a comparative study of three boundary element approaches to calculate the transient response of viscoelastic solids with unbounded domains abstract as an alternative to domain discretization methods the boundary element method bem provides a powerful tool for the calculation of dynamic structural response in frequency and time domain field equations of motion and boundary conditions are cast into boundary integral equations bie which are discretized only on the boundary fundamental solutions are used as weighting functions in the bie which fulfil the sommerfeld radiation condition i e the energy radiation into a surrounding medium is modelled correctly therefore infinite and semi infinite domains can be effectively treated by the method the soil represents such a semi infinite domain in soil structure interaction problems the response to vibratory loads superimposed to static pre loads can often be calculated by linear viscoelastic constitutive equations conventional viscoelastic constitutive equations can be generalized by taking fractional order time derivatives into account in the present paper two time domain bem approaches including generalized viscoelastic behaviour are compared with the laplace domain bem approach and subsequent numerical inverse transformation one of the presented time domain approaches uses an analytical integration of the elastodynamic bie in a time step viscoelastic constitutive properties are introduced after laplace transformation by means of an elastic viscoelastic correspondence principle the transient response is obtained by inverse transformation in each time step the other time domain approach is based on the so called convolution quadrature method in this formulation the convolution integral in the bie is numerically approximated by a quadrature formula whose weights are determined by the same laplace transformed fundamental solutions used in the first method and a linear multistep method a numerical study of wave propagation problems in 3 d viscoelastic continuum is performed for comparing the three bem formulations

a novel scheme for the solution of the time domain integral equations of electromagnetics a new method to numerically solve time domain integral equations pertinent to electromagnetic surface scattering phenomena is presented the method uses approximate prolate spheroidal wave functions and standard rao wilton glisson basis functions to effect the temporal and spatial discretization of the integral equations respectively because the temporal basis functions are noncausal an extrapolation scheme is used to construct a system of equations that can be solved by marching on in time numerical results show that the proposed method is stable and that its solutions converge exponentially fast with the time bandwidth product of the approximate prolate spheroidal wave functions to results from a frequency domain method of moments solver that uses spatial basis functions and integration rules identical to those in the time domain solver

backstepping with bounded feedbacks for time varying systems a family of time varying nonlinear systems is globally uniformly asymptotically stabilized by bounded feedbacks constructed through a new extension of the backstepping approach explicit expressions of control laws and lyapunov functions are given

a hierarchical fft algorithm hil fft for the fast analysis of transient electromagnetic scattering phenomena a fast algorithm for accelerating the time marching solution of time domain integral equations pertinent to the analysis of free space electromagnetic scattering from perfectly conducting platelike and uniformly meshed structures is presented the matrix vector multiplications required by the time marching scheme are accelerated by use of the fast fourier transform fft this acceleration is achieved in a multilevel fashion by hierarchically grouping sparse interactions to extract denser pieces that are efficiently evaluated by the fft the total computational cost and storage requirements of this algorithm scale as o n sub t n sub s log sup 2 n sub s and o n sup 1 5 respectively as opposed to o n sub t n sub s sup 2 and o n sub s sup 2 for classical time marching methods n sub s and n sub t denote the total number of spatial unknowns and time steps respectively simulation results demonstrate the accuracy and efficiency of the algorithm

an accurate scheme for the solution of the time domain integral equations of electromagnetics using higher order vector bases and bandlimited extrapolation despite the numerous advances made in increasing the computational efficiency of time domain integral equation tdie based solvers the stability and accuracy of tdie solvers remain problematic this paper introduces a new numerical method for the accurate solution of tdies for scattering from arbitrary perfectly conducting surfaces the work described in this paper uses the higher order divergence conforming basis functions of graglia et al for spatial discretization and bandlimited interpolation functions for the temporal discretization of the relevant integral equations since the basis functions used for the temporal representation are noncausal an extrapolation scheme is employed to recover the ability to solve the problem by marching on in time numerical results demonstrate that the proposed method is stable and that it exhibits superlinear convergence with regard to the spatial discretization and exponential convergence with respect to the temporal discretization

symplectic integration of constrained hamiltonian systems by composition methods recent work reported in the literature suggests that for the long term integration of hamiltonian dynamical systems one should use methods that preserve the symplectic structure of the flow in this paper the symplecticity of numerical integrators is investigated for constrained hamiltonian systems with holonomic constraints the following two results will be derived i it is shown that any first or second order symplectic integrator for unconstrained problems can be generalized to constrained systems such that the resulting scheme is symplectic and preserves the constraints based on this higher order methods can be derived by the same composition methods used for unconstrained problems ii leimkuhler and reich math comp 63 1994 pp 589 605 derived symplectic integrators based on dirac s reformulation of the constrained problem as an unconstrained hamiltonian system however although the unconstrained reformulation can be handled by direct application of any symplectic implicit runge kutta m

conservative integration of rigid body motion by quaternion parameters with implicit constraints summary an angular momentum and energy conserving time integration algorithm for rigid body rotation is formulated in terms of the quaternion parameters and the corresponding four component conjugate momentum vector via hamilton s equations the introduction of an extended mass matrix leads to a symmetric set of eight state space equations of motion the extra inertial parameter serves as a multiplier on the kinematic constraint and it is demonstrated that convergence characteristics are improved by selecting this parameter somewhat larger than the inertial moments external loads enter these equations via the set of momentum equations initially the normalization of the quaternion array is introduced via a lagrange multiplier however this lagrange multiplier can be expressed explicitly in terms of the gradient of the external load potential and elimination of the lagrange multiplier from the final format leaves only an explicit projection applied to the external load potential gradient an algorithm is developed by forming a finite increment of the hamiltonian this procedure identifies the proper selection of increments and mean values and leads to an algorithm with conservation of momentum and energy implementation conservation properties and accuracy of the algorithm are illustrated by examples with a flying box and a spinning top copyright 2012 john wiley sons ltd

explicit lie poisson integration and the euler equations we give a wide class of lie poisson systems for which explicit lie poisson integrators preserving all casimirs can be constructed the integrators are extremely simple and have widespread applicability examples are the free rigid body a moment truncation and a new fast algorithm for the sine bracket truncation of the 2d euler equations

multiple limit cycles in a continuous culture vessel with variable yield abstract determining the number of limit cycles for a continuous culture vessel system is always useful in analyzing the system we prove the conditions that guarantee there exist three limit cycles for the chemostat with variable yield that was first proposed by huang limit cycles in a continuous fermentation model j math chem 5 1990 287 296 and by pilyugin and waltman multiple limit cycles in the chemostat with variable yield math boisci 182 2003 151 166

yield trajectory tracking for hyperbolic age structured population systems abstract for population systems modeled by age structured hyperbolic partial differential equations pdes that are bilinear in the input and evolve with a positive valued infinite dimensional state global stabilization of constant yield set points was achieved in prior work seasonal demands in biotechnological production processes give rise to time varying yield references for the proposed control objective aiming at a global attractivity of desired yield trajectories multiple non standard features have to be considered a non local boundary condition a pde state restricted to the positive orthant of the function space and arbitrary restrictive but physically meaningful input constraints moreover we provide control lyapunov functionals ensuring an exponentially fast attraction of adequate reference trajectories to achieve this goal we make use of the relation between first order hyperbolic pdes and integral delay equations leading to a decoupling of the input dependent dynamics and the infinite dimensional internal one furthermore the dynamic control structure does not necessitate exact knowledge of the model parameters or online measurements of the age profile with a galerkin based numerical simulation scheme using the key ideas of the karhunen loeve decomposition we demonstrate the controller s performance

a finite difference delay modeling approach to the discretization of the time domain integral equations of electromagnetics a new method for solving the time domain integral equations of electromagnetic scattering from conductors is introduced this method called finite difference delay modeling appears to be completely stable and accurate when applied to arbitrary structures the temporal discretization used is based on finite differences specifically based on a mapping from the laplace domain to the z transform domain first and second order unconditionally stable methods are derived spatial convergence is achieved using the higher order divergence conforming vector bases of graglia et al low frequency instability problems are avoided with the loop tree decomposition approach numerical results will illustrate the accuracy and stability of the technique

asymptotic controllability implies feedback stabilization it is shown that every asymptotically controllable system can be globally stabilized by means of some discontinuous feedback law the stabilizing strategy is based on pointwise optimization of a smoothed version of a control lyapunov function iteratively sending trajectories into smaller and smaller neighborhoods of a desired equilibrium a major technical problem and one of the contributions of the present paper concerns the precise meaning of solution when using a discontinuous controller

simple and efficient integration of rigid rotations suitable for constraint solvers simple and efficient way of integrating rigid rotations is presented the algorithm is stable second order accurate and in its explicit version involves evaluation of only two exponential maps per time step the semi explicit version of the proposed scheme improves upon the long term stability while it retains the explicitness in the force evaluation the algebraic structure of both schemes makes them suitable forthe analysis of constrained multi body systems the explicit algorithm is specifically aimed at the analysis involving small incremental rotations where its modest computational cost becomes the major advantage the semi explicit scheme naturally broadens the scope of possible applications copyright 2009 john wiley sons ltd

explicit momentum conserving integrator for dynamics of rigid bodies approximating the midpoint lie algorithm we reformulate the midpoint lie algorithm which is implicit in the torque calculation to achieve explicitness in the torque evaluation this is effected by approximating the impulse imparted over the time step with discrete impulses delivered at either the beginning of the time step or at the end of the time step thus we obtain two related variants both of which are explicit in the torque calculation but only first order in the time step both variants are momentum conserving and both are symplectic consequently drawing on the properties of the composition of maps we introduce another algorithm that combines the two variants in a single time step the resulting algorithm is explicit momentum conserving symplectic and second order its accuracy is outstanding and consistently outperforms currently known implicit and explicit integrators

stabilization with relaxed controls cannot in general be stabilized using a continuous closed loop control u x even if each state separately can be driven asymptotically to the origin an example is analyzed in section 2 in this paper we examine the possibility of stabilizing such systems with a continuous closed loop relaxed control we find indeed that the family of systems stabilizable with relaxed controls is larger than the family of those stabilizable with ordinary controls an even larger class is obtained if the continuity of the closed loop at the origin is not required the latter class includes all one dimensional systems for which states can be driven asymptotically to the origin this result does not hold in two dimensional systems and we provide a counter example it should be pointed that relaxed control type stabilization is used both in theory and in practice the method is known as dither we shall comment on the similarities lyapunov functions for the system 1 help us in the construction of the continuous closed loop stabilizers in fact we find that the existence of a smooth lyapunov function is equivalent to the existence of a stabilizing closed loop which is continuous except possibly at the origin an additional condition on the lyapunov function implies the continuity at the origin as well we present these results in section 4 after a brief introduction of closed loop relaxed controls notations and terminology in section 3 prior to that in section 2 we discuss an example illustrating the power of relaxed controls in the particular case of systems linear in the controls relaxed controls can be replaced by ordinary controls this is discussed in section 5 the role of lyapunov functions in the stability and stabilization theories is of course well known examples of systems with lyapunov functions are available in the literature we display some in section 6 along with general comments on the construction applications and counterexamples including one which cannot be continuously stabilized yet possesses a nonsmooth lyapunov function closed loop stabilization with ordinary controls is analyzed extensively in the literature see sontag 8 sussmann ll and references therein lyapunov functions techniques in stabil

backstepping with bounded feedbacks an extension of the backstepping approach is proposed it allows to globally asymptotically stabilize by bounded feedbacks families of nonlinear control systems explicit expressions of control laws and lyapunov functions are given

fast numerical solution of weakly singular volterra integral equations abstract we give a fortran program for the numerical solution of abel volterra convolution equations of the second and first kind we use a fractional linear multistep method bdf4 1 2 and use fast fourier transform techniques to exploit the convolution structure a detailed description of the computation of the quadrature weights using fft is given numerical examples are included

a general result on the stabilization of linear systems using bounded controls we present two constructions of controllers that globally stabilize linear systems subject to control saturation we allow essentially arbitrary saturation functions the only conditions imposed on the system are the obvious necessary ones namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold one of the constructions is in terms of a neural network type one hidden layer architecture while the other one is in terms of cascades of linear maps and saturations

a fast time domain integral equation based scheme for analyzing scattering from dispersive objects a fast integral equation based scheme for analyzing transient scattering from inhomogeneous dispersive bodies is presented a time domain integral equation in terms of the electric flux inside the body is constructed by invoking the volume equivalence principle i e by equating the sum of the incident electric field and that radiated by equivalent volume polarization currents to the total electric field the proposed algorithm for solving this time domain integral equation incorporates a recursive convolution scheme that updates the polarization current from knowledge of the electric flux to reduce the computational cost and memory requirement of the resulting marching on in time scheme it is augmented with the plane wave time domain algorithm numerical results that validate the proposed approach and demonstrate its accuracy are presented

fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm the computational complexity of classical marching on in time mot methods for solving time domain integral equations tdies pertinent to the analysis of transient scattering phenomena involving perfectly conducting targets grows as o n sub t n sub s sup 2 n sub t and n sub s denote the number of temporal and spatial degrees of freedom dof of the electric current on the target this scaling law impedes the application of these schemes to the analysis of large scale scattering phenomena the recently developed plane wave time domain pwtd algorithm permits the rapid evaluation of transient wave fields generated by temporally bandlimited sources and hence the acceleration of marching on in time based tdie solvers previously we described a two level pwtd enhanced tdie solver for analyzing electromagnetic scattering from perfectly conducting targets the computational complexity of this algorithm scales as o n sub t n sub s sup 1 5 logn sub s here a multilevel pwtd scheme for rapidly evaluating electric fields due to temporally bandlimited electric current sources is described in addition a multilevel pwtd enhanced tdie solver for analyzing electromagnetic scattering from perfectly conducting scatterers using o n sub t n sub s log sup 2 n sub s cpu resources is outlined last the accuracy and cpu memory efficiency of this solver are demonstrated by analyzing transient scattering from electrically large bodies

analysis of transient electromagnetic scattering phenomena using a two level plane wave time domain algorithm a fast algorithm is presented for solving electric magnetic and combined field time domain integral equations pertinent to the analysis of surface scattering phenomena the proposed two level plane wave time domain pwtd algorithm permits a numerically rigorous reconstruction of transient near fields from their far field expansion and augments classical marching on in time mot based solvers the computational cost of analyzing surface scattering phenomena using pwtd enhanced mot schemes scales as o n sub i n sub s sup 2 3 log n sub s as opposed to spl oscr n sub t n sub s sup 2 for classical mot methods where n sub t and n sub s are the numbers of temporal and spatial basis functions discretizing the scatterer current numerical results that demonstrate the efficacy of the proposed solver in analyzing transient scattering from electrically large structures and that confirm the above complexity estimate are presented

a remark on the converging input converging state property suppose that an equilibrium is asymptotically stable when external inputs vanish then every bounded trajectory which corresponds to a control which approaches zero and which lies in the domain of attraction of the unforced system must also converge to the equilibrium this well known but hard to cite fact is proved and slightly generalized here

global asymptotic behavior of a chemostat model with discrete delays this paper studies the global asymptotic behavior of an exploitative competition model between n species in a chemostat the model incorporates discrete time delays to describe the delay in the conversion of nutrient consumed to viable biomass and hence includes delays simultaneously in variables of nutrient and species concentrations in the case where only two species are engaged in competition it is shown that competitive exclusion holds for any monotone growth response functions sufficient conditions are also obtained for the model to exhibit competitive exclusion in the n species case in regard to the delay effects on the qualitative outcome of competition it is demonstrated that when the delays are relatively small the predictions of the model are identical with the predictions given by corresponding models without time delays however including large delays in the model may alter the predicted outcome of competition the techniques used also work when different removal rates are permitted an

further results on stabilization of periodic trajectories for a chemostat with two species we discuss an important class of problems involving the tracking of prescribed trajectories in the chemostat model we provide new tracking results for chemostats with two species and one limiting substrate based on lyapunov function methods in particular we use a linear feedback control of the dilution rate and an appropriate time varying substrate input concentration to produce a locally exponentially stable oscillatory behavior this means that all trajectories for the nutrient and corresponding species concentrations in the closed loop chemostat that stay near the oscillatory reference trajectory are attracted to the reference trajectory exponentially fast we also obtain a globally stable oscillatory reference trajectory for the species concentrations using a nonlinear feedback control depending on the dilution rate and the substrate input concentration this guarantees that all trajectories for the closed loop chemostat dynamics are attracted to the reference trajectory finally we construct an explicit lyapunov function for the corresponding global error dynamics we demonstrate the efficacy of our method in a simulation

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

a numerical method for a partial integro differential equation a method is considered for the integration in time of a partial integro differential equation the discretization technique employed is patterned after an idea of ch lubich error bounds are derived for both smooth and nonsmooth initial data

runge kutta convolution quadrature methods for well posed equations with memory runge kutta based convolution quadrature methods for abstract well posed linear and homogeneous volterra equations non necessarily of sectorial type are developed a general representation of the numerical solution in terms of the continuous one is given the error and stability analysis is based on this representation which for the particular case of the backward euler method also shows that the numerical solution inherits some interesting qualitative properties such as positivity of the exact solution numerical illustrations are provided

on the quadrature error in operational quadrature methods for convolutions we analyze the quadrature error associated with operational quadrature methods for convolution equations the assumptions are that the convolution kernel is inl 1 and that its laplace transform is analytic and bounded in an obtuse sector of the complex plane under these circumstances the laplace transform has a slow variation property which admits a fourier analysis of the quadrature error we provide generall p error estimates assuming suitable smoothness conditions on the function under convolution

on the multistep time discretization of linear initial boundary value problems and their boundary integral equations convergence estimates in terms of the data are shown for multistep methods applied to non homogeneous linear initial boundary value problems similar error bounds are derived for a new class of time discrete and fully discrete approximation schemes for boundary integral equations of such problems e g for the single layer potential equation of the wave equation in both cases the results are obtained from convergence and stability estimates for operational quadrature approximations of convolutions these estimates which are also proved here depend on bounds of the laplace transform of the distributional convolution kernel outside the stability region scaled by the time stepsize and on the smoothness of the data

semi implicit reversible algorithms for rigid body rotational dynamics this paper presents two semi implicit algorithms based on splitting methodology for rigid body rotational dynamics the first algorithm is a variation of partitioned runge kutta prk methodology that can be formulated as a splitting method the second algorithm is akin to a multiple time stepping scheme and is based on modified crouch grossman mcg methodology which can also be expressed as a splitting algorithm these algorithms are second order accurate and time reversible however they are not poisson integrators i e non symplectic these algorithms conserve some of the first integrals of motion but some others are not conserved however the fluctuations in these invariants are bounded over exponentially long time intervals these algorithms exhibit excellent long term behavior because of their reversibility property and their approximate poisson structure preserving property the numerical results indicate that the proposed algorithms exhibit superior performance compared to some of the currently well known algorithms such as the simo wong algorithm newmark algorithm discrete moser veselov algorithm lewis simo algorithm and the liemid ea algorithm

numerical integration of lie poisson systems while preserving coadjoint orbits and energy in this paper we apply geometric integrators of the rkmk type to the problem of integrating lie poisson systems numerically by using the coadjoint action of the lie group g on the dual lie algebra mbox normalsize mathfrak g to advance the numerical flow we devise methods of arbitrary order that automatically stay on the coadjoint orbits first integrals known as casimirs are retained to machine accuracy by the numerical algorithm within the proposed class of methods we find integrators that also conserve the energy these schemes are implicit and of second order nonlinear iteration in the lie algebra and linear error growth of the global error are discussed numerical experiments with the rigid body and a finite dimensional truncation of the euler equations for a two dimensional 2d incompressible fluid are used to illustrate the properties of the algorithm

a universal construction of artstein s theorem on nonlinear stabilization abstract this note presents an explicit proof of the theorem due to artstein which states that the existence of a smooth control lyapunov function implies smooth stabilizability moreover the result is extended to the real analytic and rational cases as well the proof uses a universal formula given by an algebraic function of lie derivatives this formula originates in the solution of a simple riccati equation

higher order interpolatory vector bases for computational electromagnetics low order vector basis functions compatible with the nedelec 1980 representations are widely used for electromagnetic field problems higher order functions are receiving wider application but their development is hampered by the complex procedures used to generate them and lack of a consistent notation for both elements and bases in this paper fully interpolatory higher order vector basis functions of the nedelec type are defined in a unified and consistent manner for the most common element shapes it is shown that these functions can be obtained as the product of zeroth order nedelec representations and interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties of the vector functions are discussed and expressions for the vector functions of arbitrary polynomial order are presented sample numerical results confirm the faster convergence of the higher order functions

relaxed lyapunov criteria for robust global stabilisation of non linear systems the notion of the restricted robust control lyapunov function rclf is introduced and is exploited for the design of robust feedback stabilisers for non linear systems particularly it is shown for systems with input constraints that restricted rclfs can be easily obtained while rclfs are not available moreover it is shown that the use of restricted rclfs usually results in different feedback designs from the ones obtained by the use of the standard rclf methodology using the restricted rclfs feedback design methodology a simple controller that guarantees robust global stabilisation of a perturbed chemostat model is provided

implicit runge kutta methods for the discretization of time domain integral equations implicit runge kutta based schemes are proposed for solving the time domain integral equations of electromagnetic theory the proposed technique maps a laplace domain equation to a domain equation using the butcher tableau of the implicit runge kutta scheme a discrete time domain system is recovered by computing the inverse transform numerically the resulting technique is capable of third or fifth order accuracy in time and is stable independent of time step numerical results illustrate the accuracy and stability of the technique

symplectic integration of constrained hamiltonian systems a hamiltonian system in potential form h q p ptmp 2 f q subject to smooth constraints on q can be viewed as a hamiltonian system on a manifold but numerical computations must be performed in rn in this paper methods which reduce hamiltonian differential algebraic equations to odes in euclidean space are examined the authors study the construction of canonical parametrizations or local charts as well as methods based on the construction of ode systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an invariant manifold in each case a hamiltonian system of ordinary differential equations is produced the stability of the constraint invariants and the behavior of the original hamiltonian along solutions are investigated both numerically and analytically

backward error analysis for numerical integrators backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods this is true in particular for the integration of hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time such results are typically based on two aspects of backward error analysis i it can be shown that the modified vector fields have some qualitative properties which they share with the given problem and ii an estimate is given for the difference between the best interpolating vector field and the numerical method these aspects have been investigated recently for example by benettin and giorgilli in j statist phys 74 1994 pp 1117 1143 by hairer in ann numer math 1 1994 pp 107 132 and by hairer and lubich in numer math 76 1997 pp 441 462 in this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit taylor series expansion of the numerical method and the corresponding flow maps as in the above cited works as an application we discuss the long time integration of chaotic hamiltonian systems and the approximation of time averages along numerically computed trajectories

a numerical test of long time stability for rigid body integrators summary in the context of hamiltonian odes a necessary condition for an integrator to be symplectic or conjugate symplectic is that it nearly preserves the exact hamiltonian this paper introduces a numerical test of this necessity for rigid body methods it turns out that several rigid body integrators proposed in literature fail this test hence these integrators should be used with caution for long time simulation copyright 2011 john wiley sons ltd

an algorithm for approximate solutions to weakly filtered synchronous control systems and nonlinear renewal processes a multi time perturbation algorithm is derived to study certain synchronous control systems and nonlinear renewal processes the new methods presented here study singular perturbation problems for nonlinear volterra integro differential equations whose kernels are close to the dirac delta function the algorithm provides useful approximate solutions to weakly filtered phase locked loop circuits and to nonlinear renewal equations describing certain population dynamics

stabilization of a chemostat model with haldane growth functions and a delay in the measurements the stabilization of equilibria in chemostats with measurement delays is a complex and challenging problem and is of significant ongoing interest in bioengineering and population dynamics in this paper we solve an output feedback stabilization problem for chemostat models having two species one limiting substrate and either haldane or monod growth functions our feedback stabilizers depend on a given linear combination of the species concentrations which are both measured with a constant time delay the values of the delays are unknown instead one only knows an upper bound on the delays and we allow the upper bound to be arbitrarily large the stabilizing feedback depends on the known upper bound for the delays as well our work is based on a lyapunov krasovskii argument

time domain boundary integral equation modeling of heat transmission problems this paper investigates the numerical modeling of a time dependent heat transmission problem by the convolution quadrature boundary element method it introduces the latest theoretical development into the error analysis of the numerical scheme semigroup theory is applied to obtain stability in the spatially semidiscrete scheme functional calculus is employed to yield convergence in the fully discrete scheme we compare the results to a more traditional laplace domain approach

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells

a delayed chemostat model with general nonmonotone response functions and differential removal rates a chemostat model with general nonmonotone response functions is considered the nutrient conversion process involves time delay we show that under certain conditions when n species compete in the chemostat for a single resource that is allowed to be inhibitory at high concentrations the competitive exclusion principle holds in the case of insignificant death rates the result concerning the attractivity of the single species survival equilibrium already appears in the literature several times see h m el owaidy m ismail asymptotic behavior of the chemostat model with delayed response in growth chaos solitons fractals 13 2002 787 795 h m el owaidy a a moniem asymptotic behavior of a chemostat model with delayed response growth appl math comput 147 2004 147 161 s yuan m han z ma competition in the chemostat convergence of a model with delayed response in growth chaos solitons fractals 17 2003 659 667 however the proofs are all incorrect in this paper we provide a correct proof that also applies in the case of differential death rates in addition we provide a local stability analysis that includes sufficient conditions for the bistability of the single species survival equilibrium and the washout equilibrium thus showing the outcome can be initial condition dependent moreover we show that when the species specific death rates are included damped oscillations may occur even when there is no delay thus the species specific death rates might also account for the damped oscillations in transient behavior observed in experiments

bifurcation for a functional yield chemostat when one competitor produces a toxin abstract a model with general yield functions f i s i 1 2 of competition in the chemostat of two competitors for a single nutrient when one of the competitors produces toxin against its opponent is studied in this paper the conditions in terms of the relevant parameters for the hopf bifurcation of the three dimensional system have been proved which implies the existence of limit cycles in the 3 d system

convolution quadrature and discretized operational calculus ii operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations singular integrals power and logarithmic singularities finite part integrals multiple timescale convolution volterra integral equations wiener hopf integral equations frequency domain conditions which determine the stability of such equations can be carried over to the discretization

an accurate broad band method of moments using higher order basis functions and tree loop decomposition a method is presented that combines the low frequency accuracy of tree loop decomposition with higher order basis functions to achieve an electric field integral equation based method of moments that is accurate over a wide band of frequencies in general the use of higher order divergence conforming bases introduces more solenoidal degrees of freedom than the standard rao wilton glisson zeroth order bases these solenoidal degrees of freedom are counted and identified and an efficient method to construct the required higher order loops is presented numerical results demonstrate that the technique allows the use of a single mesh to acquire scattering results over a wide band of frequencies using arbitrary order basis functions

numerical solution of the helmholtz equation in 2d and 3d using a high order nystrom discretization we show how to solve time harmonic scattering problems by means of a high order nystrom discretization of the boundary integral equations of wave scattering in 2d and 3d the novel aspect of our new method is its use of local corrections to the discretized kernel in the vicinity of the kernel singularity enhanced by local corrections the new algorithm has the simplicity and speed advantages of the traditional nystrom method but also enjoys the advantages of high order convergence for controlling solution error we explain the practical details of implementing a scattering code based on a high order nystrom discretization and demonstrate by numerical example that a scattering code based on this algorithm can achieve high order convergence to the correct answer we also demonstrate its performance advantages over a high order galerkin code

robust solution of time domain integral equations using loop tree decomposition and bandlimited extrapolation a stabilization method that eradicates low frequency instabilities in the solution of the time domain integral equations of electromagnetic scattering is presented the method uses the loop tree decomposition originally suggested for the solution of low frequency scattering problems via the method of moments specifically a temporally differentiated form of the pertinent integral equation is tested using tree basis functions and the undifferentiated form is tested using solenoidal basis functions the underlying solution method uses bandlimited interpolation functions blifs and the higher order divergence conforming vector bases of graglia et al for the temporal and spatial discretization of the integral equations respectively an extrapolation technique has been implemented to overcome the noncausality introduced into the system by the blifs numerical results will demonstrate the stability and accuracy of the proposed technique

explicit newmark verlet algorithm for time integration of the rotational dynamics of rigid bodies we reformulate the traditional velocity based vector space newmark algorithm for the rotational dynamics of rigid bodies that is for the setting of the so 3 lie group we show that the most naive re write of the vector space algorithm possesses the properties of symplecticity and almost momentum conservation thus we obtain an explicit algorithm for rigid body dynamics that matches or exceeds performance of existing algorithms but which curiously does not seem to have been considered in the open literature so far copyright 2005 john wiley sons ltd

time domain bie analysis of large three dimensional electromagnetic scattering problems a time domain boundary integral equation bie solution of the magnetic field integral equation mfie for large electromagnetic scattering problems is presented it employs isoparametric curvilinear quadratic elements to model fields geometry and time dependence eliminating staircasing problems the approach is implicit which seems to provide both stability and permits arbitrary local mesh refinement to model geometrically difficult regions without the significant cost penalty explicit methods suffer error dependence on discretization is investigated accurate results are obtained with as few as five nodes per wavelength the performance both on large scatterers and on low radar cross section rcs scatterers is demonstrated including the six wavelength nasa almond two spheres a thirteen wavelength missile and a high q cavity

electromagnetic scattering from homogeneous dielectric bodies using time domain integral equations this paper presents a stable and accurate method to compute the electromagnetic scattering from homogeneous isotropic and nondispersive bodies using time domain integral equations tdies unlike previous tdie based scattering work the formulation presented here is based on the equations of poggio miller chang harrington wu and tsai formulation the method employs the higher order divergence conforming basis functions described by graglia et al and bandlimited interpolation functions to effect the spatial and temporal discretization of the integral equations respectively as the temporal basis functions are noncausal an extrapolation mechanism is used to modify the noncausal system of equations to a form solvable by standard marching on in time procedure this work also explains the reason for late time low frequency instabilities encountered in current tdie implementations and details a stabilization technique employed to overcome them numerical results demonstrate the accuracy and stability of the proposed technique

an energy approach to space time galerkin bem for wave propagation problems in this paper we consider dirichlet or neumann wave propagation problems reformulated in terms of boundary integral equations with retarded potential starting from a natural energy identity a space time weak formulation for 1d integral problems is briefly introduced and continuity and coerciveness properties of the related bilinear form are proved then a theoretical analysis of an extension of the introduced formulation for 2d problems is proposed pointing out the novelty with respect to existing literature results at last various numerical simulations will be presented and discussed showing unconditional stability of the space time galerkin boundary element method applied to the energetic weak problem copyright 2009 john wiley sons ltd

solutions to the discrete airy equation application to parabolic equation calculations in the case of the equidistant discretization of the airy differential equation discrete airy equation the exact solution can be found explicitly this fact is used to derive a discrete transparent boundary condition tbc for a schrodinger type equation with linear varying potential which can be used in parabolic equation simulations in underwater acoustics and for radar propagation in the troposphere we propose different strategies for the discrete tbc and show an efficient implementation finally a stability proof for the resulting scheme is given a numerical example in the application to underwater acoustics shows the superiority of the new discrete tbc

uniform error estimates of galerkin methods for monotone abel volterra integral equations on the half line we consider galerkin methods for monotone abel volterra integral equations of the second kind on the half line the l2 theory follows from kolodner s theory of monotone hammerstein equations we derive the loo theory from the l2 theory by relating the l2and loo spectra of operators of the form x b ax to one another here denotes convolution and b e l1 and a e loo as an extra condition we need b t o t a 1 with a 0 we also prove the discrete analogue in particular we verify that the galerkin matrix satisfies the discrete conditions

runge kutta convolution quadrature for the boundary element method abstract time domain boundary element formulations can be established either directly in time domain or via laplace or fourier domain somewhere in between are the convolution quadrature based boundary element formulations which utilize the laplace domain fundamental solution but establish a time stepping procedure up to now in applications mostly backward differential formulas of second order are used as the underlying multistep method however in recent mathematical literature also runge kutta methods have been applied here the use of runge kutta methods is explained in detail and some numerical studies are given in these studies the backward difference based procedures are compared to runge kutta methods for a non smooth problem an l 2 norm of the error is used as the basis of comparison the convergence of which is investigated theoretically as well the results confirm that the usage of the new techniques is preferable with regard to less numerical oscillations in the solution and better representation of wave fronts

an analysis of the modified l1 scheme for time fractional partial differential equations with nonsmooth data we introduce a modified l1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous c

nonsmooth data error estimates for approximations of an evolution equation with a positive type memory term we study the numerical approximation of an integro differential equation which is intermediate between the heat and wave equations the proposed discretization uses convolution quadrature based on the first and second order backward difference methods in time and piecewise linear finite elements in space optimal order error bounds in terms of the initial data and the inhomogeneity are shown for positive times without assumptions of spatial regularity of the data

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

geometric integration on spheres and some interesting applications geometric integration theory can be employed when numerically solving odes or pdes with constraints in this paper we present several one step algorithms of various orders for odes on a collection of spheres to demonstrate the versatility of these algorithms we present representative calculations for reduced free rigid body motion a conservative ode and a discretization of micromagnetics a dissipative pde we emphasize the role of isotropy in geometric integration and link numerical integration schemes to modern differential geometry through the use of partial connection forms this theoretical framework generalizes moving frames and connections on principal bundles to manifolds with nonfree actions

systems based decomposition schemes for the approximate solution of multi term fractional differential equations we give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi term fractional differential equations using the caputo form of the fractional derivative the schemes we compare are based on conversion of the original problem into a system of equations we review alternative approaches and consider how the most appropriate numerical scheme may be chosen to solve a particular equation

electromagnetic scattering from dispersive dielectric scatterers using the finite difference delay modeling method a new method for solving the time domain integral equations of electromagnetic scattering from dispersive dielectric bodies is proposed the method is based on the finite difference delay modeling scheme which accomplishes the temporal discretization by substituting a z domain finite difference approximation for the continuous time laplace domain s parameter the inverse z transform used for computing the temporal kernels is calculated numerically using the trapezoidal rule and inverse discrete fourier transform arbitrary dispersive models can be simulated without extraneous convolution using the new technique the normal field equations are augmented to the standard tangential equations to solve the low frequency instability problem

momentum conserving symplectic integrators abstract in this paper we show that symplectic partitioned runge kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of hamiltonian differential equations by extended point transformation we also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of lie poisson systems on certain submanifolds of the general matrix group gl n

mechanical integrators derived from a discrete variational principle many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion in this paper we present a procedure to construct time stepping algorithms that approximate the flow of continuous odes for mechanical systems by discretizing hamilton s principle rather than the equations of motion the discrete equations share similarities to the continuous equations by preserving invariants including the symplectic form and the momentum map we first present a formulation of discrete mechanics along with a discrete variational principle we then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of noether s theorem we then use the discrete mechanics formulation to develop a procedure for constructing symplectic momentum mechanical integrators for lagrangian systems with holonomic constraints we apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators

boundary integral solvers for an evolutionary exterior stokes problem this paper proposes and analyzes a full discretization of the exterior transient stokes problem with dirichlet boundary conditions the method is based on a single layer boundary integral representation using galerkin semidiscretization in the space variables and multistep convolution quadrature in time convergence estimates are based on a laplace domain analysis which translates into a detailed study of the exterior brinkman problem some numerical experiments are provided

fractional linear multistep methods for abel volterra integral equations of the second kind fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular volterra integral equations the proposed methods are convergent of the order of the underlying multistep method also in the generic case of solutions which are not smooth at the origin the stability properties stability region a stability a a stability are closely related to those of the underlying linear multistep method