analysis of a predator prey model with modified leslie gower and holling type ii schemes with stochastic perturbation abstract in this paper we consider a predator prey model with modified leslie gower and holling type ii schemes with stochastic perturbation we show there is a unique positive solution to the system with positive initial value and mainly investigate the long time behavior of the system condition for the system to be extinct is given and persistent condition is established at last numerical simulations are carried out to support our results

on the convergence of a finite element method for a nonlinear hyperbolic conservation law we consider a space time finite element discretization of a time dependent nonlinear hyperbolic conservation law in one space dimension burgers equation the finite element method is higher order accurate and is a petrov galerkin method based on the so called streamline diffusion modification of the test functions giving added stability we first prove that if a sequence of finite element solutions converges boundedly almost everywhere as the mesh size tends to zero to a function u then u is an entropy solution of the conservation law this result may be extended to systems of conservation laws with convex entropy in several dimensions we then prove using a compensated compactness result of murat tartar that if the finite element solutions are uniformly bounded then a subsequence will converge to an entropy solution of burgers equation we also consider a further modification of the test functions giving a method with improved shock capturing finally we present the results of some numerical experiments

a time space flux corrected transport finite element formulation for solving multi dimensional advection diffusion reaction equations abstract we present a time space flux corrected transport fct finite element formulation for the multi dimensional time dependent advection diffusion reaction equation monotonic solutions can be achieved with the presented method while large time steps with courant number c r 1 are used numerical verification as well as a grid convergence analysis are carried out for 1d and 2d benchmark problems a 1d burgers equation with various reynolds numbers is solved with the time space fct method in addition a 2d transport problem with and without a nonlinear reaction inside of a cavity is considered finally the newly developed time space fct formulation is applied for modeling biofilm growth problems based on a continuum mathematical model it turns out that the time space fct method helps to reduce numerical dissipation and guarantees comparable numerical dispersion of the solution at the same time comparing to numerical solutions obtained by a time space finite incremental calculus fic method

compact subgroups of linear algebraic groups abstract the general problem underlying this article is to give a qualitative classification of all compact subgroups gl n f where f is a local field and n is arbitrary it is natural to ask whether is an open compact subgroup of h e where h is a linear algebraic group over a closed subfield e f we show that indeed has this form up to finite index and a finite number of abelian subquotients when is zariski dense in a connected semisimple group we give a precise openness result for the closure of the commutator group of in the case char f 0 the answers have long been known by results of chevalley and weyl the motivation for this work comes from the positive characteristic case where such results are needed to study galois representations associated to function fields we also derive openness results over a finite number of local fields

time discontinuous stabilized space time finite elements for timoshenko beams stabilized space time e nite element methods for the transient computational analysis of the elastodynamics of timoshenko beams have been developed the underlying time discontinuous galerkin formulation is implicit unconditionally stable higher order accurate and robust the employed interpolations are continuous in space and time inside time slabs but discontinuous in time between adjacent time slabs to suppress spurious numerical oscillations near discontinuities or high gradients galerkin least squares stabilization has been applied further improvement of the numerical representation of the transient elastic wave propagation phenomena has been achieved by specially designed galerkin gradient least squares operators because of the reduced phase and amplitude errors of these e nite elements much coarser spatial meshes may be used without loss of accuracy the resulting reduction of the number of unknowns leads to a signie cant decrease of computer time furthermore a globaladaptivetime stepping strategy has beenemployed basedon thetemporal jump residual of thetimee niteelement method as error estimator numerical examples of transientelastic wave propagation in timoshenko beams involving a wide spectrum of wave numbers and frequencies demonstrate the good performance of the developed methods

dynamics of a predator prey model we describe the bifurcation diagram of limit cycles that appear in the first realistic quadrant of the predator prey model proposed by r m may stability and complexity in model ecosystems princeton university press princeton nj 1974 in particular we give a qualitative description of the bifurcation curve when two limit cycles collapse on a semistable limit cycle and disappear moreover we show that locally asymptotic stability of a positive equilibrium point does not imply global stability for this class of predator prey models

periodic solution for a non autonomous lotka volterra predator prey model with random perturbation abstract this paper studies a stochastic non autonomous lotka volterra predator prey model we prove that there exists at least one positive periodic solution under some simple and reasonable conditions in addition we obtain sufficient conditions for persistence in mean and extinction for the stochastic non autonomous system our method and results are new and the ideas could be used to study other types of non autonomous stochastic predator prey systems

bi discontinuous time step integration algorithms part 2 second order equations abstract in this paper bi discontinuous time step integration algorithms for linear second order equations are investigated the second order equations are manipulated directly both the initial and final displacement and velocity are only weakly enforced at the beginning and at the end of the time interval as a result there may be discontinuous jumps for both the displacement and velocity at the two ends of the time interval if the displacement is assumed to be a polynomial of degree n 1 within the time interval there are n 4 unknowns in the formulation it is found that the order of accuracy would be at least n 1 for displacement and n for velocity in general it is shown that the relevant algorithmic parameters for the second order equations can be related to the algorithmic parameters presented in part 1 of this paper for the first order equations unconditionally stable higher order accurate time step integration algorithms with controllable numerical dissipation equivalent to the generalized pade approximations with an order of accuracy 2 n 3 can then be constructed systematically the weighting parameters and the corresponding weighting functions are given explicitly it is also shown that the accuracy of the particular solutions is compatible with the accuracy of the homogeneous solutions if the proposed weighting functions are employed

higher order accurate time step integration algorithms by post integration techniques in this paper a framework to construct higher order accurate time step integration algorithms based on the post integration techniques is presented the prescribed initial conditions are naturally incorporated in the formulations and can be strongly or weakly enforced the algorithmic parameters are chosen such that unconditionally a stable higher order accurate time step integration algorithms with controllable numerical dissipation can be constructed for linear problems besides it is shown that the order of accuracy for non linear problems is maintained through the relationship between the present formulation and the runge kutta method the second order differential equations are also considered numerical examples are given to illustrate the validity of the present formulation copyright 2001 john wiley sons ltd

enriched space time finite element method a new paradigm for multiscaling from elastodynamics to molecular dynamics summary the main objective of this paper is to present an enriched version of the space time fem method to incorporate multiple temporal scale features with a focus on dynamics problems the method is established by integrating the basic framework of the space time discontinuous galerkin method with the extended finite element method two versions of the method have been developed one at the continuum scale elastodynamics and the other focuses on the dynamics at the molecular level after an initial outline of the formulation we explore the incorporation of different types of enrichments based on the length scale of interest the effects of both continuous and discontinuous enrichments are demonstrated through numerical examples involving wave propagations and dynamic fracture in harmonic lattice the robustness of the method is evaluated in terms of convergence and the ability to capture the fine scale features it is shown that the enriched space time fem leads to an improvement in the convergence properties over the traditional space time fem for problems with multiple temporal features it is also highly effective in integrating atomistic with continuum representations with a coupled framework copyright 2012 john wiley sons ltd

on semigroups of matrices with traces in a subfield abstract it is shown that an irreducible semigroup of n n complex matrices with real spectra is simultaneously similar to a semigroup of real matrices weaker results are obtained for semigroups of matrices over a general field with traces in a subfield

discontinuous galerkin methods for ordinary differential equations a class of galerkin methods derived from discontinuous piecewise polynomial spaces is analyzed for polynomials of degree k these methods lead to a family of one step schemes generating approximations up to order 2k 2 for the solution of an ordinary differential equation

hamilton s law of variable mass system and time finite element formulations for time varying structures based on the law summary traditional principles of mechanics are primarily conceived for constant mass systems which are only valid if mass is gained or lost at null velocity with respect to an inertial reference frame for variable mass systems thus the numerical algorithms for time varying structures based on these principles are only suitable for this special case in this paper hamilton s law of variable mass system is derived based on meshchersky s fundamental equation and two classes of novel time finite element formulations for linear systems with arbitrary continuous time varying parameters are developed based on the previous law the formulations are verified extensively through numerical examples in which the convergence and effectiveness of algorithms are evaluated numerical examples demonstrate that compared with the algorithms for time varying structures that developed based on traditional principles of mechanics the proposed algorithms provide extended capabilities in both time varying mass problems that mass is gained or lost at any velocity such as rocket problem and moving mass problems such as vehicle bridge interaction problem besides the time varying stiffness and damping problems the proposed algorithms have a wider range of application in particular hamilton s law of variable mass system provides a solid theoretical foundation for further research on the algorithm design for time varying structures copyright 2014 john wiley sons ltd

global stability for a class of predator prey systems this paper deals with the question of global stability of the positive locally asymptotically stable equilibrium in a class of predator prey systems the dulac s criterion is applied and liapunov functions are constructed to establish the global stability

stochastic persistence and stationary distribution in a holling tanner type prey predator model abstract in this paper we study a stochastic predator prey model with beddington deangelis type functional response and logistic growth for predators the deterministic model is already well studied and we recall some important results here we construct the stochastic model from the deterministic model by introducing multiplicative noise terms into the growth equations of prey and predator populations for the stochastic model we show that the system admits unique positive global solution starting from the positive initial value then we prove that the system is strongly persistent in mean when the intensity of environmental forcing is less than some threshold magnitudes finally we show that the system has a stationary distribution under certain parametric restrictions numerical simulations are carried out to substantiate the analytical results

a comprehensive unified set of single step algorithms with controllable dissipation for dynamics part ii algorithms and analysis in part i a comprehensive unified set of single step algorithms with controllable dissipation based on the generalized galerkin formulation was presented in this paper the details of four lower order algorithms using parametrized temporal finite element are presented the analyses of the numerical stability dissipation and dispersion are also provided in comparison with the established algorithms the present algorithms facilitate controllable dissipation and yield results of higher order accuracy examples are also included to illustrate the validity of the algorithms

consistent discontinuous finite elements in elastodynamics finite element discontinuities with respect to time have recently been extremely used in elastodynamic problems due to their natural utilization in combination with adaptive methods and their efficiency in discontinuity capturing techniques for non smooth problems in this work we present some theoretical aspects and numerical results concerning the use of spatial discontinuities in a consistent finite element method for the same class of problems we first review some formulations for the elastostatic problem and prove two korn like inequalities which are very useful for the derivation of convergence rates in sobolev norms next we present formulations for the dynamic case along with comments on their properties and estimates of convergence rates for smooth solutions followed by numerical investigations of a typically non smooth problem involving classical and emerging variational formulations we also show some numerical experiments with finite element spaces enriched by discontinuous functions other than piecewise lagrangian polynomials

periodic solution for stochastic non autonomous multispecies lotka volterra mutualism type ecosystem we give sufficient conditions for the existence of positive periodic solution we give sufficient conditions for the global attractiveness of the periodic solution we prove that all species will be extinct if the intensity of noise is large enough the results are illustrated by computer simulations the paper characterizes some qualitative dynamic properties of a stochastic non autonomous multi species mutualism model with continuous periodic parameters using khasminskii theory of stability with suitable lyapunov functions and m matrices sufficient conditions are established to guarantee existence of positive periodic solutions to the system we also provide conditions for the global attractiveness of the latter or extinction of all species for sufficiently high volatility levels results are finally supported by numerical computations

energy conserving decaying implicit time stepping scheme for nonlinear dynamics of three dimensional beams undergoing finite rotations in this work we present an extension of time integration energy conserving scheme which introduces desirable properties of controllable energy decay as well as numerical dissipation of high frequency contribution to total response finite element implementation details are given for the chosen model problem of geometrically exact beam undergoing finite rotations several numerical simulations illustrate a very satisfying performance of the proposed time stepping scheme

saddle point principles and numerical integration methods for second order hyperbolic equations abstract this work describes a family of functionals whose stationarity often saddle point condition leads to well known so called variational formulations for structural dynamics such as the weak hamilton ritz formulation and the continuous discontinuous galerkin formulation and in turn to methods for the numerical integration of the equations of motion it is shown that all the time integration methods based on variational formulations do descend from such functionals moreover starting from the described family of functionals it is possible to construct new families of time integration methods which might exhibit computational advantages over the corresponding ones derived from variational formulations only

use of post processing to increase the order of accuracy of the trapezoidal rule at time integration of linear elastodynamics problems in this paper we suggest a very simple and effective post processing procedure to increase the order of accuracy in time for numerical results obtained by the trapezoidal rule we first derive a new exact closed form a priori error estimator for time integration of linear elastodynamics equations by the trapezoidal rule with non uniform time increments based on this error estimator we suggest a new post processing procedure containing additional time integration of elastodynamics equations by the trapezoidal rule with few time increments that systematically improves the order of accuracy of numerical results with the increase in the number of additional time increments used for post processing for example the use of just one additional time increment for post processing after time integration with any number of uniform time increments renders the order of accuracy of numerical results equal to 10 3 numerical examples of the application of the new techniques to a system with a single degree of freedom and to a multi degree system confirm the corresponding increase in the order of convergence of numerical results after post processing because the same trapezoidal rule is used for basic computations and post processing the new technique retains all of the properties of the trapezoidal rule requires no writing of a new computer program for its implementation and can be easily used with any existing commercial and research codes for elastodynamics

a discontinuous galerkin finite element method for dynamic and wave propagation problems in non linear solids and saturated porous media a time discontinuous galerkin finite element method dgfem for dynamics and wave propagation in non linear solids and saturated porous media is presented the main distinct characteristic of the proposed dgfem is that the specific p3 p1 interpolation approximation which uses piecewise cubic hermite s polynomial and linear interpolations for both displacements and velocities in the time domain is particularly proposed consequently continuity of the displacement vector at each discrete time instant is exactly ensured whereas discontinuity of the velocity vector at the discrete time levels still remains the computational cost is then obviously saved particularly in the materially non linear problems as compared with that required for the existing dgfem both the implicit and explicit algorithms are developed to solve the derived formulations for linear and materially non linear problems numerical results illustrate good performance of the present method in eliminating spurious numerical oscillations and in providing much more accurate solutions over the traditional galerkin finite element method using the newmark algorithm in the time domain copyright 2003 john wiley sons ltd

unique stationary distribution and ergodicity of a stochastic logistic model with distributed delay abstract dynamics of a stochastic logistic model with distributed delay are considered we first transfer a scalar stochastic logistic model with strong kernel or weak kernel into an equivalent stochastic system through the linear chain technique then we obtain the sufficient and necessary conditions for extinction and persistence of the species with probability one moreover in the case of persistence we prove that there exists a unique stationary distribution by the markov semigroups theory the results show that the stronger white noise results in the extinction of the species and the weaker white noise guarantees the existence of a unique stationary distribution though for the deterministic model with strong kernel or weak kernel the average delay may induce the existence of a group of small amplitude periodic solutions

error estimates and automatic time step control for nonlinear parabolic problems i in this note we extend results by one of the authors on time discretization error estimates and related automatic time step control for stiff ordinary differential equations to the case of a nonlinear parabolic problem the method for time discretization is the so called discontinuous galerkin method based on using piecewise polynomials of degree q geqq 0 we consider in this note the case q 0 corresponding to a variant of the backward euler method we prove a new almost optimal error estimate and present a related new algorithm for automatic time step control this algorithm is very simple but yet is efficient and gives control of the global error

accuracy of one step integration schemes for damped forced linear structural dynamics summary this paper proposes an energy based measure for the evaluation of the local truncation error of two level one step integration schemes the measure applies to multiple degree of freedom systems and does not necessarily require modal reduction to a scalar model it naturally handles the structural damping and external forcing terms that are generally and mistakenly neglected in error analyses and it segregates the error associated with the free and forced response components of the problem to illustrate the approach two examples associated with the application of the trapezoidal scheme and of a high order scheme proposed in the literature are analyzed the latter reveals the shortcomings of the standard approach that is based on the undamped unforced linear oscillator and therefore highlights the need for the proposed framework indeed the scheme order of accuracy is below expectation when structural damping or external forcing is considered in the numerically dissipative setting developments on the basis of the time discontinuous galerkin tdg method are then proposed to recover the scheme high order accuracy additionally they show the similarity that exists between schemes related to the tdg method and the ones obtained by integration by parts of the equation of motion copyright 2014 john wiley sons ltd

high order hierarchical a and l stable integration methods numerical integration of stiff first order systems of differential equations is considered it is shown how a c continuous polynomial time discretization in conjunction with a weighted residual method can be used to derive methods corresponding to the diagonal and first subdiagonal pade approximants in this manner a stable schemes of order 2k and l stable schemes of order 2k l are obtained at least for k 4 the methods are hierarchical in the sense that a scheme with a given accuracy embraces the equations of all lower order methods that have the same stability type

implementation and adaptivity of a space time finite element method for structural dynamics abstract in this paper we study the implementation and adaptivity of a space time finite element method for the analysis of two dimensional problems in structural dynamics the method also known as the time discontinuous galerkin method or the dg method employs finite elements discretizations in space and time simultaneously with bilinear basis functions that are continuous in space and discontinuous in time based on using the zienkiewicz zhu error estimate in space and the jumps discontinuities of displacements and velocities in the total energy norm as a local error estimate in time an h adaptive procedure which updates the spatial mesh and the time step size automatically so as to control the estimated errors within specified tolerances is proposed numerical examples are presented showing that the considered dg method is of second order accuracy in space in l 2 and third order accuracy in time and the adaptive procedure is capable of updating the spatial meshes and the time steps when necessary making the solutions reliable and the computation efficient

postintegration methods for numerical integration of non linear dynamic equations of motion the paper presents a family of methods with postintegration for numerical integration of non linear dynamic equations of motion of discrete systems both explicit and implicit algorithms of these methods are conditionally stable but can give solutions of accuracy of a few orders higher than the known methods with preintegration corresponding to them a general analysis of stability and accuracy of single step methods is carried out the postintegration methods have been tested on a single degree of freedom non linear system

an a posteriori error estimate and adaptive timestep control for a backward euler discretization of a parabolic problem a simple algorithm for adaptive timestep control is presented for a backward euler discretization of a linear parabolic problem the algorithm is based on an a posteriori error estimate involving the computed approximate solution it is proved that with only very rough a priori information on the exact solution the algorithm will choose a sequence of timesteps for which the error will be controlled up to a constant uniformly in time on a given tolerance level

structural dynamic analysis by a time discontinuous galerkin finite element method this paper studies a time discontinuous galerkin finite element method for structural dynamic problems by which both displacements and velocities are approximated as piecewise linear functions in the time domain and may be discontinuous at the discrete time levels a new iterative solution algorithm which involves only one factorization for each fixed time step size and a few iterations at each step is presented for solving the resulted system of coupled equations by using the jumps of the displacements and the velocities in the total energy norm as error indicators an adaptive time stepping procedure for selecting the proper time step size is described numerical examples including both single dof and multi dof problems are used to illustrate the performance of these algorithms comparisons with the exact results and or the results by the newmark integration scheme are given it is shown that the time discontinuous galerkin finite element method discussed in this study possesses good accuracy third order and stability properties its numerical implementation is not difficult and the higher computational cost needed in each time step is compensated by use of a larger time step size

shape functions for mixed p version finite elements in the time domain abstract the advantages of mixed formulations are well known in that they simplify the form of the governing equations and can lead to potentially significant savings in computational effort for given accuracy likewise the p version of the finite element method is known to offer savings in computational effort for some problems in this paper shape functions for a mixed p version finite element formulation in the time domain are developed these shape functions produce a high percentage of zeros in certain coefficient matrices the mixed formulation in time used here is based on a weak form of hamilton s extended principle the present formulation allows for accurate and efficient solution of time domain problems the accuracy and power of this formulation are shown for initial value problems such as time marching and for directly obtaining the periodic response of linear systems the method is shown to yield a family of accurate unconditionally stable algorithms for initial value problems based on a simple linear oscillator

time discontinuous galerkin methods with energy decaying correction for non linear elastodynamics in this paper a new time discontinuous galerkin tdg formulation for non linear elastodynamics is presented the new formulation embeds an energy correction which ensures truly energy decaying thus allowing to achieve unconditional stability that as shown in the paper is not guaranteed by the classical tdg formulation the resulting method is simple and easily implementable into existing finite element codes moreover it inherits the desirable higher order accuracy and high frequency dissipation properties of the classical formulation numerical results illustrate the very good performance of the proposed formulation copyright 2010 john wiley sons ltd

an efficient time discontinuous galerkin procedure for non linear structural dynamics this paper presents an effective time discontinuous galerkin procedure for non linear dynamics the procedure is based on a cheap iterative algorithm which is rather different from previous efforts in fact the algorithm is designed so that the corrected solutions inherit the desired stability and dissipative properties and the iterations serve only to improve accuracy some numerical tests illustrate the good performance of the present procedure which appears to be competitive with the available implementations of time discontinuous galerkin methods

time discontinuous material point method for transient problems abstract this paper presents a time discontinuous material point method tdmpm for transient problems such as the wave propagation and impact responses in solids by dividing the continuous time domain into discrete time intervals the weak form of the tdmpm is established by considering the discrete grid based governing equations constraint and discontinuity conditions the displacement and velocity fields in a time interval are interpolated with the piecewise cubic and linear functions respectively by substituting the assumed displacement and velocity fields into the weak form a novel computational framework for the grid displacements and velocities at the discrete time instants is constructed in the new formulations the displacement field at each time instant remains to be continuous whereas the velocity field at the time instant becomes discontinuous these unique features ensure the tdmpm could properly capture the discontinuous characteristics and control the spurious numerical oscillations two numerical examples under the impact loading are used to verify the proposed method two representative impact problems are then presented for further verification and demonstration besides the corresponding contact algorithm adopted in the tdmpm is shown to be capable of capturing the correct contact behavior with higher fidelity and less computational cost than the mpm the presented results illustrate that the tdmpm could successfully control the spurious numerical oscillations associated with transient simulations

weighting parameters for unconditionally stable higher order accurate time step integration algorithms part 2 second order equations in this paper unconditionally stable higher order accurate time step integration algorithms suitable for linear second order differential equations based on the weighted residual method are presented the second order equations are manipulated directly as in part 1 of this paper instead of specifying the weighting functions the weighting parameters are used to control the algorithm characteristics the algorithms are at least nth order accurate if the numerical solution for displacement is approximated by a polynomial of degree n 1 with n undetermined coefficients by choosing the weighting parameters carefully the order of accuracy can be improved the generalized pade approximations for the second order equations are considered the ultimate spectral radius is an algorithmic parameter by relating the approximate solutions to the equivalent formulations presented in part 1 of this paper the required weighting parameters are found explicitly any set of linearly independent functions can be used to construct the corresponding weighting functions from the weighting parameters the stabilizing weighting functions for the weighted residual method are found explicitly to ensure higher order accuracy in the general solution the accuracy of the particular solution due to excitation is also examined copyright 1999 john wiley sons ltd

dynamics of a stochastic lotka volterra model perturbed by white noise abstract this paper continues the study of mao et al investigating two aspects of the equation d x t diag x 1 t x n t b a x t d t x t d w t t 0 the first of these is to slightly improve results in x mao s sabais e renshaw asymptotic behavior of stochastic lotka volterra model j math anal 287 2003 141 156 concerning with the upper growth rate of the total quantity i 1 n x i t of species by weakening hypotheses posed on the coefficients of the equation the second aspect is to investigate the lower growth rate of the positive solutions by using lyapunov function technique and using a changing time method we prove that the total quantity i 1 n x i t always visits any neighborhood of the point 0 and we simultaneously give estimates for this lower growth rate

symbolic computation of fundamental solution matrices for linear time periodic dynamical systems abstract a new technique which employs both picard iteration and expansion in shifted chebyshev polynomials is used to symbolically approximate the fundamental solution matrix for linear time periodic dynamical systems of arbitrary dimension explicitly as a function of the system parameters and time as in previous studies the integration and product operational matrices associated withe chebyshev polynomials are employed however the need to algebraically solve for the chebyshev coefficients of the fundamental solution matrix is completely avoided as only matrix multiplications and additions are utilized since these coefficients are expressed as homogeneous polynomials of the system parameters closed form approximations to the true solutions may be obtained also because this method is not based on expansion in terms of a small parameter it can successfully be applied to periodic systems whose internal excitation is strong two formulations are proposed the first is applicable to general time periodic systems while the second approach is useful when the system equations contain a constant matrix three different example problems including a double inverted pendulum subjected to a periodic follower force are included and cpu time and convergence results are discussed

adaptive finite element procedures for linear and non linear dynamics this paper discusses implementation and adaptivity of the discontinuous galerkin dg finite element method as applied to linear and non linear structural dynamic problems by the dg method both displacements and velocities are approximated as piecewise bilinear functions in space and time and may be discontinuous at the discrete time levels both implicit and explicit iterative algorithms for solving the resulted system of coupled equations are derived they are third order accurate and while the implicit procedure is unconditionally stable the explicit one is conditionally stable an h adaptive procedure based on the zienkiewicz zhu error estimate using the spr technique is applied numerical examples are presented to show the suitability of the dg method for both linear and non linear structural dynamic analysis copyright 1999 john wiley sons ltd

stationary distribution and extinction of a stochastic ratio dependent predator prey system with stage structure for the predator abstract in this paper we deal with a stochastic predator prey model with stage structure for predator population and ratio dependent functional response the proposed mathematical model consists of a system of three stochastic differential equations to stimulate the interactions between prey population immature predator and mature predator population we first establish sufficient conditions for the existence and uniqueness of the positive solutions by constructing an appropriate lyapunov function then we extend the existence of stationary distribution under certain parametric restrictions we also obtain the sufficient conditions for extinction of the predator populations finally numerical simulations have been carried out to validate our analytical findings

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

a time integration algorithm for linear transient analysis based on the reproducing kernel method in this paper a new algorithm based on the reproducing kernel method is presented for linear transient analysis using very large time steps in the conventional time integration methods the required accuracy is always achieved by either decreasing the size of time steps or increasing the order of approximations the proposed method however does not require a restriction on the size of the time step the required approximating functions for integration of governing equations are determined using the concept of reproduction of the solution within each time step a recurrence relationship is obtained through solving the governing equation by the point collocation based reproducing kernel method within the time step in collocation points the values of external excitation can be obtained either through the approximation between the values at the beginning and the end of each time step or assigning the exact values of the external force hence one advantage of this new method is the independency of the sampling rates of external excitation from the size of the time step the proposed method is applied for solving first and second order differential equations and advantages of the method are illustrated through a number of numerical examples

unconditionally stable higher order newmark methods by sub stepping procedure abstract in this paper a procedure to construct unconditionally stable higher order step by step time integration algorithms based on the newmark method is presented the second order accurate newmark method is extended to third fourth and even higher order by linearly combining the numerical results evaluated at some specific sub step locations the weighting factors and the sub step locations are algorithmic parameters they are chosen to eliminate the leading truncation error terms and to control the stability characteristics the ranges of parameters for unconditionally stable third and fourth order algorithms are given besides by using complex time steps a third order l stable and a fourth order a stable non dissipative algorithms are derived furthermore the accuracy of the excitation responses is shown to be improved by the present procedure as well to maintain high order accuracy the excitation may need to be modified the computational procedure is discussed it is found that there are no changes in the implementation of the underlying newmark method and no extra computations of high order gradients as the numerical results at the sub step locations are obtained independent of one another the evaluations can be computed in parallel comparisons with other higher order algorithms are presented

a unified set of single step algorithms part 1 general formulation and applications a general algorithm for a single step time marching scheme for use in dynamic or diffusion equations is presented this algorithm is easily programmed in its universal form for all orders of approximation and covers most of the currently used schemes as well as presenting many new possibilities in many cases it presents a computationally advantageous form over conventional procedures this is particularly so when compared with the newmark algorighm and its variants

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

computational algorithms for transient analysis the burden of weight and consequences towards formalizing discrete numerically assigned dna algorithmic markers wp family abstract different from the way we have been looking in the past at developments encompassing modal type and a wide class of step by step time stepping approaches for transient and dynamic field problems and significantly different from the way these have been developed and described in standard text books over the years this paper outlines the theoretical ideas and basis towards providing a generalized methodology and formulations leading to integral and a wide class of integration operators for the resulting solution of these transient equation systems it is herein postulated that integral operators and a wide class of integration operators pertain to and emanate from the same family with the burden which is being carried by a virtual field or weighted function field specifically introduced for the time discretization is strictly enacted in a mathematically consistent manner so as to first permit obtaining the adjoint operator of the original semi discretized equation system subsequently the selection or burden carried by the virtual or weighted function fields originally introduced to facilitate the time discretization process determines the formal development and outcome of exact integral operators approximate integral operators and a wide class of integration operators thus the present developments not only serve as a prelude towards the formal developments for exact integral operators but also demonstrate that the resulting approximate integral operators and a wide class of integration operators and known methods are simply subsets of the generalizations of a standardized w p family and emanate from the principles presented herein the burden of weight and the resulting consequences towards subsequently enabling a formal basis for the selection of discrete numerically assigned dna algorimmic markers to particularly identitfy a wide class of algorithms is described for transient field problems and dynamic field problems respectively

optimal periodic gain fractional delayed state feedback control for linear fractional periodic time delayed systems this paper develops the fundamentals of optimal tuning periodic gain fractional delayed state feedback control for a class of linear fractional order periodic time delayed systems although there exist techniques for the state feedback control of linear periodic time delayed systems by discretization of the monodromy operator there is no systematic method to design state feedback control for linear fractional periodic time delayed fptd systems this paper is devoted to defining and approximating the monodromy operator for a steady state solution of fptd systems it is shown that the monodromy operator cannot be achieved in a closed form for fptd systems and hence the short memory principle along with the fractional chebyshev collocation method is used to approximate the monodromy operator the proposed method guarantees a near optimal solution for fptd systems with fractional orders close to unity the proposed technique is illustrated in examples specifically in finding optimal linear periodic gain fractional delayed state feedback control laws for the fractional damped mathieu equation and a double inverted pendulum subjected to a periodic retarded follower force with fractional dampers in which it is demonstrated that the use of time periodic control gains in the fractional feedback control generally leads to a faster response

energy decaying scheme for non linear beam models abstract this paper is concerned with the time integration of the equations of motion describing the non linear dynamic response of beams desirable characteristics of integration schemes for structural dynamics problems include unconditional stability accuracy and high frequency numerical dissipation several schemes exist that present all these features when applied to linear problems though the application of those schemes to non linear problems is often successful proofs of unconditional stability are rarely derived a finite difference integration scheme is derived in this paper for the non linear dynamic response of beams though of a finite difference nature the proposed scheme mimics the integration scheme obtained by applying the time discontinuous galerkin methodology to a single degree of freedom linear oscillator a formal proof of unconditional stability for the non linear problem is derived based on an energy decay argument numerical examples using the proposed scheme are given and the results are compared with the predictions of other available schemes the accuracy of the proposed scheme and its high frequency numerical dissipation characteristics are demonstrated in these examples

numerical investigation of constrained direct solutions using hamilton s law this paper presents numerical results obtained for a dynamic system that is modeled using hamilton s law of varying action and for which geometric constraints are enforced explicitly these results are compared with numerical results for the same system that were obtained from ordinary differential equations differential algebraic equations and a formulation based on hamilton s law in which the constraints were enforced implicitly this comparison had two objectives first to show that numerical results from the formulation based on hamilton s law and explicit constraints were equal or superior to results from other formulations second to verify that explicit constraints must be enforced at the end of the time step both objectives were successfully accomplished

space time topology optimization for one dimensional wave propagation abstract a space time extension of the topology optimization method is presented the formulation with design variables in both the spatial and temporal domains is used to create structures with an optimized distribution of material properties that can vary in time the method is outlined for one dimensional transient wave propagation in an elastic rod with time dependent young s modulus by two simulation examples it is demonstrated how dynamic structures can display rich dynamic behavior such as wavenumber frequency shifts and lack of energy conservation the optimization method s potential for creating structures with novel dynamic behavior is illustrated by a simple example it is shown that an elastic rod in which the optimized stiffness distribution is allowed to vary in time can be much more efficient in prohibiting wave propagation compared to a static bandgap structure optimized designs in form of spatio temporal laminates and checkerboards are generated and discussed the example lays the foundation for creating designs with more advanced functionalities in future work

extended framework of hamilton s principle for continuum dynamics abstract hamilton s principle is the variational principle for dynamical systems and it has been widely used in mathematical physics and engineering however it has a critical weakness termed end point constraints which means that in the weak form we cannot use the given initial conditions properly by utilizing a mixed formulation and sequentially assigning initial conditions this paper presents a novel extended framework of hamilton s principle for continuum dynamics to resolve such weakness the primary applications lie in an elastic and a j 2 viscoplastic continuum dynamics the framework is simple and initiates the development of a space time finite element method with the proper use of initial conditions non iterative numerical algorithms for both elasticity and j 2 viscoplasticity are presented

weighting parameters for time step integration algorithms with predetermined coefficients in this paper the effect of using the predetermined coefficients in constructing time step integration algorithms is investigated both first and second order equations are considered the approximate solution is assumed to be in a form of polynomial in the time domain it can be related to the truncated taylor s series expansion of the exact solution therefore some of the coefficients can be predetermined from the known initial conditions if there are m predetermined coefficients and r unknown coefficients in the approximate solution the unknowns can be solved by the weighted residual method the weighting parameter method is used to investigate the resultant algorithm characteristics it is shown that the formulation is consistent with a minimum order of accuracy m r the maximum order of accuracy achievable is m 2r unconditionally stable algorithms exist if m r for first order equations and m 1 r for second order equations hence the dahlquist s theorem is generalized algorithms equivalent to the pade approximations and unconditionally stable algorithms equivalent to the generalized pade approximations are constructed the corresponding weighting parameters and weighting functions for the pade and generalized pade approximations are given explicitly copyright 2000 john wiley sons ltd

asymptotic behaviour of the stochastic lotka volterra model this paper examines the asymptotic behaviour of the stochastic extension of a fundamentally important population process namely the lotka volterra model the stochastic version of this process appears to have far more intriguing properties than its deterministic counterpart indeed the fact that a potential deterministic population explosion can be prevented by the presence of even a tiny amount of environmental noise shows the high level of difference which exists between these two representations

discontinuous galerkin finite element in time for solving periodic differential equations the dynamics of rotating systems can be modeled by ordinary differential equations with periodic time varying coefficients and forcing discontinuous galerkin dg methods which are known to be higher order accurate a stable and locally conservative are used to solve such equations in this work dg methods are shown to find the steady state periodic solution of a constant coefficient structural dynamics problem with periodic forcing and b helicopter rotor rigid blade flapping equation which is a parametric ordinary differential equation with periodic time varying coefficients and periodic forcing convergence of the numerical solution obtained is studied for both the above equations while h and p are refined separately it is found that the dg method works well for these problems and allows the development of an element level error estimate which can be used to monitor convergence and hence develop adaptive schemes it is clear from the results that even though higher order polynomials result in much faster convergence with reduced computational effort it also results in deterioration of the numerical performance for p 12 two adaptive strategies based on the element level displacement jump norm i e h and hp refinement are proposed and demonstrated on problems with higher harmonic forcing primary controls and parametric variations

solution of linear elastodynamics problems with space time finite elements on structured and unstructured meshes abstract a new implicit approach for the solution of linear elastodynamics problems on structured and unstructured meshes with space time finite elements is suggested new weak formulations are based on time continuous tcg and discontinuous tdg galerkin methods it was shown that for structured meshes the new tcg method has a higher order of accuracy by a factor of two than that of the standard tcg method at the same number of degrees of freedom the accuracy of the new tcg method is one order higher than the accuracy of the standard tdg method the introduction of new weighting functions for the weak formulations of the tcg and tdg methods allows control of the high frequency behavior without degradation of accuracy of the methods a variable number of elements in the time direction as well as a variable order of polynomial time approximations of finite elements for time slabs allows the implementation of a coupled space time adaptive procedure using structured or unstructured meshes numerical examples for 1 d problems demonstrate the effectiveness of the technique

stationary distribution and periodic solutions for stochastic holling leslie predator prey systems abstract the stochastic autonomous and periodic predator prey systems with holling and leslie type functional response are investigated for the autonomous system we prove that there exists a unique stationary distribution which is ergodic by constructing a suitable lyapunov function under relatively small white noise the result shows that stationary distribution doesn t rely on the existence and the stability of the positive equilibrium in the undisturbed system furthermore for the corresponding non autonomous system we show that there exists a positive periodic markov process under relatively weaker condition finally numerical simulations illustrate our theoretical results

a space time discontinuous galerkin method for the solution of the wave equation in the time domain in recent years the focus of research in the field of computational acoustics has shifted to the medium frequency regime and multiscale wave propagation this has led to the development of new concepts including the discontinuous enrichment method its basic principle is the incorporation of features of the governing partial differential equation in the approximation in this contribution this concept is adapted for the simulation of transient problems governed by the wave equation we present a space time discontinuous galerkin method with lagrange multipliers where the shape approximation in space and time is based on solutions of the homogeneous wave equation the use of hierarchical wave like basis functions is enabled by means of a variational formulation that allows for discontinuities in both the spatial and the temporal discretizations numerical examples in one space dimension demonstrate the outstanding performance of the proposed method compared with conventional space time finite element methods copyright 2008 john wiley sons ltd

a particular integral bem time discontinuous fem methodology for solving 2 d elastodynamic problems abstract this study proposes a time discontinuous galerkin finite element method fem for solving second order ordinary differential equations in the time domain the equations are formulated using a particular integral boundary element method bem in the space domain for elastodynamic problems the particular integral bem technique depends only on elastostatic displacement and traction fundamental solutions without resorting to commonly used complex fundamental solutions for elastodynamic problems based on the time discontinuous galerkin fem the unknown displacements and velocities are approximated as piecewise linear functions in the time domain and are permitted to be discontinuous at the discrete time levels this leads to stable and third order accurate solution algorithms for ordinary differential equations numerical results using the time discontinuous galerkin fem are compared with results using a conventional finite difference method the houbolt method both methods are employed for a particular integral bem analysis in elastodynamics this comparison reveals that the time discontinuous galerkin fem is more stable and more accurate than the traditional finite difference methods

space time proper generalized decompositions for the resolution of transient elastodynamic models in this paper we investigate ability of proper generalized decomposition pgd to solve transient elastodynamic models in space time domain classical methods use time integration schemes and an incremental resolution process we propose here to use standard time integration methods in a non incremental strategy as a result pgd gives a separated representation of the space time solution as a sum of tensorial products of space and time vectors that we interpret as space time modes recent time integration schemes are based on multi field formulations in this case separated representation can be constructed using state vectors in space and same vectors in time however we have experienced bad convergence order using this decomposition furthermore temporal approximation must be the same for all fields thus we propose an extension of classical separated representation for multi field problems this multi field pgd mf pgd uses space and time vectors that are different for each field calculation of decomposition is done using a monolithic approach in space and time potentially allowing the use of different approximations in space and time finally several simulations are performed with the transient elastodynamic problem with one dimension in space different approximations in time are investigated newmark scheme single field time discontinuous galerkin method and two fields time continuous and discontinuous galerkin methods

formulation and analysis of variational methods for time integration of linear elastodynamics abstract a general framework for variational formulations of linear elastodynamics is introduced variational formulations which originate from the principle of minimum potential energy the hellinger reissner principle and the hu washizu principle are considered they are characterized by the weak enforcement of the velocity displacement relation and of the initial displacement and velocity special cases are obtained by a priori enforcement of these conditions the time integration algorithms obtained discretizing in the space and time domains are studied with reference to accuracy stability and high frequency behaviour it is shown that some of these algorithms are very competitive with respect to classical finite difference methods the only drawback of variational methods seems to be the large size of the set of simultaneous equations which they generate effective solution procedures aiming at minimizing both the storage and the computational effort will be discussed in a separate paper

explicit time domain approaches based on numerical green s functions computed by finite differences the exga family the present paper describes a new family of time stepping methods to integrate dynamic equations of motion the scalar wave equation is considered here however the method can be applied to time domain analyses of other hyperbolic e g elastodynamics or parabolic e g transient diffusion problems the algorithms presented require the knowledge of the green s function of mechanical systems in nodal coordinates the finite difference method is used here to compute numerically the problem green s function however any other numerical method can be employed e g finite elements finite volumes etc the green s matrix and its time derivative are computed explicitly through the range 0 dt with either the fourth order runge kutta algorithm or the central difference scheme in order to improve the stability of the algorithm based on central differences an additional matrix called step response is also calculated the new methods become more stable and accurate when a sub stepping procedure is adopted to obtain the green s and step response matrices and their time derivatives at the end of the time step three numerical examples are presented to illustrate the high precision of the present approach

space time finite element methods for the sensitivity analysis of contact impact response of axisymmetric composite structures abstract space time finite element methods are applied to the sensitivity analysis of frictional contact impact response of axisymmetric composite structures the structures are assumed to consist of an arbitrary number of perfectly bonded homogeneous anisotropic layers only small displacements are considered and the material of each layer is assumed to be hyperelastic the sensitivity coefficients measure the sensitivity of the response to variations in material parameters of the structure a displacement finite element model is used for the spatial discretization the temporal integration is performed by using the time discontinuous galerkin method least squares stabilizing operators are added to the governing equations to enhance the stability by smoothing out the high frequency modes without degrading the accuracy a quasi explicit iterative technique is used for generating the response and evaluating the sensitivity coefficients the normal contact conditions are incorporated within the iterative process numerical results are presented for the sensitivity analysis of contact impact response of a composite spherical cap impacting a rigid plate

a weak weak formulation for large displacements beam statics a finite volumes approximation this paper addresses the problem of the numerical solution of beam statics undergoing large displacements a kinematic analysis outlines the beam geometrical model through the definition of its lagrangian co ordinate and strain parameters a definition of the stress parameters a constitutive law and an expression for the strain energy of the beam are then provided under the hypothesis of small strain the equations governing the beam equilibrium are introduced and their weak form is derived these equations are then proved to be equivalent to the primal and mixed form of principle of virtual work the numerical approximation is introduced by applying the bidiscontinuous finite elements method on the linearized weak form the weak weak formulation is attained by using the lowest interpolation order both for test and trial functions on two staggered decompositions of the space domain some numerical examples prove the capability of present formulation in handling actual problems

asymptotic properties of hybrid diffusion systems in response to the increasing needs for control and optimization of hybrid systems this work is concerned with such asymptotic properties as recurrence also known as weak stochastic stability in the literature and ergodicity of regime switching diffusions using liapunov functions necessary and sufficient conditions for positive recurrence are developed then ergodicity of positive recurrent regime switching diffusions is obtained by constructing cycles using the associated discrete time markov chains

a monolithic approach to fluid structure interaction using space time finite elements abstract the paper presents a simultaneous solution procedure for fluid structure interaction problems the structural motion is described by geometrically nonlinear elastodynamics the fluid is modeled by the incompressible navier stokes equations the space time finite element method is applied to both continua leading to an almost uniform discretization in which velocity variables are used for fluid and structure in order to enforce momentum conservation and geometric continuity at the interface a weighted residual formulation of coupling conditions is introduced the discretized model equations for fluid structure and coupling conditions are formulated in a single equation system the nonlinear system and the fluid mesh movement are solved in a single iteration loop two dimensional examples of membrane flutter and vortex excited plate vibrations demonstrate the efficiency of the methodology

a generalization of brauer s theorem on splitting fields to semigroups abstract let f be either an algebraic number field or a p adic field and a a central simple algebra over f suppose a is spanned by a multiplicative semigroup a with the property that the minimal polynomial of every g splits over f then a represents the trivial class in the brauer group of f

an a priori error estimate for a temporally discontinuous galerkin space time finite element method for linear elasto and visco dynamics the work of shaw was supported in part by the engineering and physical sciences research council ep h011072 1 this support is gratefully acknowledged

complex time step newmark methods with controllable numerical dissipation in this paper unconditionally stable higher order accurate time step integration algorithms with controllable numerical dissipation are presented the algorithms are based on the newmark method with complex time steps the ultimate spectral radius the sub step locations j and the weighting factors j are the algorithmic parameters for an algorithm that is 2n 1 th order accurate the sub step locations which may be complex are shown to be the roots of an nth degree polynomial the polynomial is given explicitly in terms of n and the weighting factors are then obtained by solving a system of n simultaneous equations it is further shown that the order of accuracy is increased by one for the non dissipative algorithms with 1 the stability properties of the present algorithms are studied it is shown that if the ultimate spectral radius is set between 1 and 1 the eigenvalues of the numerical amplification matrix are complex with magnitude less than or equal to unity the algorithms are therefore unconditionally c stable when the ultimate spectral radius is set to 0 or 1 the algorithms are found to be equivalent to the first sub diagonal and diagonal pade approximations respectively the present algorithms are more general as the numerical dissipation is controllable and are very suitable for parallel computers the accuracy of the excitation responses is found to be enhanced by the present complex time step procedure to maintain high order accuracy the excitation may need some modifications 1998 john wiley sons ltd

a novel family of explicit time marching techniques for structural dynamics and wave propagation models abstract in this work a novel family of explicit time marching techniques is presented for hyperbolic models the main features of the proposed procedure are i the family is based only on single step displacement velocity relations being very simple to implement ii as usual in explicit analyses it requires no system of equations to be dealt with being very efficient iii it is truly self starting iv it is second order accurate v it enables controllable algorithmic dissipation in the higher modes vi it provides reduced period elongation errors in the lower modes vii it is conditionally stable and exhibits high stability limits with its critical sampling frequency varying between 3 571 and 4 i e its upper limit is twice that of the central difference method the proposed technique is very effective providing the main positive attributes that are requested from an efficacious explicit time marching procedure numerical results are presented along the paper illustrating the excellent performance of the novel method

collocation methods with controllable dissipation for linear dynamics abstract this paper introduces a unified set of collocation methods for linear dynamics in which a parameter is used to control the position of the collocation points in this manner both a stable algorithms of order 2 p and l stable algorithms of order 2 p 1 are derived from collocation polynomials of degree p in addition algorithms with intermediate accuracy and stability properties are made available the effects of varying the algorithmic parameter are investigated with particular reference to the numerical dissipation of spurious high frequency modes numerical tests are reported which support the theoretical analysis and demonstrate the performance of the proposed algorithms

on the convergence of shock capturing streamline diffusion finite element methods for hyperbolic conservation laws we extend our previous analysis of streamline diffusion finite element methods for hyperbolic systems of conservation laws to include a shock capturing term adding artificial viscosity depending on the local absolute value of the residual of the finite element solution and the meh size with this term present we prove a maximum norm bound for finite element solutionsof burgers equation an thus complete an earlier convergence proof for this equation we further prove using entropy variables that a strong limit of finite element solutions is a weak solution of the system of conservation laws and satisfies the entropy inequality asociated with the entropy variables results of some numerical experiments for the time dependent compressible euler equations in two dimensions are also reported

a unified set of single step asymptotic annihilation algorithms for structural dynamics when solving the equations of structural dynamics using direct time integration meth6ds algorithmic damping is useful in controlling spurious high frequency oscillations ideally an algorithm should possess asymptotic annihilation of the highfrequency response i e spurious oscillations in the high frequencies are eliminated after one time step in this paper a new class of asymptotic annihilation algorithms for structural dynamics is presented that possesses little numerical dissipation in the low frequency regime the algorithms are based upon using finite elements in time the displacement and velocity fields are interpolated independently using time discontinuous functions the equations of motion displacement velocity compatibility and the time continuity of displacement and velocity are satisfied weakly asymptotic annihilation is achieved by choosing the displacement and velocity fields to be of equal order algorithms of any desired order of temporal accuracy can be obtained by appropriate choice of the finite element interpolations in time an analysis of the proposed class of algorithms is given proving the asymptotic annihilation property and the spectral equivalence of the algorithms to the upper diagonal of the pad6 approximation table results from finite difference analyses are presented showing the spectral behavior of the algorithms as well as their dissipation and dispersion properties in the low frequency regime

time finite element methods for structural dynamics summary time finite element methods are developed for the equations of structural dynamics the approach employs the time discontinuous galerkin method and incorporates stabilizing terms having least squares form these enable a general convergence theorem to be proved in a norm stronger than the energy norm results are presented from finite difference analyses of the time discontinuous galerkin and least squares methods with various temporal interpolations and commonly used finite difference methods for structural dynamics these results show that for particular interpolations the time finite element method exhibits improved accuracy and stability

a comprehensive unified set of single step algorithms with controllable dissipation for dynamics part i formulation a new class of time integration algorithms for dynamics is presented the algorithms are based on finite element in time the equations of motion are expressed in first order form and the variables are interpolated via time polynomials the unknown nodal variables are solved by the generalized galerkin method algorithms of any desired order of time accuracy can be achieved by proper choice of the finite element interpolation in time the salient feature of the algorithms is that it can be either asymptotic annihilating or non dissipative besides any desired algorithmic damping between these two extreme cases could be obtained the algorithms are analysed in detail the spectral characteristics equivalent to the upper diagonal diagonal of the pade approximations and other rational approximations are given