strichartz estimates for the schrodinger propagator in wiener amalgam spaces abstract in this paper we study the strichartz estimates for the schrodinger propagator in the context of wiener amalgam spaces which unlike the lebesgue spaces control the local regularity of a function and its decay at infinity separately this separability makes it possible to perform a finer analysis of the local and global behavior of the propagator our results improve some of the classical ones in the case of large time

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

the optimal projection equations for reduced order state estimation the singular measurement noise case the optimal projection equations for reduced order state estimation are generalized to allow for singular i e colored measurement noise the noisy and noise free measurements serve as inputs to dynamic and static estimators respectively the optimal solution is characterized by necessary conditions which involve a pair of oblique projections corresponding to reduced estimator order and singular measurement noise intensity

on the asymptotic stability ofθ methods for delay differential equations stability regions of methods for the linear delay differential test equations begin gathered y t py t qy t tau t 0 hfill y t varphi t t in tau 0 hfill end gathered where is a positive constant are presented in the case thatp andq are real constant coefficients necessary and sufficient conditions on the stepsize for the stability of a method are obtained furthermore whenp andq are complex coefficients sufficient conditions for the stability of the methods are also given

global well posedness and scattering for the defocusing l 2 critical nonlinear schrodinger equation when d 2 in this article we prove that the defocusing cubic nonlinear schrodinger initial value problem is globally well posed and scattering for u0 l2 r2 the proof uses the bilinear estimates of planchon and vega and a frequency localized interaction morawetz estimate similar to the high frequency estimate of colliander keel staffilani takaoka and tao and especially the low frequency estimate of dodson

numerical solution of delay differential equations by uniform corrections to an implicit runge kutta method in this paper we develop a class of numerical methods to approximate the solutions of delay differential equations they are essentially based on a modified version in a predictor corrector mode of the one step collocation method atn gaussian points these methods applied to ode s provide a continuous approximate solution which is accurate of order 2n at the nodes and of ordern 1 uniformly in the whole interval in order to extend the methods to delay differential equations the uniform accuracy is raised to the order 2n by some a posteriori corrections numerical tests and comparisons with other methods are made on real life problems

on the asymptotic stability properties of runge kutta methods for delay differential equations in this paper asymptotic stability properties of runge kutta r k methods for delay differential equations ddes are considered with respect to the following test equation begin eqnarray left begin array ccl y t a y t b y t tau 0 3cm y t g t end array right hskip 0 3cm begin array r t 0 0 3cm tau leq t leq 0 end array end eqnarray where a b in bbb r tau 0 and g t is a continuous real valued function in the last few years stability properties of r k methods applied to ddes have been studied by numerous authors who have considered regions of asymptotic stability for any positive delay and thus independent of the specific value of tau

a fully discrete spectral method for delay differential equations in this paper a new lanczos tau method for solving linear functional differential equations is introduced the scheme has infinite order of accuracy both in time and in the delayed argument the high accuracy in time is obtained without increasing the computational work and memory space which is needed for a one step explicit difference scheme the article demonstrates how to implement the algorithm in a robust and efficient manner and to treat problems with piecewise continuous initial function numerical results illustrating the behavior of the method when faced with difficult problems are presented and the numerical results are compared to those obtained by using two other methods

stepsize control and continuity consistency for state dependent delay differential equations abstract derivative discontinuities occur frequently in the solutions of delay differential equations even when the functions defining them are all c unless correctly treated such discontinuities can undermine the continuity assumptions made by ode software in this paper we discuss some specific difficulties arising from this aspect in the treatment of state dependent problems a necessary property continuity consistency is introduced along with a class of methods for which it is satisfied

computer solutions of state dependent delay differential equations abstract an adaptation of the runge kutta verner 5 6 formula pair is used to construct a numerical method for the solution of state dependent delay differential equations with nonvanishing lag a fifth degree divided difference newton backward interpolation polynomial is used to find the location of the derivative jump discontinuities of the solution in order to maintain the sixth order accuracy of the rkv pair the value of the solution at the delay is approximated by a three point hermite polynomial this new method is tested on some real life problems a fortran program called sysdel is available from the authors

on the strichartz estimates for orthonormal systems of initial data with regularity abstract the classical strichartz estimates for the free schrodinger propagator have recently been substantially generalised to estimates of the form j j e i t f j 2 l p t l x q l for orthonormal systems f j j of initial data in l 2 firstly in work of frank lewin lieb seiringer and later by frank sabin the primary objective is identifying the largest possible as a function of p and q and in contrast to the classical case for such estimates the critical case turns out to be p q d 1 d d 1 d 1 we consider the case of orthonormal systems f j j in the homogeneous sobolev spaces h s for s 0 d 2 and we establish the sharp value of as a function of p q and s except possibly an endpoint in certain cases furthermore at the critical case p q d 1 d 2 s d d 1 d 1 d 2 s for general s we show the veracity of the desired estimates when p if we consider frequency localised estimates and the failure of the non localised estimates when p this exhibits the difficulty of upgrading from frequency localised estimates in this context again in contrast to the classical setting

numerical solution by lmms of stiff delay differential systems modelling an immune response we consider the application of linear multistep methods lmms for the numerical solution of initial value problem for stiff delay differential equations ddes with several constant delays which are used in mathematical modelling of immune response for the approximation of delayed variables the nordsieck s interpolation technique providing an interpolation procedure consistent with the underlying linear multistep formula is used an analysis of the convergence for a variable stepsize and structure of the asymptotic expansion of global error for a fixed stepsize is presented some absolute stability characteristics of the method are examined implementation details of the code difsub dde being a modification of the gear s difsub are given finally an efficiency of the code developed for solution of stiff ddes over a wide range of tolerances is illustrated on biomedical application model

numerical treatment of stochastic delay differential equations a global error bound abstract motivated by the error analysis of numerical solution for stochastic delay differential equations a new result indicating the dependency of convergence rate of the scheme on two separate quantities p and q where the former is the global order of underlying scheme and the latter is that of interpolation approximating the history of system is presented as an objective instance the stochastic runge kutta srk schemes of global order 0 5 1 and 1 5 in the strong convergence sense are utilized to formulate three one step schemes to approximate the solution of the problem moreover the mean square stability of these schemes will be asymptotically developed finally the theoretical results will be confirmed by some test problems

sharp strichartz estimates on nontrapping asymptotically conic manifolds we obtain the strichartz inequalities u l q t l r x 0 1 m c u 0 l 2 m for any smooth n dimensional riemannian manifold m which is asymptotically conic at infinity with either short range or long range metric perturbation and nontrapping where u is a solution to the schrodinger equation iu t 1 2 m u 0 and 2 q r are admissible strichartz exponents 2 q n r n 2 this corresponds with the estimates available for euclidean space except for the endpoint q r 2 2 n n 2 when n 2 these estimates imply existence theorems for semi linear schrodinger equations on m by adapting arguments from cazenave and weissler and kato this result improves on our previous result which was an l 4 t x strichartz estimate in three dimensions it is closely related to results of staffilani tataru burq robbiano zuily and tataru who consider the case of asymptotically flat manifolds

derivation of continuous explicit two step runge kutta methods of order three we describe a construction of continuous extensions to a new representation of two step runge kutta methods for ordinary differential equations this representation makes possible the accurate and reliable estimation of local discretization error facilitates the efficient implementation of these methods in variable stepsize environment and adapts readily to the numerical solution of a class of delay differential equations a number of numerical tests carried out on the obtained methods of order 3 with quadratic interpolants show their efficiency and robust performance which allow them to compete with the state of the art dde23 code from matlab

metaplectic representation on wiener amalgam spaces and applications to the schrodinger equation we study the action of metaplectic operators on wiener amalgam spaces giving upper bounds for their norms as an application we obtain new fixed time estimates in these spaces for schrodinger equations with general quadratic hamiltonians and strichartz estimates for the schrodinger equation with potentials v x x 2

improved inhomogeneous strichartz estimates for the schrodinger equation abstract we study inhomogeneous strichartz estimates for the schrodinger equation for dimension n 3 using a frequency localization we obtain some improved range of strichartz estimates for the solution of inhomogeneous schrodinger equation except dimension n 3

regularity decay and best constants for dispersive equations we show that the strichartz l2w l2 estimates for solutions to the pseudo differential equations xu i tuand x 1u i tu are equivalent a necessary and sufficient condition for decay and regularity for solutions to the equation x u i tu is given

remarks on endpoint strichartz estimates for schrodinger equations with the critical inverse square potential abstract the purpose of this paper is to study the validity of global in time strichartz estimates for the schrodinger equation on r n n 3 with the negative inverse square potential x 2 in the critical case n 2 2 4 it turns out that the situation is different from the subcritical case n 2 2 4 in which the full range of strichartz estimates is known to be hold more precisely splitting the solution into the radial and non radial parts we show that i the radial part satisfies a weak type endpoint estimate which can be regarded as an extension to higher dimensions of the endpoint strichartz estimate with radial data for the two dimensional free schrodinger equation ii other endpoint estimates in lorentz spaces for the radial part fail in general iii the non radial part satisfies the full range of strichartz estimates

qmc methods for the solution of delay differential equations in this paper the quasi monte carlo methods for runge kutta solution techniques of differential equations which were developed by stengle lecot coulibaly and koudiraty are extended to delay differential equations of the form y t f t y t y t t the retarded argument is approximated by interpolation after which the conventional quasi monte carlo runge kutta methods can be applied we give a proof of the convergence of this method and its order in a general form which does not depend on a specific quasi monte carlo runge kutta method finally a numerical investigation shows that similar to ordinary differential equations this quasi randomized method leads to an improvement for heavily oscillating delay differential equations compared even to high order runge kutta schemes

natural continuous extensions of runge kutta methods the present paper develops a theory of natural continuous extensions nces for the discrete approximate solution of an ode given by a runge kutta process these nces are defined in such a way that the continuous solutions furnished by the one step collocation methods are included

some remarks on strichartz estimates for homogeneous wave equation abstract we give several remarks on strichartz estimates for homogeneous wave equation with special attention to the cases of l x estimates radial solutions and initial data from the inhomogeneous sobolev spaces in particular we give the failure of the endpoint estimate l t 4 l x for n 2

continuous parallel block methods and their applications continuous numerical methods have many applications in the numerical solution of discontinuous ordinary differential equations delay differential equations delay differential algebraic equations neutral delay differential equations integro differential equations etc this paper deals with a continuous extension for the discrete approximate solution of ordinary differential equations generated by a class of parallel block methods existence and uniqueness for the continuous extension are discussed convergence and absolute stability of the continuous parallel block methods for ordinary differential equations are studied as applications we adopt the continuous parallel block methods to solve delay differential equations neutral delay differential equations and delay differential algebraic equations and obtain sufficient and necessary conditions for the continuous parallel block methods to be asymptotically stable several numerical experiments are given to illustrate the performance of the continuous parallel block methods

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

the optimal projection equations for model reduction and the relationships among the methods of wilson skelton and moore first order necessary conditions for quadratically optimal reduced order modeling of linear time invariant systems are derived in the form of a pair of modified lyapunov equations coupled by an oblique projection which determines the optimal reduced order model this form of the necessary conditions considerably simplifies previous results of wilson 1 and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of moore 2 the possible existence of multiple solutions of the optimal projection equations is demonstrated and a relaxation type algorithm is proposed for computing these local extrema a component cost analysis of the model error criterion similar to the approach of skelton 3 is utilized at each iteration to direct the algorithm to the global minimum

some new strichartz estimates for the schrodinger equation abstract we deal with fixed time and strichartz estimates for the schrodinger propagator as an operator on wiener amalgam spaces we discuss the sharpness of the known estimates and we provide some new estimates which generalize the classical ones as an application we present a result on the wellposedness of the linear schrodinger equation with a rough time dependent potential

numerical solution of retarded initial value problems local and global error and stepsize control retarded initial value problems are routinely replaced by an initial value problem of ordinary differential equations along with an appropriate interpolation scheme hence one can control the global error of the modified problem but not directly the actual global error of the original problem in this paper we give an estimate for the actual global error in terms of controllable quantities further we show that the notion of local error as inherited from the theory of ordinary differential equations must be generalized for retarded problems along with the new definition we are led to developing a reliable basis for a step selection scheme

numerical treatment of delay differential equations by hermite interpolation a class of numerical methods for the treatment of delay differential equations is developed these methods are based on the wellknown runge kutta fehlberg methods the retarded argument is approximated by an appropriate multipoint hermite interpolation the inherent jump discontinuities in the various derivatives of the solution are considered automatically problems with piecewise continuous right hand side and initial function are treated too real life problems are used for the numerical test and a comparison with other methods published in literature

d convergence of general linear methods for stiff delay differential equations abstract this paper is concerned with the numerical solution of delay differential equations ddes we focus on the error behaviour of general linear methods for stiff ddes a restricted type of interpolation procedure is considered for general linear methods d convergence properties of general linear methods with this interpolation procedure are investigated some extant results are unified

linear multistep methods for functional differential equations a new way to define linear multistep methods for functional differential equations is presented and some of their properties are analyzed the asymptotic behavior of the global discretization error is investigated finally milne s device is generalized to functional differen tial equations the effect of the nonsmoothness of the exact solution is taken into account 1 introduction consider the functional differential equation fde 1 x t f t xt to t t xto o where x t e ri 0 e c t 0 rn fe c to t x c t 0 rn ri and xt 9 x t 9 for 0 e t 0 t e to t it is well known that the solution x of 1 is usually not smooth that is even if f and 4 are c functions x may have jump discontinuities in its derivatives the occurrence of these jump discontinuities may lead to order breakdown for numerical methods if no special provisions are made it seems that all available techniques require information about the location of the jumps if the delay is not state dependent then these locations may be known a priori if not then they may be calculated numerically see feldstein and neves 6 when the location of a jump discontinuity is known two different techniques are available i make the location of the jump a grid point and restart in practice this

solution of time varying delay systems using an adaptive collocation method abstract an adaptive legendre gauss radau collocation method is presented for solving time varying delay systems by dividing the domain of the problem into a uniform mesh the delay system is replaced with a sequence of initial value problems the method developed in this paper adjusts both the mesh spacing and the degree of the polynomial on each mesh interval in order a specified error tolerance is satisfied we investigate some error estimates on the whole domain of interest some numerical examples are given and comparisons with other methods are made to demonstrate the efficiency and accuracy of the proposed method

variable step variable order 3 stage hermite birkhoff obrechkoff dde solver of order 4 to 14 abstract this article presents a solver for delay differential equations ddes called hbo414dde based on a hybrid variable step variable order 3 stage hermite birkhoff obrechkoff ode solver of order 4 to 14 the current version of our method solves ddes with state dependent non vanishing small vanishing and asymptotically vanishing delays except neutral type and initial value ddes delayed values are computed using hermite interpolation small delays are dealt with by extrapolation and discontinuities are located by a bisection method hbo414dde was tested on several problems and results were compared with those of known solvers like sysdel and the recent matlab dde solver ddesd and statistics show that it gives most of the time a smaller relative error than the other solvers for the same number of function evaluations

approximating the linear quadratic optimal control law for hereditary systems with delays in the control a factorization approach is presented for deriving approximations to the optimal feedback gains for the linear regulator quadratic cost problem associated with time varying functional differential equations with control delays the approach is based on a discretization of the state penalty which leads to a simple structure for the feedback control law general properties of the volterra factors of hilbert schmidt operators are then used to obtain convergence results for the controls trajectories and feedback kernels two algorithms are derived from the basic approximation scheme including a fast algorithm in the time invariant case a numerical example is also considered

strichartz estimates via the schrodinger maximal operator we consider the schrodinger operator e it delta acting on initial data f in dot h s we show that an affirmative answer to a question of carleson concerning the sharp range of s for which lim t to 0 e it delta f x f x a e x in mathbb r n would imply an affirmative answer to a question of planchon concerning the sharp range of q and r for which e it delta is bounded in l x q mathbb r n l r t mathbb r when n 2 we unconditionally improve the range for which the mixed norm estimates hold

the optimal projection equations for reduced order state estimation first order necessary conditions for optimal steady state reduced order state estimation for a linear time invariant plant in the presence of correlated disturbance and nonsingular measurement noise are derived in a new and highly simplified form in contrast to the lone matrix riccati equation arising in the full order kalman filter case the optimal steady state reduced order estimator is characterized by three matrix equations one modified riccati equation and two modified lyapunov equations coupled by a projection whose rank is precisely equal to the order of the estimator and which determines the optimal estimator gains this coupling is a graphic reminder of the suboptimality of proposed approaches involving either model reduction followed by full order estimator design or full order estimator design followed by estimator reduction techniques the results given here complement recently obtained results which characterize the optimal reduced order model by means of a pair of coupled modified lyapunov equations 7 and the optimal fixed order dynamic compensator by means of a coupled system of two modified riceati equations and two modified lyapunov equations 6

stability analysis of linearly implicit one step interpolation methods for stiff retarded differential equations the aim of this paper is to analyze the stability properties of the modified problem and of linearly implicit runge kutta interpolation methods for stiff retarded differential equations with respect to the model equation y t py qy t tau quad t geqq 0 y t varphi t quad t leqq 0 where p and q are co mplex constants and tau 0 is a constant d necessary and sufficient conditions for the stability of the modified problem i e the retarded term y t tau of the model equation is replaced by an interpolation polynomial and of linearly implicit one step interpolation methods are presented these conditions allow us to compute the boundary locus of the stability region it is shown that any modified problem and any linearly implicit runge kutta lagrange method with appropriate stability function are p beta stable and that linearly implicit runge kutta lagrange methods possess better stability properties than linearly implicit runge kutta hermite methods furthermore

2h step spline method for the solution of delay differential equations abstract an approximate spline is constructed for the solution of initial value problems in delay differential equations the existence and uniqueness of the approximate spline solution are investigated samples of our computational experiments are given and compared with other methods

on the boundary strichartz estimates for wave and schrodinger equations abstract we consider the l t 2 l x r estimates for the solutions to the wave and schrodinger equations in high dimensions for the homogeneous estimates we show l t 2 l x estimates fail at the critical regularity in high dimensions by using stable levy process in r d moreover we show that some spherically averaged l t 2 l x estimate holds at the critical regularity as a by product we obtain strichartz estimates with angular smoothing effect for the inhomogeneous estimates we prove double l t 2 type estimates

spline collocation methods for solving delay differential equations the convergence and stability analysis of seventh c 3 spline collocation methods if applied to delay differential equations ddes will be considered letting the interior collocation points x i 1 c j x i 1 c jh j 1 3 be dependent on three parameters c 1 c 2 c email protected 0 1 it will be shown that these methods for ddes possess convergence rate of order seven if 0 7279115

on the solutions of a system of linear retarded and advanced differential equations by the bessel collocation approximation in this study we present a numerical approximation for the solutions of the system of high order linear retarded and advanced differential equations with variable coefficients under the mixed conditions this method is based on taking the truncated bessel expansion of the functions in the retarded and advanced differential equation system by the aid of the matrix operations and the collocation points the problem is transformed into a matrix equation with the unknown bessel coefficients by solving this matrix equation the unknown coefficients of the approximate solutions are computed in addition examples that illustrate the pertinent features of the method are presented and the results of this investigation are discussed

the optimal projection equations for static and dynamic output feedback the singular case oblique projections have been shown to arise naturally in both static and dynamic optimal design problems for static controllers an oblique projection was inherent in the early work of levine and athans while for dynamic controllers an oblique projection was developed by hyland and bernstein this note is motivated by the following natural question what is the relationship between the oblique projection arising in optimal static output feedback and the oblique projection arising in optimal fixed order dynamic compensation we show that in nonstrictly proper optimal output feedback there are indeed three distinct oblique projections corresponding to singular measurement noise singular control weighting and reduced compensator order moreover we unify the levine athans and hyland bernstein approaches by rederiving the optimal projection equations for combined static dynamic nonstrictly proper output feedback in a form which clearly illustrates the role of the three projections in characterizing the optimal feedback gains even when the dynamic component of the nonstrictly proper controller is of full order the controller is characterized by four matrix equations which generalize the standard lqg result

extended limiting forms of optimum observers and lqg regulators the general problem of optimal estimation in singular observation systems is considered by generalizing the limiting procedure of friedland explicit and closed expressions for the transfer matrices of the optimum observer as well as for the optimal control law in the singular measurement lqg problem are derived the number of integrators that are required for the implementation of the singular optimum observer is shown for square systems to be equal to the number of the system s zeros the closed form of the optimum observer given here is advantageous as compared with existing procedures structural properties of the singular observation problem are discussed and the role of the zeros of the system in such problems is clarified the dual results for the case of stochastic cheap control are also given

optimal observers for systems with colored noises the problem of optimal full order observers for continuous time linear systems with colored process and measurement noises is considered in such cases optimal estimation of the state involves augmenting the system thus a higher order observer is required the structure of a full order observer is assumed and necessary conditions for the optimal observer are derived the conditions are given for the general case where the intensity of the white noise component of the measurement noise may be singular the solution consists of a modified riccati equation and a lyapunov equation coupled by two projection matrices in the singular case and one projection matrix in the nonsingular case

fractal strichartz estimate for the wave equation abstract we consider strichartz estimates for the wave equation with respect to general measures which satisfy certain growth conditions in r 3 1 we obtain the sharp estimate and in higher dimensions improve the previous results

asymptotic stability concentration and oscillation in harmonic map heat flow landau lifshitz and schrodinger maps on mathbb r 2 we consider the landau lifshitz equations of ferromagnetism including the harmonic map heat flow and schrodinger flow as special cases for degree m equivariant maps from mathbb r 2 to mathbb s 2 if m 3 we prove that near minimal energy solutions converge to a harmonic map as t asymptotic stability extending previous work gustafson et al duke math j 145 3 537 583 2008 down to degree m 3 due to slow spatial decay of the harmonic map components a new approach is needed for m 3 involving among other tools a normal form for the parameter dynamics and the 2d radial double endpoint strichartz estimate for schrodinger operators with sufficiently repulsive potentials which may be of some independent interest when m 2 this asymptotic stability may fail in the case of heat flow with a further symmetry restriction we show that more exotic asymptotics are possible including infinite time concentration blow up and even eternal oscillation

differential algebraic equations with after effect in this paper we are concerned with the solution of delay differential algebraic equations these are differential algebraic equations with after effect or constrained delay differential equations the general semi explicit form of the problem consists of a set of delay differential equations combined with a set of constraints that may involve retarded arguments even simply stated problems of this type can give rise to difficult analytical and numerical problems the more tractable examples can be shown to be equivalent to systems of delay or neutral delay differential equations our purpose is to highlight some of the complexities and obstacles that can arise when solving these problems and to indicate problems that require further research

direct solution to the general reduced order stochastic observation problem the stochastic optimal state observation problem is considered for a general linear continuous time invariant system with unmeasurable stationary inputs and measurement outputs that may be at least in part perfect a general solution to the problem is obtained by processing the perfect measurements through a specific differentiation transformation scheme in order to extract the maximum accurate information on the system states using this information the original system is transformed to a new reduced order model whose measurements are corrupted by a white noise of non singular intensity matrix a minimum order full state estimator to the original system is then constructed by combining the outputs of any full order observer to the reduced order model and the perfect combinations of the system states that were derived by the differentiation transformation scheme a solution to the general singular kalman filtering problem is then obtained by minimizing the variance of the estimation error of the obser

mixed norm strichartz type estimates for hypersurfaces in three dimensions in their work ikromov and muller fourier restriction for hypersurfaces in three dimensions and newton polyhedra princeton university press princeton 2016 proved the full range l p l 2 fourier restriction estimates for a very general class of hypersurfaces in mathbb r 3 which includes the class of real analytic hypersurfaces in this article we partly extend their results to the mixed norm case where the coordinates are split in two directions one tangential and the other normal to the surface at a fixed given point in particular we resolve completely the adapted case and partly the non adapted case in the non adapted case the case when the linear height h text lin phi is below two is settled completely

local linearization method for numerical integration of delay differential equations in this paper a new approach for the numerical computation of delay differential equations ddes is introduced the essential idea consists of obtaining numerical integrators that use a code expressly developed for linear ddes in contrast with the conventional approach of using a code for ordinary differential equations specifically two numerical schemes of this new class of integrators are proposed and their numerical viability analyzed it includes the estimation of the convergence rate the evaluation of the computational cost of the schemes and a simulation study it is proved that these one step explicit integrators converge uniformly with order two to the solution of nonlinear ddes and are able to integrate stiff equations in a satisfactory way with low computational cost

strichartz estimates for the euler equations in the rotational framework abstract we consider the initial value problems of the incompressible euler equations in the rotational framework we obtain the optimal range of the strichartz estimate for the linear group associated with the coriolis forces as an application we prove that the lifespan of the solution can be taken arbitrarily large provided that the speed of rotation is sufficiently high

the optimal reduced order estimator for systems with singular measurement noise the optimal reduced order estimator is completely characterized by necessary conditions resulting from the optimal projection equations the solution consists of one riccati equation and two lyapunov equations coupled by two projections explicit expressions for all of the estimator parameters are given the relation between the reduced order singular estimator and the full order optimal singular estimator which is of reduced order itself is investigated it is shown that under certain conditions the optimal estimator is recovered from the reduced order estimator

sensitivity analysis for dynamic systems with time lags many problems in bioscience for which observations are reported in the literature can be modelled by suitable functional differential equations incorporating time lags other terminology delays or memory effects parameterized by scientifically meaningful constant parameters p or and variable parameters for example control functions u t it is often desirable to have information about the effect on the solution of the dynamic system of perturbing the initial data control functions time lags and other parameters appearing in the model the main purpose of this paper is to derive a general theory for sensitivity analysis of mathematical models that contain time lags in this paper we use adjoint equations and direct methods to estimate the sensitivity functions when the parameters appearing in the model are not only constants but also variables of time to illustrate the results the methodology is applied numerically to an example of a delay differential model

p stability properties of runge kutta methods for delay differential equations recently the author and others have studied a class of runge kutta methods for ddes these methods make use of certain constrained meshes which allow to get optimal order results in this paper we study their asymptotic stability properties using the test equation begin gathered y t ay t by t tau t 0 hfill y t g t for tau leqq t leqq 0 hfill end gathered wherea b 0 andg t is continuous and complex valued it is known thaty t 0 ast if b re a in particular we show that any astable one step collocation method for odes inherits the same property when it is applied to ddes with a constrained mesh i e it is p stable

asymptotic expansions for the midpoint rule applied to delay differential equations let u be the solution of the delay differential equation begin gathered u t f t u t u t tau quad text for t 0 hfill u t varphi t qquad text for tau leqq t leqq 0 hfill end gathered for t j jh let u h t j be the approximation for u t j obtained by the explicit midpoint rule initialized by one euler step it is shown that if f and varphi are sufficiently differentiable and tau h in mathbb n there are functions e k such that for j even u h left t j right u left t j right sum k 1 r h 2k e k left t j right o left h 2r 2 right this expansion of the global discretization error exists in spite of the fact that the solution u usually has jump discontinuities in its derivatives at the points t n tau n in mathbb n thus repeated extrapolation to the limit richardson extrapolation may be applied to improve the approximation furthermore it is show

a new interpretation of the fault detection filter part 2 the optimal detection filter park and rizzoni 1993 obtained closed form expressions for detection filters i e the structure of all detection filters for a given fault direction was defined an important consequence of these results is that they permit the formation of the optimal detection filter problem for optimization with respect to process and measurement noises the necessary conditions for the existence of the optimal detection filter are obtained and a numerical solution technique is shown to be feasible by virtue of the uniqueness of the detection filter gains from an optimization point of view the problem can be regarded as optimal estimation with some structural constraints on the observer gain this problem is solved for both the continuous time and the discrete time cases

stepsize control for delay differential equations using a pair of formulae abstract the effect of the local approximation error on the stepsize control at one step methods which are used for the numerical solution of delay differential equations are considered it is shown how to get a reliable estimate for the local error and so a working stepsize control by using a pair of formulae

limiting forms of optimal observers halevi and palmor 1986 presented closed form expressions for optimal observers for singular measurement systems two assumptions were made there i the measurement is totally singular and ii the number of outputs is less than or equal to the number of stochastic inputs these two assumptions are relaxed in this paper and the optimal observer for the most general singular measurement case is given

convergence analysis of the solution of retarded and neutral delay differential equations by continuous numerical methods we have recently developed a generic approach to solving retarded and neutral delay differential equations ddes the approach is based on the use of an explicit continuous runge kutta formula and employs defect control the approach can be applied to problems with state dependent delays and vanishing delays in this paper we determine convergence properties for the numerical solution associated with methods that implement this approach we first analyze such properties for retarded ddes and then the analysis is extended to the case of neutral ddes ndes such an extension is particularly important since ndes have received little attention in the literature the main result we establish is that the global error of the numerical solution can be efficiently and reliably controlled by directly monitoring the magnitude of the associated defect our analysis can also be applied directly to other numerical methods which use defect error control based on runge kutta formulas for ddes

on the stability of exponential fitting bdf algorithms we present bdf type formulas capable of the exact integration with only round off errors of differential equations whose solutions are linear combinations of an exponential with parameter l and ordinary polynomials for l 0 new formulas reduces to the classical bdf formulas plots of their 0 stability regions in terms of l are provided plots of their regions absolute stability that include all the negative real axis are provided numerical examples shows the efficiency of the proposed codes specially when we are integrating stiff oscillatory problems

inhomogeneous strichartz estimates for the schrodinger equation we study strichartz estimates for the solution of the cauchy problem associated with the inhomogeneous free schrodinger equation in the case when the inital data is equal to zero proving some new estimates for certain exponents and giving counterexamples for some others