an invariant imbedding method for computation of eigenlengths of singular two point boundary value problems in this article we have described an invariant imbedding method for calculating the smallest eigenlength of a singular tpbvp with the singularity at the origin the invariant imbedding yields a first order nonlinear equation called a riccati equation and also gives the initial conditions at the origin for this equation with the aid of theorem 8 in section 3 we numerically integrate the riccati equation to blowup which gives our computed eigenlength in closing we would like to comment on the numerical merits of the integration to blowup technique on the basis of the examples presented it appears that this technique combined with the available numerical integrators with variable step size is capable of producing accurate results the feature of a variable step size is essential as the value of z approaches the actual eigenlength however it is desirable to have a priori estimate or bounds of the eigenlength similar to those of boland and nelson 2 for the nonsingular case the singular system however presents difficulties due to the lack of sign conditions on the coefficient matrices in obtaining such bounds hopefully an investigation of the matrix r z will yield these results

quasilinear multistep methods and variable step predictor corrector methods for neutral functional differential equations the theory of quasilinear multistep methods and predictor corrector methods for neutral functional differential equations is developed the theorem about the asymptotic behaviour of the global discretization error of quasilinear methods which forms a theoretical basis for step control strategy for predictor corrector methods for neutral equations is given some problems of implementation of these methods are also addressed and the results of numerical experiments are presented

variable step size predictor corrector schemes for second kind volterra integral equations in this paper a family of implicit multistep methods for the solution of volterra integral equations is derived these methods together with an explicit euler predictor permit the use of a variable step size when solving integral equations means of controlling the error and stability by varying the step size and the order of the method are described extensive numerical results are presented

linear conservation laws for odes abstract some physical systems described by odes have quantities that are conserved as the system evolves runge kutta formulas and linear multistep methods preserve linear conservation laws automatically a code for the integration of odes involves a number of algorithms in addition to the basic formula it is shown here that popular codes preserve linear conservation laws if they are used properly an application is made to differential algebraic equations of index 2 arising in the solution of the navier stokes equation by the method of lines

a multi purpose system for the numerical integration of ode s in this paper we describe the development of an experimental multi purpose integration system strongly modular the system allows the use of wide classes of multistep methods for solving ode s singulsr perturbation problems as well as implicit differential algebraic equations we focus our attention on a new family of collocation methods for nonstiff problems while these methods are strongly related to the stiffly stable bdf methods they are particularly well suited for the implementation of a type insensitive integration algorithm compound discretizations in which certain variables are integrated with the first method and others with the second can also be used a stability analysis is given and some implementation issues are discussed

time scales to equipartition in the fermi pasta ulam problem finite size effects and thermodynamic limit we investigate numerically the common and the pure fpu models as well as some higher order generalizations we consider initial conditions in which only low frequency normal modes are excited and perform a very accurate systematic study of the equilibrium time as a function of the number n of particles the specific energy e and the parameters and while at any fixed n the equilibrium time is found to be a stretched exponential in 1 e in the thermodynamic limit i e for n at fixed e we observe a crossover to a power law concerning the usually disregarded dependence of teq on and we find it is nontrivial and propose and test a general law a central role is played by the comparison of the fpu models with the toda model

on the quantization of the three particle toda lattice the authors compare the einstein brillouin keller quantization procedure and the canonical quantization of a three particle toda chain with periodic boundary conditions in particular the transition from very low energies at which the system may be approximated by harmonic oscillators to intermediate energies is investigated this is the regime of a general integrable nonlinear system for which they find a poissonian statistics for the energy levels in the limit of very high energies they exploit the fact that the system may be described essentially by a triangular billiard and thus can derive some exact results

behind and beyond the matlab ode suite abstract the paper explains the concepts of order and absolute stability of numerical methods for solving systems of first order ordinary differential equations ode of the form describes the phenomenon of problem stiffness and reviews explicit runge kutta methods and explicit and implicit linear multistep methods it surveys the five numerical methods contained in the matlab ode suite three for nonstiff problems and two for stiff problems to solve the above system lists the available options and uses the odedemo command to demonstrate the methods one stiff ode code in matlab can solve more general equations of the form m t y f t y provided the mass option is on

a singular nonlinear boundary value problem membrane response of a spherical cap the nonlinear boundary value problem y n x 2 32y 2 lambda 2 8 quad 0 0 nu 0 are constants models the large deflection membrane response of a spherical cap and has been solved numerically by various techniques in the engineering literature we prove existence and uniqueness of a solution of this problem provide qualitative information about the solution and use this information to suggest an approach to numerical computation

scattering of lattice solitons from a mass interface a synergetic approach abstract a special mathematical method a synergetic approach combining analytical and numerical procedures is developed to study how a toda lattice soliton is scattered from a mass interface the method is based on the inverse scattering transform and the spectral theorem for self adjoint operators the laws for the scattering processes are given in a series of diagrams

stiffness characteristic and the applications of the method of lines on nonuniform grids the method of lines is investigated for the numerical solution of the stream function and vorticity form of the navier stokes equations on nonuniform grids stiffness characteristics of a linear one dimensional model equation are examined to establish the feasibility of applying the method to the vorticity equation in two dimensions the governing equations are transformed from the physical domain with a highly variable grid to a computational domain with a uniform grid the method of lines is used to solve only the vorticity equation and the successive over relaxation technique is used to solve the stream function equation it is observed that the transformed governing equations become stiffer with increased concentration of grid points and also as the number of grid points increases it is also observed that the differencing technique affects the stiffness characteristics the use of forward differencing is not feasible and backward differencing is preferable to central differencing for high reynolds numbers the results of specific applications for the solution of flow in curved wall diffusers and a driven cavity demonstrate that the method of lines under certain circumstances is feasible for the numerical solution of physical problems on domains covered with variable grids

a systematic approach to the soliton equations of a discrete eigenvalue problem the ablowitz ladik al problem is a linear vector difference equation whose isospectral flow equations include several important soliton equations e g the discrete nonlinear schrodinger equation iqn qn 1 2qn qn 1 qn 2 qn 1 qn 1 there is an established procedure for describing the soliton hierarchy of the more familiar akns ablowitz kaup newell and segur problem it is based on the notion of a generator for the hierarchy in this paper the soliton equations of the al hierarchy are described and characterized by a generator pair a new continuous spectral problem is introduced and the akns hierarchy is embedded in its hierarchy as a specialization

on the exponential fitting of composite multiderivative linear multistep methods exponentially fitted composite multiderivative linear multistep methods of orders up to 5 are derived a numerical investigation of these formulae indicates that they may be a stable for all choices of the fitting parameters an important aspect of the class of integration formulae considered is that each formula contains a built in local error estimate a facility lacking in most other exponentially fitted formulae some preliminary numerical results are given and these show that for problems for which exponential fitting is appropriate the new formulae provide a large saving in computational effort compared with conventional ones

numerical treatment of o d es the theory ofa methods almost all commonly used methods for o d es and their most miscellaneous compositions area methods i e they can be reduced toz o z j az j 1 h x j 1 z j 1 z j h z j s a s s j 1 1 m this paper presents a general theory fora methods and discusses its practical consequences an analysis of local discretization error l d e accumulation results in a general order criterium and reveals which part of the l d e effectively influences the global error this facilitates the comparison of methods and generalizes considerably the concept of error constants it is shown as a consequence that the global error cannot be safely controlled by the size of the l d e and that the conventional error control may fail in important cases furthermore butcher s effective order methods the concept of nordsieck forms and gear s interpretation of lineark step schemes as relaxation methods are generalized the stability of step changing is shortly discussed

strict limit types for monotone convolution we introduce the notion of strict domain of attraction in the context of monotone probability theory and prove that a law has a nonempty strict domain of attraction if and only if it is strictly stable we also show that the monotonic weak law of large numbers is equivalent to the classical and the free weak laws of large numbers finally a necessary condition for a measure to belong to the strict domain of attraction of a nondegenerate strictly stable law is given in terms of the absolute moments of

stable laws and domains of attraction in free probability theory in this paper we determine the distributional behavior of sums of free in the sense of voiculescu identically distributed inflnitesimal random variables the theory is shown to parallel the classical theory of independent random variables though the limit laws are usually quite difierent our work subsumes all previously known instances of weak convergence of sums of free identically distributed random variables in particular we determine the domains of attraction of stable distributions in the free theory these freely stable distributions are studied in detail in the appendix where their unimodality and duality properties are demonstrated

nonlinear fourier analysis for the infinite interval korteweg de vries equation i an algorithm for the direct scattering transform the nonlinear fourier analysis of wave motion governed approximately by the korteweg de vries kdv equation on the infinite line is the central point of discussion we assume that the wave amplitude is recorded in the form of a discrete space or time series which is determined either by experimental measurement or by computer simulation of the physical system of interest we develop numerical data analysis procedures based upon the scattering transform solution to the kdv equation as given by gardner et al 1 we are motivated by the observation that historically the fourier transform has been ubiquitously used to spectrally analyze linear wave data here we develop methods for employing the scattering transform as a tool to similarly analyze nonlinear wave data specifically we develop numerical methods to evaluate the direct scattering transform dst of a space or time series the approach thus provides a basis for analyzing and interpreting nonlinear wave behavior in the wavenumber or frequency domain the dst spectrum separates naturally into soliton and radiation components and may be simply interpreted in terms of the large time asymptotic state of the infinite line kdv equation

variable order adams codes abstract variable step size variable order vsvo adams codes are very effective for solving initial value problems for first order systems of ordinary differential equations the theory of fixed order codes is classical but when the order is varied there is no theory explaining fundamental issues with realistic assumptions about order and step size selection we prove convergence approximate locally the behavior of the error and justify standard error estimators

type insensitive ode codes based on implicit a stable formulas a special concept of stiffness is appropriate for implict a stable formulas it is possible to recognize this kind of stiffness economically and reliably using information readily available during the integration of an ode using this development a variety of effective ode solvers could be made insensitive to the type of problem i e the code would automatically recognize and alter automatically its algorithm at any step depending on whether the problem is stiff there

design sensitivity analysis and optimization of dynamic response abstract the paper presents methods of design sensitivity analysis and optimization of dynamic response of mechanical and structural systems a key feature of the paper is the development of procedures to handle point wise state variable constraints difficulties with a previous treatment where such constraints were transformed to equivalent integral constraints are noted and explained from theoretical as well as engineering standpoints an alternate treatment of such constraints is proposed developed and evaluated in this treatment each point wise state variable constraint is replaced by several constraints that are imposed at all the local max points for the original constraint function the differential equations of motion are formulated in the first order form so as to handle more general problems the direct differentiation and adjoint variable methods of design sensitivity analysis to deal with the point wise constraints are presented with the adjoint variable methods there are two ways of calculating design sensitivity coefficients the first approach uses an impulse load and the second approach uses a step load for the corresponding adjoint equation since the adjoint variable methods are better for a large class of problems an efficient computational algorithm with these methods is presented in detail optimum results for several problems are obtained and compared with those available in the literature the new formulation works extremely well as precise optimum designs are obtained

automatic differentiation of numerical integration algorithms automatic differentiation ad is a technique for automatically augmenting computer programs with statements for the computation of derivatives this article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations odes in particular the ramifications of the fact that ad is applied not only to the solution of such an algorithm but to the solution procedure itself this subtle issue can lead to surprising results when ad tools are applied to variable stepsize variable order ode integrators the computation of the final time step plays a special role in determining the computed derivatives we investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives

hp adaptive time integration based on the bdf for viscous flows this paper presents a procedure based on the backward differentiation formulas of order 1 to 5 to obtain efficient time integration of the incompressible navier stokes equations the adaptive algorithm performs both stepsize and order selections to control respectively the solution accuracy and the computational efficiency of the time integration process the stepsize selection h adaptivity is based on a local error estimate and an error controller to guarantee that the numerical solution accuracy is within a user prescribed tolerance the order selection p adaptivity relies on the idea that low accuracy solutions can be computed efficiently by low order time integrators while accurate solutions require high order time integrators to keep computational time low the selection is based on a stability test that detects growing numerical noise and deems a method of order p stable if there is no method of lower order that delivers the same solution accuracy for a larger stepsize hence it guarantees both that 1 the used method of integration operates inside of its stability region and 2 the time integration procedure is computationally efficient the proposed time integration procedure also features a time step rejection and quarantine mechanisms a modified newton method with a predictor and dense output techniques to compute solution at off step points

sensitivity analysis of flexible multibody systems using composite materials components in this paper a general formulation for the computation of the first order analytical sensitivities based on the direct method using automatic differentiation of flexible multibody systems is presented the direct method for sensitivity calculation is obtained by differentiating the equations that define the response of the flexible multibody systems of composite materials with respect to the design variables which are the ply orientations of the laminated in order to appraise the benefits of the approach suggested and to highlight the risks of the procedure the analytical sensitivities are compared with the numerical results obtained by using the finite difference method for the beam composite material elements the section properties and their sensitivities are found using an asymptotic procedure that involves a two dimensional 2 d finite element analysis of their cross section the equations of the sensitivities are obtained by automatic differentiation and integrated in time simultaneously with the equations of motion of the multibody systems the equations of motion and the sensitivities of the flexible multibody system are solved and the accelerations velocities and the sensitivities of accelerations and velocities are integrated in time using a multi step multi order integration algorithm through the application of the methodology to two simple flexible multibody systems the difficulties and benefits of the procedure with respect to finite difference approaches or to the direct implementation of the analytic sensitivities are discussed copyright 2008 john wiley sons ltd

run time estimation of the spectral radius of jacobians it is useful for ordinary differential equation ode solvers to include an estimator of the spectral radius of the jacobian matrix of the system of ode s since this determines the numerical stability of the method hence a knowledge of spectral radius enables the run time selection of a more efficient integrator some techniques for estimating spectral radius are described and compared they include methods suitable for use with any ode solvers but which require additional computation other methods are described which are suitable with runge kutta or rosenbrock methods and which require little extra computation

a note on the relative efficiency of adams methods abstract in several pieces of the most effective mathematical software for non stiff and mildly stiff initial value problems in ordinary differential equations the classical adams methods are implemented in a particular mode one reason for the choice of mode is sometimes explained in terms of the associated scaled absolute stability regions for the methods in this note it is shown that the same choice of mode remains appropriate from this point of view when certain families of generalized adams type methods are considered

an algorithm for starting multistep methods the present paper deals with the problem of starting multistep methods we take into consideration an adams bashforth moulton pece pair with the predictor of order q and the corrector of order q 1 to start this method q 1 starting values are necessary in addition to y0 a well known result from theory says that the order of convergence of the whole integration is q 1 if all those starting values are accurate of that same order present production codes start with a predictor of order 1 and a corrector of order 2 at the first step and then proceed step by step each time raising the order by 1 until all the necessary starting values have been obtained but in this manner all the starting errors keep of order 3 and so the whole integration converges no faster than that order this drawback is normally compensated for by taking very small step sizes in the starting phase the general algorithm we propose furnishes at a reasonably low cost the necessary number q 1 of starting values each of the appropriate order q 1 whatever q might be it is independent of the particular multistep formula considered and is mainly designed to be used for high values of q q 10 where the alternative strategies are too expensive or do not exist at all the numerical results reported show the validity of our approach 2003 elsevier science ltd all rights reserved

numerical approaches for solution of differential equations on manifolds numerical approaches for the solution of vector fields differential equations defined on a manifold have attracted wide attention over the past few years this paper first reviews the various numerical approaches available in the literature for the solution of vector fields namely parameterization approach constraint stabilization approach and perturbation approach pa in the process the paper also makes the following useful contributions an expanded analysis and a new perturbation scheme for the pa and a new way of choosing integration error tolerances for the parameterization approach a comparison of all the approaches is carried out both by means of a crude cost analysis as well as by studying their numerical performance on examples of vector fields arising from constrained mechanical systems cms based on this comparison recommendations are made for a proper choice of a suitable approach overall the pa performs better than the other approaches

approximation of weak adjoints by reverse automatic differentiation of bdf methods we shed light on the relation between the discrete adjoints of multistep backward differentiation formula bdf methods and the solution of the adjoint differential equation to this end we develop a functional analytic framework based on a constrained variational problem and introduce the notion of weak adjoint solutions of ordinary differential equations we devise a petrov galerkin finite element fe interpretation of the bdf method and its discrete adjoint scheme obtained by reverse internal numerical differentiation we show how the fe approximation of the weak adjoint is computed by the discrete adjoint scheme and prove its convergence in the space of normalized functions of bounded variation we also show convergence of the discrete adjoints to the classical adjoints on the inner time interval finally we give numerical results for non adaptive and fully adaptive bdf schemes the presented framework opens the way to carry over techniques on global error estimation from fe methods to bdf methods

nonlinear normal modes for the toda chain abstract the toda chain is a nonlinear mass spring chain which can in principle be integrated analytically by the action angle theory of classical mechanics we show that the transformation from physical variables to action variables can be implemented very simply on a computer we study the correspondence between action variables and the motions of the chain and find the role of the action variables to be very similar to the role of normal mode amplitudes in the harmonic chain this similarity leads to a partly rigorous partly heuristic normal mode analysis for the toda chain new results are obtained when this normal mode analysis is applied to certain perturbations of the toda chain such as the fermi pasta ulam chain or the toda chain with a mass impurity

solving nonstiff higher order odes directly by the direct integration method the variable order and stepsize direct integration di method for solving systems of nonstiff higher order odes directly is reviewed mention is also made of the current inroads in solving stiff higher order odes directly the advantages of the di method over the corresponding adams method when used to solve the same problem as a first order system are pointed out numerical results are presented comparing the two methods in terms of execution times for each tolerance used and execution times for each solution of the same range of accuracy

coherence and chaos in the driven damped sine gordon equation measurement of the soliton spectrum abstract a numerical procedure is developed which measures the sine gordon soliton and radiation content of any field t which is periodic in space the procedure is applied to the field generated by a damped driven sine gordon equation this field can be either temporally periodic locked to the driver or chaotic in either case the numerical measurement shows that the spatial structure can be described by only a few spatially localized soliton wave train modes the numerical procedure quantitatively identifies the presence number and properties of these soliton wave trains for example an increase of spatial symmetry is accompanied by the injection of additional solitons into the field

decompositions of the free additive convolution we introduce and study a new type of convolution of probability measures denoted and called the s free additive convolution which is defined by the subordination functions associated with the free additive convolution we derive an alternating decomposition of for compactly supported and using another convolution called orthogonal additive convolution this decomposition leads to two types of complete alternating decompositions of the free additive convolution more importantly we develop an operatorial approach to the subordination property and introduce the associated notion of s free independence moreover we establish relations between convolutions associated with the main notions of noncommutative independence free monotone and boolean finally our result leads to natural decompositions of the free product of rooted graphs

the solitons of zabusky and kruskal revisited perspective in terms of the periodic spectral transform abstract we consider the numerical soliton experiments of zabusky and kruskal 1 and address their results in light of the spectral structure of the periodic korteweg devries kdv equation we derive and exploit a simple relationship between the infinite line transfer matrix and the monodromy matrix of periodic theory this establishes the connection at all stages of our analysis between the infinite line soliton and the soliton limit of periodic theory we develop analytical and numerical procedures for determining the spectral structure of a periodic wave train and discuss how the methods may be viewed as a generalization of ordinary periodic fourier analysis finally we introduce the concept of a reference level which is defined to be the level upon which the solitons propagate these results together with judicious numerical experiments allow us to reassess the work of zabusky and kruskal from the standpoint of periodic theory for the kdv equation we find their system to be soliton dominated and our estimates of the soliton amplitudes and phase speeds agree well with those given in their classic paper we also find that even in the presence of periodic boundary conditions the solitons behave like infinite line solitons in that they undergo phase shifts upon collision with each other our estimate of the fpu recurrence time which includes the effect of phase shifting is within one percent of their observed value

the double cnoidal wave of the korteweg de vries equation an overview earlier work of the author on the spatially periodic solutions of the korteweg de vries equation is here extended via an in depth treatment of a special case the double cnoidal wave is the simplest generalization of the ordinary cnoidal wave discovered by korteweg and de vries in 1895 in the limit of small amplitude the double cnoidal wave is the sum of two noninteracting linear sine waves in the oppositie limit of large amplitude it is the sum of solitary waves of two different heights repeated periodically over all space although special the double cnoidal wave is important because it is but the particular case n 2 of a broad family of solutions known variously as n polycnoidal waves finite gap finite zone solutions waves on a circle or n phase wave trains it has been shown by others that the set of n polycnoidal waves gives the general initial value solution to the korteweg de vries equation this present work is the core of a three part treatment of the double cnoidal wave this part the overview presents graphic examples in all the important parameter regimes explains how collision phase shifts alter the average speed of the two wave phases from the free velocities of the two solitary waves describes the different branches or modes of the double cnoidal wave it is possible to have many solitary waves on each spatial period provided they are of only two distinct sizes and contrasts the results of this work with the very limited numerical calculations of previous authors the second part describes how the problem of numerically calculating the double cnoidal wave can be reduced down to solving four algebraic equations by perturbation theory the third part explains how the so called modular transformation of the riemann theta functions is important in interpreting n polycnoidal waves

the fermi pasta ulam problem and its underlying integrable dynamics an approach through lyapunov exponents fpu models in dimension one are perturbations either of the linear model or of the toda model perturbations of the linear model include the usual beta model perturbations of toda include the usual alpha beta model in this paper we explore and compare two families or hierarchies of fpu models closer and closer to either the linear or the toda model by computing numerically for each model the maximal lyapunov exponent chi more precisely we consider statistically typical trajectories and study the asymptotics of chi for large n the number of particles and small varepsilon the specific energy e n and find for all models asymptotic power laws chi simeq c varepsilon a c and a depending on the model the asymptotics turns out to be in general rather slow and producing accurate results requires a great computational effort we also revisit and extend the analytic computation of chi introduced by casetti livi and pettini originally formulated for the beta model with great evidence the theory extends successfully to all models of the linear hierarchy but not to models close to toda

conservation laws and the numerical solution of odes ii abstract many physical systems described by an initial value problem for a system of ordinary differential equations odes conserve physical quantities such as the net charge or total energy as the system evolves typical codes for the numerical solution of the odes will not conserve these quantities and this can lead to solutions which are not even qualitatively correct one way to impose conservation laws is to perturb the numerical solution at each step of the integration a simple theory is developed in this paper which tells how this should be done so as to guarantee convergence of codes based on one step methods it is also easy to interpret the effect on the accuracy of the perturbations

varying the componentwise order of the multistep methods in solving odes and its absolute stability most variable order variable stepsize codes for solving odes vary the order along the interval of integration but keep the order constant across the system however some codes that do partitioning of the odes into stiff and nonstiff subsystems vary the order for each equation in the system in this paper the reasons for varying the order componentwise are given and some of its salient features are illustrated using a numerical example also given is the direct proof of the existence of a region of absolute stability when different order adams explicit methods are used to solve a system of odes

explicit parallel two step runge kutta nystrom methods abstract the aim of this paper is to design a class of two step runge kutta nystrom methods of arbitrarily high order for the special second order equation y t f y t for use on parallel computers starting with an s stage implicit two step runge kutta nystrom method of order p with k p 2 implicit stages we apply the highly parallel predictor corrector iteration process in p ec me mode in this way we obtain an explicit two step runge kutta nystrom method that has order p for all m and that requires k m 1 right hand side evaluations per step of which each k evaluation can be computed in parallel by a number of numerical experiments we show the superiority of the parallel predictor corrector methods proposed in this paper over both sequential and parallel methods available in the literature

recurrence phenomena in soliton propagation in a lattice with impurities the scattering of a soliton from a mass impurity in a morse or toda one dimensional lattice with periodic boundary conditions is studied numerically the energy of the soliton as a function of time exhibits either a fast decay chaotic behaviour or a recurrence or an intermediate metastable behaviour which consists of a time period when recurrence takes place followed by a fast decay a model is developed to explain the recurrence and predict the recurrence time a semiquantitative argument is also presented to explain the metastable case finally the generality of this type of behaviour is discussed

strategies for solving index one dae with non negative constraints liquid liquid extraction modeling leads to solve an index one dae system for the sake of robustness it is desirable to account for non negative constraints based on the dassl architecture a classical index one dae solver we propose and compare three different strategies to implement these bound constraints each of these strategies corresponds to a different newton modification clipping damping or interior point method the comparisons are made on two test cases the robertson ode problem and an example from liquid liquid extraction modeling

efficient solution of nonlinear ordinary differential equations expressed in s system canonical form ordinary differential equations can be recast into a nonlinear canonical form called an s system evidence for the generality of this class comes from extensive empirical examples that have been recast and from the discovery that sets of differential equations and functions recognized as among the most general are special cases of s systems identification of this nonlinear canonical form suggests a radically different approach to numerical solution of ordinary differential equations by capitalizing on the regular structure of s systems efficient formulas for a variable order variable step taylor series method are developed the minimum improvement in efficiency over conventional runge kutta methods is shown to be more than 20 to 60 percent for equations expressed in s system form tests with implemented methods demonstrate that 10 to 100 fold reductions in time are actually realized for solution of s systems to examine the potential benefit of recasting ordinary differential equations and then sol

birkhoff coordinates for the toda lattice in the limit of infinitely many particles with an application to fpu abstract in this paper we study the birkhoff coordinates cartesian action angle coordinates of the toda lattice with periodic boundary condition in the limit where the number n of the particles tends to infinity we prove that the transformation introducing such coordinates maps analytically a complex ball of radius r n in discrete sobolev analytic norms into a ball of radius r n with r r 0 independent of n if and only if 2 then we consider the problem of equipartition of energy in the spirit of fermi pasta ulam we deduce that corresponding to initial data of size r n 2 0 r 1 and with only the first fourier mode excited the energy remains forever in a packet of fourier modes exponentially decreasing with the wave number finally we consider the original fpu model and prove that energy remains localized in a similar packet of fourier modes for times one order of magnitude longer than those covered by previous results which is the time of formation of the packet the proof of the theorem on birkhoff coordinates is based on a new quantitative version of a vey type theorem by kuksin and perelman which could be interesting in itself

rkn type parallel block pc methods with lagrange type predictors abstract this paper describes the construction of block predictor corrector methods based on runge kutta nystrom correctors our approach is to apply the predictor corrector method not only at step points but also at off step points block points so that in each step a whole block of approximations to the exact solution at off step points is computed in the next step these approximations are used to obtain a high order predictor formula using lagrange interpolation by suitable choice of the abscissas of the off step points a much more accurately predicted value is obtained than by predictor formulas based on last step values since the block of approximations at the off step points can be computed in parallel the sequential costs of these block predictor corrector methods are comparable with those of a conventional predictor corrector method furthermore by using runge kutta nystrom corrector methods the computation of the approximation at each off step point is also highly parallel application of the resulting block predictor corrector methods to a few widely used test problems reveals that the sequential costs are reduced by a factor ranging from 4 to 50 when compared with the best sequential methods from the literature

using nonconvergence of iteration to partition odes given is a variation of partitioning odes when using the multistep method essentially the nonconvergence property is made use of for partitioning the criteria used are explained by using a test system of two linear equations and generalisation is made to a general system of first order odes finally some numerical results are given that include systems greater than two

ac driven damped toda lattice abstract the dynamic properties of the ac driven damped toda lattice are studied using numerical simulations the threshold effect of the drive parameter and the stability conditions are investigated by varying the all essential parameters the collisions of solitons under the influence of the external drive are explored the drive scheme is also adapted to the continuum system

computational synergetics and mathematical innovation it is demonstrated how the computer used in a heuristic mode has greatly augmented our understanding of the mathematics of nonlinear dynamical process examples are given of recent work in soliton mathematics mainly but also nonlinear iterative mappings and low order nonlinear ordinary differential equation systems the role of good graphics in enhancing the discovery and retention of new mathematical properties of equations is illustrated several examples are given of the present frontier of research in two space dimensional nonlinear evolutionary problems

construction of variable stepsize multistep formulas a systematic way of extending a general fixed stepsize multistep formula to a minimum storage variable stepsize formula has been discovered that encompasses fixed coefficient interpolatory variable coefficient variable step and fixed leading coefficient as special cases in particular it is shown that the interpolatory stepsize changing technique of nordsieck leads to a truly variable stepsize multistep formula which has implications for local error estimation and formula changing and it is shown that the variable step stepsize changing technique applicable to the adams and backward differentiation formulas has a reasonable generalization to the general multistep formula in fact it is shown how to construct a variable order family of variable coefficient formulas finally it is observed that the first dahlquist barrier does not apply to adaptable multistep methods if storage rather than stepnumber is the key consideration

numerical solutions of the korteweg di vries equation using the periodic scattering transform m representation abstract we develop a numerical procedure based upon the periodic scattering transform for determining the spatial and temporal evolution of solutions to the korteweg de vries kdv equation with periodic boundary conditions in this approach solutions to kdv are found in terms of the representation i e the wave amplitude is given by a trace formula consisting of a linear superposition of hyperelliptic functions j x t 1 j n which are governed by n nonlinearly coupled ordinary differential equations our approach consists of the numerical integration of these equations using a second order taylor series expansion with variable integration step size it is well known that the j are generalizations of elliptic functions to the hyperelliptic class nevertheless we find that the j x t bear little or no resemblance to elliptic functions except in the small amplitude linear limit or for the case of a single soliton we study in particular the most difficult numerical case which occurs for the n soliton wave train solution of the periodic kdv equation

ground state of the quantum periodic toda lattice in the large n limit in gutzwiller s formulation of the quantum periodic toda lattice quantization conditions are expressed in terms of floquet s characteristic exponents which are equal to the zeros of a hill type determinant we have derived an integral equation for the density distribution of those zeros for the ground state in the large n limit the large n asymptotic expansions of the conserved quantities given by this density distribution are found to be fairly good approximations to the exact values we have also carried out the semiclassical quantization by the ebk formulation and the semiclassical eigenvalues turned out to be very close to the exact values

the nonlinear dynamics of the damped and driven toda chain 1 energy bifurcation diagrams abstract the dynamics of a periodically driven damped toda chain with periodic boundary conditions is investigated with the help of energy cross sections it can be shown that besides periodic motions of different periodicity quasiperiodic and aperiodic chaotic motions exist the method of energy bifurcation diagrams is introduced which allows a survey of the resonance properties of the chain

multi variable subordination distributions for free additive convolution abstract let k be a positive integer and let d c k denote the space of joint distributions for k tuples of selfadjoint elements in c probability space the paper studies the concept of subordination distribution of with respect to for d c k where is the operation of free additive convolution on d c k the main tools used in this study are combinatorial properties of r transforms for joint distributions and a related operator model with operators acting on the full fock space multi variable subordination turns out to have nice relations to a process of evolution towards infinite divisibility on d c k that was recently found by belinschi and nica arxiv 0711 3787 most notably one gets better insight into a connection which this process was known to have with free brownian motion

an averaging theorem for fpu in the thermodynamic limit consider an fpu chain composed of n gg 1 particles and endow the phase space with the gibbs measure corresponding to a small temperature beta 1 given a fixed k we construct k packets of normal modes whose energies are adiabatic invariants i e are approximately constant for times of order beta 1 a a 0 for initial data in a set of large measure furthermore the time autocorrelation function of the energy of each packet does not decay significantly for times of order beta the restrictions on the shape of the packets are very mild all estimates are uniform in the number n of particles and thus hold in the thermodynamic limit n rightarrow infty beta 0

implicit schemes for differential equations abstract implicit high order accurate temporal integration schemes are developed to reduce partial differential equations of evolution type to a sequence of ordinary differential equations involving the spatial directions only the ordinary differential equations are solved using various accurate integration methods the schemes are free from extraneous boundary and or initial conditions and are stable for large courant number

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

the projected gradient methods for least squares matrix approximations with spectral constraints the problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure this paper describes a general procedure in using the projected gradient method it is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicitly this gives rise to the construction of a descent flow that can be followed numerically the explicit form also facilitates the computation of the second order optimality conditions examples of applications are discussed with slight modifications the procedure can be extended to solve least squares problems for general matrices subject to singular value constraints

efficient calculation of chemically reacting flow a semi implicit finite volume formulation is used to jrvstudy flow with chemical reactions in this formulation the source terms resulting from the chemical reaction are treated implicitly and the resulting system of partial differential equations is sovled using two time stepping schemes the first is based on the runge kutta method and the second on an adams predictor corrector method results show that improvements in computational efficiency depend to a large extent on the manner in which the source term is treated furthermore analysis and computation indicate that the rungekutta method is more efficient than the adams methods for these systems of differential equations finally an adaptive time stepping scheme is developed to study problems involving shock ignition calculations for a hydrogen air system agree well with other methods

trapping and fusion of solitons in a nonuniform toda lattice abstract in this paper we consider strongly localized solitons leading to high energy events in a toda lattice with springs of distinct nonlinearity the trapping and superposition of solitons at a single soft spring embedded in a hard host lattice is investigated numerically we further show the generation of high energy solitons at an interface between a hard and a soft lattice the behaviour of a soliton incident on an inhomogeneity is approached analytically the interaction of solitons in an inhomogeneous toda lattice is compared to that in a homogeneous one

variable order variable step methods for second order initial value problems abstract we develop a variable order variable step algorithm for solving second order initial value problems y f t y y 0 y 0 given this algorithm employs a family of implicit two step methods in which the differential equation is evaluated at a number of off step points we have derived the order conditions for this family of methods in previously published work and we have also shown that the remaining free parameters of the method may be chosen to given p stability here the emphasis is on the combination of second fourth sixth and eighth order methods from this family in a variable order variable step algorithm the aim is to choose particular p stable methods from the family in such a way that a saving can be made in the amount of computational effort involved when they are combined together

the nonlinear dynamics of the damped and driven toda chain ii fourier and lyapunov analysis of tori abstract the dynamics of a periodically damped and driven toda chain with periodic boundary conditions is investigated with the help of spectral energy bifurcation diagrams and lyapunov diagrams it can be shown that quasiperiodic motions on tori tk of dimension k 3 exist the technical problems concerning the fourier and lyapunov analysis of the motions on tori tk k 2 are addressed

on noncharacteristic boundary conditions for discrete hyperbolic initial boundary value problems abstract the stability properties of some discrete hyperbolic initial boundary value problems involving two dependent variables and noncharacteristic boundary conditions are examined approximations based on the lax wendroff leap frog and crank nicolson methods are considered the most accurate approximations are found to be the least stable the results for the model problems can be used in the stability analysis of larger problems where characteristics occur singly or in pairs in particular discretizations of the shallow water and gas dynamics equations which involve boundary conditions written in natural variables are considered some computations demonstrating the stability results are discussed an appendix discussing numerical stability verification techniques is included

design of software for odes we discuss the design of software that is easy to use for simple problems but still capable of solving complicated problems nowadays a design should accommodate related but fundamentally different tasks such as solving daes and ddes in addition to ivps and bvps for odes major issues are illustrated with the matlab ode suite and selected solvers written in fortran 90 and maple

solitons cnoidal waves and nonlinear interactions in shallow water ocean surface waves abstract we analyze shallow water surface wave data from the adriatic sea using a nonlinear generalization of fourier analysis based upon the periodic inverse scattering transform in the function representation for the korteweg de vries kdv equation while linear fourier analysis superposes sine waves the nonlinear fourier approach superposes cnoidal waves the travelling wave solution to kdv plus their mutual nonlinear interactions a new procedure is presented for the nonlinear low pass and band pass filtering of measured wave trains we apply the approach to a measured time series and discuss the dynamics of solitons and the physics of the nonlinear interactions in terms of global spatio temporal phase shifts amongst the cnoidal waves

an efficient approach for the numerical simulation of multibody systems the field of kinematics and dynamics of mechanical systems has progressed from a manual graphics art to a highly developed discipline in analytical geometry and dynamics various general purpose formulations for the dynamic analysis of constrained mechanical systems cms lead to mixed differential algebraic equations daes called the euler lagrange equations during the past fifteen years many contributions have been made to the theory of computational kinematics and dynamics of cms also called mutibody dynamics the recent advances in computer hardware and software have tremendously revolutionized the analysis of cms there are various numerical approaches for solving general vector fields in the previous paper appl math comput 92 1998 153 193 a complete and detailed analysis of various approaches for the numerical solution of vector fields is given in this paper we extend the algorithms presented in the above mentioned reference to solve the euler lagrange equations of motion for cms the numerical experiments suggest that perturbation approach performs better than the other approaches

multiderivative methods for periodic initial value problems a family of two step multiderivative methods is developed for the numerical solution of periodic initial value problems of the form bf y t bf f t bf y with initial conditions bf y t 0 bf y 0 bf y t 0 bf y 0 given the methods are developed by showing that the solution of the single test equation y t lambda 2 y t lambda real satisfies a recurrence relation in which exponential functions are involved the methods of the family are then derived by replacing the exponential terms by pade approximants the methods are analyzed for accuracy and periodicity properties the error constants and intervals of periodicity are contained in an appendix those pade approximants for which the degree of the numerator do not exceed the degree of the denominator are seen to be p stable suitable predictor corrector combinations of the methods are analyzed for use in pece mode the methods are compared with existing methods and they are tested on four problems from the

a two loop sparse matrix numerical integration procedure for the solution of differential algebraic equations application to multibody systems abstract in this paper a two loop implicit sparse matrix numerical integration tlismni procedure for the solution of constrained rigid and flexible multibody system differential and algebraic equations is proposed the proposed method ensures that the kinematic constraint equations are satisfied at the position velocity and acceleration levels in this method a sparse lagrangian augmented form of the equations of motion that ensures that the constraints are satisfied at the acceleration level is first used to solve for all the accelerations and lagrange multipliers the independent coordinates and velocities are then identified and integrated using htt or newmark formulas expressed in this paper in terms of the independent accelerations only the constraint equations at the position level are then used within an iterative newton raphson procedure to determine the dependent coordinates the dependent velocities are determined by solving a linear system of algebraic equations in order to effectively exploit efficient sparse matrix techniques and have minimum storage requirements a two loop iterative method is proposed equally important the proposed method avoids the use of numerical differentiation which is commonly associated with the use of implicit integration methods in multibody system algorithms numerical examples are presented in order to demonstrate the use of the new integration procedure

automatic solution of sturm liouville problems using the pruess method abstract we develop algorithms based on coefficient approximation for the automatic solution of regular and singular sturm liouville problems

error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations we prove new optimal a priori error estimates for a class of implicit one step methods for stiff ordinary differential equations obtained by using the discontinuous galerkin method with piecewise polynomials of degree zero and one starting from these estimates we propose a new algorithm for automatic time step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations

strong convergence rates for backward euler on a class of nonlinear jump diffusion problems we generalise the current theory of optimal strong convergence rates for implicit euler based methods by allowing for poisson driven jumps in a stochastic differential equation sde more precisely we show that under one sided lipschitz and polynomial growth conditions on the drift coefficient and global lipschitz conditions on the diffusion and jump coefficients three variants of backward euler converge with strong order of one half the analysis exploits a relation between the backward and explicit euler methods

the fermi pasta ulam problem and its underlying integrable dynamics this paper is devoted to a numerical study of the familiar fpu model precisely we here discuss revisit and combine together two main ideas on the subject i in the system at small specific energy e e n two well separated time scales are present in the former one a kind of metastable state is produced while in the second much larger one such an intermediate state evolves and reaches statistical equilibrium ii fpu should be interpreted as a perturbed toda model rather than as is typical as a linear model perturbed by nonlinear terms in the view we here present and support the former time scale is the one in which fpu is essentially integrable its dynamics being almost indistinguishable from the toda dynamics the toda actions stay constant for fpu too while the usual linear normal modes do not the angles fill their almost invariant torus and nothing else happens the second time scale is instead the one in which the toda actions significantly evolve and statistical equilibrium is possible we study both fpu like initial states in which only a few degrees of freedom are excited and generic initial states extracted randomly from an approximated microcanonical distribution the study is based on a close comparison between the behavior of fpu and toda in various situations the main technical novelty is the study of the correlation functions of the toda constants of motion in the fpu dynamics such a study allows us to provide a good definition of the equilibrium time i e of the second time scale for generic initial data our investigation shows that is stable in the thermodynamic limit i e the limit of large n at fixed e and that by reducing e ideally the temperature approximately grows following a power law e a with a 5 2

a toda lattice model for dna thermally generated solitons in this paper we calculate the thermal equilibrium number of solitons in dna as a function of absolute temperature and the number of base pairs these calculations are effected by modeling dna as a toda lattice with parameters chosen to match experimentally measured properties of dna we find that a significant number of solitons is generated at physiological temperature 310 k moreover at low temperature the dependence of the number of solitons on the temperature follows a t13 law

a quasi periodic route to chaos in a near integrable pde abstract pattern formation and transitions to chaos are described for the damped ac driven one dimensional periodic sine gordon equation in a nonlinear schrodinger regime a generic quasi periodic route to intermittent chaos is exhibited in detail using a range of dynamical systems diagnostics in addition a nonlinear spectral transform is exploited i to identify and quantify coordinates of space time attractors in terms of a small number of soliton modes of the underlying integrable system ii to use these analytic coordinates to identify homoclinic orbits as possible sources of chaos and iii to demonstrate the significance of energy transfer between coherent and extended states in this chaotic system

numerical inverse spectral transform for the periodic sine gordon equation theta function solutions and their linearized stability abstract the inverse spectral transform for integrable nonlinear ordinary and partial differential equations such as the toda lattice korteweg de vries sine gordon and nonlinear schrodinger equations provides explicit algorithms to generate exact solutions under periodic or quasiperiodic boundary conditions these oscillatory wavetrains may be prescribed a priori to consist of a nonlinear superposition of n phases j x t j x j t 0 j j 1 n where the wave is 2 periodic independently in each phase this paper exhibits the numerical implementation of the inverse spectral solution of the sine gordon equation the general construction is outlined and then implemented for n 1 2 and 3 we compute 1 the exact theta function solutions 2 the floquet spectrum of x periodic solutions 3 the labelling of linearized instabilities of n phase solutions in terms of spectral data and 4 the linearized growth rate in each unstable mode the associated surfaces q n x t are displayed to illustrate a variety of spatial and dynamical phenomena in the oscillatory solution space of this integrable system

numerical investigations on global error estimation for ordinary differential equations four techniques of global error estimation which are richardson extrapolation rs zadunaisky s technique zd solving for the correction sc and integration of principal error equation ipee have been compared in different integration codes dopri5 dvode dstep theoretical aspects concerning their implementations and their orders are first given second a comparison of them based on a large number of tests is presented in terms of cost and precision sc is a method of choice for one step methods it is much more precise and less costly than rs and leads to the same precision as zd for half its cost ipee can provide the order of the error for a cheap cost in codes based on one step methods in multistep codes only rs and ipee have been implemented since they are the only ones whose theoretical justification has been extended to this case there rs still provides a more reliable estimation than ipee however as these techniques are based on variations of the global error irrespective of the numerical method used they fail to provide any more usefull information once the numerical method has reached its limit of accuracy due to the finite arithmetic

comments on strong convergence rates for backward euler on a class of nonlinear jump diffusion problems j comput appl math 205 2007 949 956 abstract in this note we correct an inaccuracy of eq 13 in the proof of theorem 1 in the paper higham and kloeden 2007 we also give a complete proof of the theorem by applying the burkholder davis gundy inequality