways of measuring the amount of order disorder in such systems and there seems to be no direct correspondence between the measures

numerical solutions of the laplace s equation abstract this paper uses the sinc methods to construct a solution of the laplace s equation using two solutions of the heat equation a numerical approximation is obtained with an exponential accuracy we also present a reliable algorithm of adomian decomposition method to construct a numerical solution of the laplace s equation in the form a rapidly convergence series and not at grid points numerical examples are given and comparisons are made to the sinc solution with the adomian decomposition method the comparison shows that the adomian decomposition method is efficient and easy to use

improving the variable ordering of obdds is np complete ordered binary decision diagrams are a useful representation of boolean functions if a good variable ordering is known variable orderings are computed by heuristic algorithms and then improved with local search and simulated annealing algorithms this approach is based on the conjecture that the following problem is np complete given an obdd g representing f and a size bound s does there exist an obdd g respecting an arbitrary variable ordering representing f with at most s nodes this conjecture is proved

the complex wavenumber eigenvalues of laplace s tidal equations for oceans bounded by meridians corrigendum an error in the article o connor 1995 was brought to my attention by charles sozou this changes the values of the eigenvalues calculated in table 1 and the geophysical conclusions are modified the introduction and 2 on laplace s tidal equations are correct the mistake occured in 3 when the legendre function expansion was substituted into the set of laplace s tidal equations the term a in the expansion for the tide height in equation 12 must be zero otherwise equation 7 cannot be satisfied i did not see this at the time because i neglected to include the a term when substituting the expansions 12 into equation 7 this is why no a term appears in equation 18 the corrections are to omit the a terms in equations 12 and 17

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

spectrum and extended states in a harmonic chain with controlled disorder effects of the thue morse symmetry along the lines of previous work we give the general framework together with a detailed and rigorous study of the spectrum and born von karman eigenstates of a 1d harmonic chain with controlled disorder determined by the thue morse sequence the spectrum is a cantor like set we prove numerically that its measure is zero and calculate its bouligand minkowski dimension box dimension we prove that the value of the ids on each of the gaps is 2k 1 3 2 p withk andp integers finally we also prove that points in a dense subset of the spectrum give rise to extended states an exceptional property due to the symmetry of the thue morse substitution which can have important applications to multilayered structures and we illustrate this situation

a comparison between the sinc galerkin and the modified decomposition methods for solving two point boundary value problems one of the new techniques used in solving boundary value problems involving ordinary differential equations is the sinc galerkin method this method has been shown to be a powerful numerical tool for finding fast and accurate solutions a less known technique that has been around for almost two decades is the decomposition method in this paper we solve boundary value problems of higher order using these two methods and then compare the results it is shown that the sinc galerkin method in many instances gives better results

collocation methods for poisson s equation in this paper we provide an analysis on the collocation methods cm which uses a large scale of admissible functions such as orthogonal polynomials trigonometric functions radial basis functions and particular solutions etc the admissible functions can be chosen to be piecewise i e different functions are used in different subdomains the key idea is that the collocation method can be regarded as the least squares method involving integration approximation and optimal convergence rates can be easily achieved based on the traditional analysis of the finite element method the key analysis is to prove the uniformly vh elliptic inequality and some inverse inequalities used this paper explores the interesting fact that for the collocation methods given in this paper the integration rules only affect on the uniformly vh elliptic inequality but not on the solution accuracy the advantage of the cm is to formulate easily the associated algebraic equations which can be solved from the collocation equations directly by the least squares method thus to greatly reduce the condition number of the associated matrix moreover the new effective condition number is proposed to provide a better upper bound of condition number and to show a good stability for real problems solved by the collocation methods note that the boundary approximation method in li z c li combined methods for elliptic equations with singularities interfaces and infinities kluwer academic publishers boston london 1998 is a special case of the cm where the admissible functions satisfy the equations exactly numerical experiments are also carried for poisson s problem to support the analysis made

chaotic trajectories in the standard map the concept of anti integrability a rigorous proof is given in the standard map associated with a frenkel kontorowa model for the existence of chaotic trajectories with unbounded momenta for large enough coupling constant k k0 these chaotic trajectories with finite entropy per site are coded by integer sequences mi such that the sequence bi mi 1 mi 1 2mi be bounded by some integer b the bound k0 in k depends on b and can be lowered for coding sequences mi fulfilling more restrictive conditions the obtained chaotic trajectories correspond to stationary configurations of the frenkel kontorowa model with a finite non zero photon gap called gap parameter in dimensionless units this property implies that the trajectory or the configuration ui can be uniquely continued as a uniformly continuous function of the model parameter k in some neighborhood of the initial configuration a non zero gap parameter implies that the lyapunov coefficient is strictly positive when it is defined in addition the existence of dilating and contracting manifolds is proven for these chaotic trajectories exotic trajectories such as ballistic trajectories are also proven to exist as a consequence of these theorems the concept of anti integrability emerges from these theorems in the anti integrable limit which can be only defined for a discrete time dynamical system the coordinates of the trajectory at time i do not depend on the coordinates at time i 1 thus at this singular limit the existence of chaotic trajectories is trivial and the dynamical system reduces to a bernoulli shift it is well known that the kam tori of symplectic dynamical originates by continuity from the invariant tori which exists in the integrible limit under certain conditions in a similar way it appears that the chaotic trajectories of dynamical systems originate by continuity from those which exists at the anti integrable limits also under certain conditions

a rational spectral method for the kdv equation on the half line a petrov galerkin method using orthogonal rational functions is proposed for the korteweg de vries kdv equation on the half line with initial boundary values the nonlinear term and the right hand side term are treated by chebyshev rational interpolation explicitly and the linear terms are computed with the galerkin method implicitly such an approach is applicable using fast algorithms numerical results are presented for problems with both exponentially and algebraically decaying solutions respectively highlighting the performance of the proposed method

a collocated c0 finite element method reduced quadrature perspective cost comparison with standard finite elements and explicit structural dynamics summary we demonstrate the potential of collocation methods for efficient higher order analysis on standard nodal finite element meshes we focus on a collocation method that is variationally consistent and geometrically flexible converges optimally embraces concepts of reduced quadrature and leads to symmetric stiffness and diagonal consistent mass matrices at the same time it minimizes the evaluation cost per quadrature point thus reducing formation and assembly effort significantly with respect to standard galerkin finite element methods we provide a detailed review of all components of the technology in the context of elastodynamics that is weighted residual formulation nodal basis functions on gauss lobatto quadrature points and symmetrization by averaging with the ultra weak formulation we quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort such as higher order elastostatic analysis furthermore we illustrate the potential of collocation for efficient higher order explicit dynamics throughout this work we advocate a straightforward implementation based on simple modifications of standard finite element codes we also point out the close connection to spectral element methods where many of the key ideas are already established copyright 2014 john wiley sons ltd

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

block iterative algorithms for solving hermite bicubic collocation equations in this paper we present a so called local elimination technique by which a nonsymmetric system arising in the discretization of the poisson equation with a hermite bicubic collocation approximation is reduced to a block tridiagonal and block symmetric system a class of block iterative algorithms is developed for solving the resulting system the convergence rates of algorithms are discussed

convergence of adomian s method applied to differential equations this paper deals with a new proof of convergence of adomian s method applied to differential equations we also give new formulae and properties and we suggest a simple computational form for adomian s polynomials

a fast poisson solver of arbitrary order accuracy in rectangular regions in this paper we propose a direct method for the solution of the poisson equation in rectangular regions it has an arbitrary order accuracy and low cpu requirements which makes it practical for treating large scale problems the method is based on a pseudospectral fourier approximation and a polynomial subtraction technique fast convergence of the fourier series is achieved by removing the discontinuities at the corner points using polynomial subtraction functions these functions have the same discontinuities at the corner points as the sought solution in addition to this they satisfy the laplace equation so that the subtraction procedure does not generate nonperiodic nonhomogeneous terms the solution of a boundary value problem is obtained in a series form in o n log n floating point operations where n2 is the number of grid nodes evaluating the solution at all n2 interior points requires o n2 log n operations

numerical analysis of singularities in two dimensions part 1 computation of eigenpairs a numerical method is described for the computation of eigenpairs which characterize the exact solution of linear second order elliptic partial differential equations in two dimensions in the vicinity of singular points the singularities may be caused by re entrant corners and abrupt changes in boundary conditions or material properties such singularities are of great interest from the point of view of failure initiation the eigenpairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes for this reason failure theories directly or indirectly involve the eigenpairs and their amplitudes this paper addresses the problem of determining the eigenpairs numerically on the basis of the steklov formulation numerical results are presented for several cases importantly the method is applicable to three dimensional cases

from multi site to on site transfer matrix models for self similar chains we show how transfer matrix models on chains that are self similar renormalizable with respect to a substitution rule can be transformed from multi site models in which transfer matrices depend on the nature of a finite number of neighboring sites to on site models in which transfer matrices depend on the nature of one site only we present sufficient conditions and show that these conditions are satisfied in the case of quasiperiodic chains of two symbols that are renormalizable with respect to an invertible substitution rule we illustrate the application of our results to tight binding schrodinger equations modeling the electronic behavior of self similar chains of atoms and to models describing the transmission of light through self similarly stacked multilayers

properties of complexity measures for prams and wrams abstract the computation of boolean functions by parallel computers with shared memory prams and wrams is considered in particular complexity measures for parallel computers like critical and sensitive complexity are compared with other complexity measures for boolean functions like branching program depth and length of prime implicants and clauses the relations between these complexity measures and their asymptotic behaviour are investigated for the classes of boolean functions monotone functions and symmetric functions

the nonapproximability of obdd minimization the size of ordered binary decision diagrams obdds is determined by the chosen variable ordering a poor choice may cause an obdd to be too large to fit into the available memory the decision variant of the variable ordering problem is known to be np complete we strengthen this result by showing that unless p np for each constant c 1 there is no polynomial time approximation algorithm with the performance ratio c for the variable ordering problem i e no polynomial time algorithm that guarantees the computation of a variable ordering so that the resulting obdd size is larger than the minimum size by a factor of at most c this result justifies also from a theoretical point of view the use of heuristics for the variable ordering problem

enriched collocation finite element method a model problem abstract the standard collocation finite element method requires a fine mesh in the vicinity of singularities the concept of enriched elements is introduced to rectify this limitation the additional degree of freedom is handled by introducing a constraint equation relating the stress intensity factor to the nodal values in the interior the numerical results indicate that the recommended strategy is viable and leads to a significant improvement in the accuracy

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

lower bounds for depth restricted branching programs abstract we present a new method for proving lower bounds on the complexity of branching programs and consider k times only branching programs while exponential and nearly exponential lower bounds on the complexity of one time only branching programs were proved for many problems there are still missing methods of proving lower bounds for k times only programs k 1 we prove exponential lower bounds for k times only branching programs which have the additional restriction that the input bits are read k times yet blockwise and in each block in the same order this is done both for the algebraic decision problem polyn d n n prime d n whether a given mapping g f n f n is a polynomial over f n of degree at most d and for the corresponding monotone problem over quadratic boolean matrices as a consequence we obtain a sharp bound of order n log n on the communication complexity of polyn n 0 1 2

adaptive refinement for arbitrary finite element spaces with hierarchical bases abstract an adaptive refinement algorithm is presented and interpreted as the selective enrichment of a finite element space through the hierarchical basis each elemental division corresponds exactly to the inclusion of a small number of new basis functions while existing basis functions remain unchanged which bases to add i e which division to perform are determined so that those that make the largest contribution to the function are included first the space of c p th degree piecewise polynomials over triangles are used for illustration but the techniques apply to any family of function spaces that can be represented with a hierarchical basis numerical examples are presented to show that the technique can regain the optimal smooth order of convergence for the solution of partial differential equations with nonsmooth solutions

numerical solution of singular poisson problems via the sinc galerkin method the galerkin method is used with sinc functions for the basis elements to approximate the solution of the poisson problem particular attention is paid to singular problems a first or higher order partial derivative of the solution is undefined on the boundary the linear system is derived and a spectral decomposition technique of solution is used explicit parameter selections that enhance the performance of the method are given the method is thoroughly tested with the numerical results presented

the decomposition method for initial value problems abstract in this paper initial value problems for the heat wave and plate equations in which the vectorial space variable varies in the whole r d will be handled using the reliable decomposition method the solutions are obtained in the form of rapidly convergent power series with elegantly computable terms showing that the new technique is reliable powerful and promising

the sinc galerkin method for parameter dependent self adjoint problems the sinc galerkin method is being applied to a growing number of diverse problems in ordinary and partial differential equations including both forward and inverse parameter recovery problems as a result of these continuing extensions the treatment of parameter dependent problems needs to be thoroughly investigated two specific questions considered here are the incorporation of various nonhomogeneous boundary conditions and the treatment of a variable parameter the latter topic is particularly important for inverse problems that arise when numerically estimating physical parameters the point of view taken emphasizes the maintenance of the classical exponential convergence rate the techniques described are suitable both for the direct problem and for the parameter estimation problem numerical results are presented to substantiate the accuracy of the method

numerical method for the solution of special nonlinear fourth order boundary value problems the sinc galerkin method is used to approximate solution of nonlinear problems this work deals with the sinc galerkin method for solving nonlinear fourth order differential equations with homogeneous and nonhomogeneous boundary conditions the scheme is tested on three nonlinear problems and a comparison with finite difference methods and fourth order multiderivative method is made it is shown that the sinc galerkin method yields better results

the symmetric sinc galerkin method yields adi model problems abstract we show that when a symmetric sinc galerkin method is used to solve a poisson problem the resulting sylvester matrix equation is a discrete adi model problem we employ a new alternating direction scheme known as the alternating direction sinc galerkin adsg method on illustrative partial differential equation boundary value problems to document the exponential convergence rate that can be achieved unlike classical adi schemes direct numerical application of adsg avoids the computation of iteration parameters matrix eigenvalues and eigenvectors as well as the use of the kronecker product and sum

finding the optimal variable ordering for binary decision diagrams the ordered binary decision diagram is a canonical representation for boolean functions presented by r e bryant 1985 as a compact representation for a broad class of interesting functions derived from circuits however the size of the diagram is very sensitive to the choice of ordering on the variables hence for some applications such as differential cascode voltage switch dcvs trees it becomes extremely important to find the ordering leading to the most compact representation an algorithm for this problem with time complexity o n sup 2 3 sup n is presented this represents an improvement over the previous best algorithm

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

the space time sinc gallerkin method for parabolic problems this paper contains the formulation of a space time sinc galerkin method for the numerical solution of the parabolic partial differential equation in one space dimension the space time adjective means that the galerkin technique is employed simultaneously in time and space salient features of the method include exponential rate of convergence ease of assembly of the discrete system a global approximation and the ability to handle singular problems two methods of solution for the discrete system are offered and numerical results for test problems selected from the literature are included

on the complexity of vlsi implementations and graph representations of boolean functions with application to integer multiplication lower bound results on boolean function complexity under two different models are discussed the first is an abstraction of tradeoffs between chip area and speed in very large scale integrated vlsi circuits the second is the ordered binary decision diagram obdd representation used as a data structure for symbolically representing and manipulating boolean functions the lower bounds demonstrate the fundamental limitations of vlsi as an implementation medium and that of the obdd as a data structure it is shown that the same technique used to prove that any vlsi implementation of a single output boolean function has area time complexity at sup 2 omega n sup 2 also proves that any obdd representation of the function has omega c sup n vertices for some c 1 but that the converse is not true an integer multiplier for word size n with outputs numbered 0 least significant through 2n 1 most significant is described for the boolean function representing either output i 1 or output 2n i 1 where 1

a two dimensional adaptive mesh generation method abstract an automatic two dimensional adaptive mesh generation method is persented the method is designed so that a small portion of the mesh can be modified without disturbing a large number of adjacent mesh points the method can be used with or without boundary fitted coordinate generation procedures on the generated mesh a differential equation can be discretized by using classical difference formulas designed for uniform meshes as well as the difference formulas developed in this work both cases are illustrated by applying the method to the hiemenz flow for which the exact solution of the navier stokes equation is known 1 and to a two dimensional viscous internal flow model problem

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

fast algorithms for high order spline collocation systems in this paper we present a complete eigenvalue analysis for arbitrary order c 1 spline collocation methods applied to the poisson equation on a rectangular domain with dirichlet boundary conditions based on this analysis we develop some fast algorithms for solving a class of high order spline collocation systems which arise from discretizing the poisson equation

a tensor product generalized adi method for elliptic problems on cylindrical domains with holes abstract we consider solving second order linear elliptic partial differential equations together with dirichlet boundary conditions in three dimensions on cylindrical domains nonrectangular in x and y with holes we approximate the partial differential operators by standard partial difference operators if the partial differential operator separates into two terms one depending on x and y and one depending on z then the discrete elliptic problem may be written in tensor product form as tz i i axy u f we consider a specific implementation which uses a method of planes approach with unequally spaced finite differences in the xy direction and symmetric finite difference in the z direction we establish the convergence of the tensor product generalized alternating direction implicit iterative method applied to such discrete problems we show that this method gives a fast and memory efficient scheme for solving a large class of elliptic problems

discrete breathers in aperiodic diatomic fpu lattices with long range order this study presents various aspects of discrete breathers in diatomic fpu lattices with masses varying according to the aperiodic fibonacci and thue morse sequences we investigate the existence of time periodic breathers starting from the anti continuous limit and consider excitations of isolated light atoms hence obtaining the domain of unique existence for these breathers we also perform a linear stability analysis by studying the floquet operator the found exact solutions are used slightly perturbed as initial conditions for long time simulations of the breathers these breathers turn out to be robust finally we consider how initial excitations of two consecutive light atoms evolve depending on the properties of the phase space for the two atom system at the anti continuous limit we obtain different behaviour of these excitations especially we find that the aperiodic lattices can support localized excitations with a continuous frequency distribution for the timescales we consider while a periodic lattice is unable to these excitations are referred to as chaotic breathers c 2000 elsevier science b v all rights reserved

high order methods for elliptic partial differential equations with singularities this paper reports on an experimental study of the effectiveness of high order numerical methods applied to linear elliptic partial differential equations whose solutions have singularities or similar difficulties e g boundary layers sharp peaks three specific hypotheses are established with high levels of statistical confidence to support the general conclusion there is a strong correlation between the order of a method and its efficiency higher order is better

a method for locating symmetric homoclinic orbits using symbolic dynamics in this paper we present a method which can identify and locate symmetric homoclinic orbits in a homoclinic tangle formed by the intersecting stable and unstable manifolds of a symmetric 2d map the method consists of a systematic search in parameter space and determination of the order in which these orbits arise using symbolic dynamics each orbit corresponds to a unique sequence and it is computed by iterating the map along the unstable manifold to match a specific symmetry at the middle of the orbit an application of the method to the determination of multibreather solutions of 1d lattices is discussed

a fast parallel algorithm for the poisson equation on a disk a parallel algorithm for solving the poisson equation with either dirichlet or neumann conditions is presented the solver follows some of the principles introduced in a previous fast algorithm for evaluating singular integral transforms by daripa et al here we present recursive relations in fourier space together with fast fourier transforms which lead to a fast and accurate algorithm for solving poisson problems within a unit disk the algorithm is highly parallelizable and our implementation is virtually architecture independent theoretical estimates show good parallel scalability of the algorithm and numerical results show the accuracy of the method for problems with sharp variations on inhomogeneous term finally performance results for sequential and parallel implementations are presented c 2001 academic press

spectral analysis of hermite cubic spline collocation systems in this paper we present an eigenvalue analysis of the first order and second order hermite cubic spline collocation differentiation matrices with arbitrary collocation points some important features are explored and compared with some other discrete methods such as finite difference methods

spectral multigrid methods for elliptic equations ii a detailed description of spectral multigrid methods is provided this includes the interpolation and coarse grid operators for both periodic and dirichlet problems the spectral methods for periodic problems use fourier series and those for dirichlet problems are based upon chebyshev polynomials an improved preconditioning for dirichlet problems is given numerical examples and practical advice are included

performance of gem2 on the ellpack problem population the pilot global element program gem2 mk1 0 has been used to solve the two dimensional problems in the ellpack population of partial differential equation test problems the performance achieved is summarized in this paper special attention is paid to those problems which are singular or near singular and for which comparisons have been published by houstis and rice 12

sinc and the numerical solution of fifth order boundary value problems sinc methods are a family of self contained methods of approximation which have several advantages over classical methods of approximation in the case of the presence of end point singularities in this paper we present a fast and accurate numerical scheme for the fifth order boundary value problems with two point boundary conditions the method is then tested on linear and nonlinear examples and a comparison with sixth degree b spline functions is made it is shown that the sinc galerkin method yields better results

finite size effects on instabilities of discrete breathers abstract when the two arcs of the continuous phonon spectrum of the floquet matrix of a discrete breather overlap on the unit circle the breather solution in the infinite lattice might be stable while the corresponding solutions in finite systems appear to be unstable more precisely when the model parameters vary the breather in the finite system exhibits a large number of collisions between the floquet eigenvalues belonging to the phonon spectrum these collisions correspond to complex cascades of instability thresholds followed near after by re entrant stability thresholds we interpret this complex structure on the basis of the band analysis of the matrix of the second variation of the action then we can predict that in the limit of an infinite system the number of instability and stability thresholds in the cascade diverges but simultaneously the maximum amplitude of the instabilities vanishes so that the breather in the infinite system recovers its linear stability as long as all its other localized modes remain stable this is the situation which is required in cretegny et al physica d 119 1998 73 87 for having inelastic phonon scattering with two channels we also analyze the size effects when a floquet eigenvalue associated with a localized mode collides with the floquet continuous phonon spectrum with different krein signature in contrast to the previous case the infinite system is unstable after the collision

a population of linear second order elliptic partial differential equations on rectangular domains part 1 abstract we present a population of 56 linear two dimensional elliptic partial differential equations pdes suitable for evaluating numerical methods and software forty two of the pdes are parameterized which allows much larger populations to be made 189 specific cases are presented here along with solutions some are only approximate many of the pdes are artificially created so as to exhibit various mathematical behaviors of interest the others are taken from real world problems in various ways the population has been structured by introducing measures of complexity of the operator boundary conditions solution and problem the pdes are first presented in mathematical terms along with contour plots of the 189 specific solutions machine readable descriptions are given in part 2 mrc technical summary report no 2079 many of the pdes involve lengthy expressions and about a dozen involve extensive tabulations of approximate solutions