on stability of lms methods and characteristic roots of delay differential equations we investigate the use of linear multistep lms methods for computing characteristic roots of systems of linear delay differential equations ddes with multiple fixed discrete delays these roots are important in the context of stability and bifurcation analysis we prove convergence orders for the characteristic root approximations and analyze under what condition for the steplength the discrete integration scheme retains certain delay independent stability properties of the original equations unlike existing results we concentrate on the recovery of both stability and instability we illustrate our findings with a number of numerical test results

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

an extension of fourier analysis for the n torus in the magnetic field and its application to spectral analysis of the magnetic laplacian we solved the schrodinger equation for a particle in a uniform magnetic field in the n dimensional torus we obtained a complete set of solutions for a broad class of problems the torus tn rn is defined as a quotient of the euclidean space rn by an arbitrary n dimensional lattice the lattice is not necessary either cubic or rectangular the magnetic field is also arbitrary however we restrict ourselves within potential free problems the schrodinger operator is assumed to be the laplace operator defined with the covariant derivative we defined an algebra that characterizes the symmetry of the laplacian and named it the magnetic algebra we proved that the space of functions on which the laplacian acts is an irreducible representation space of the magnetic algebra in this sense the magnetic algebra completely characterizes the quantum mechanics in the magnetic torus we developed a new method for fourier analysis for the magnetic torus and used it to solve the eigenvalue problem of the laplacian

aharonov bohm quantum systems on a punctured 2 torus this paper deals with aharonov bohm a b quantum systems on a punctured two dimensional torus from the geometric and the operator theoretic point of view first flat connections on the u 1 bundles over the punctured 2 torus are studied which serve as vector potentials for a b effect magnetic fields it is proved that the moduli space of flat connections is identified with the n 1 dimensional torus tn 1 if the punctured torus has n 0 pinholes at which solenoids are assumed to penetrate the 2 torus for a given point of tn 1 an associated flat connection is constructed in terms of the weierstrass zeta function on c together with an inhomogeneous linear function on r2 a b quantum systems are defined in terms of position operators and momentum operators coupled with the a b potentials necessary and sufficient conditions are given for two a b quantum systems to be unitarily equivalent further the a b hamiltonian is defined and analysed from the viewpoint of operator theory the deficiency indices of the a b hamiltonian are determined to be n m n m where m is the number of solenoids whose fluxes are not quantized finally the eigenvalue problem is studied for the a b hamiltonian with all fluxes quantized to obtain eigenvalues together with eigenfunctions which are described in terms of the weierstrass sigma functions

momentum operators with gauge potentials local quantization of magnetic flux and representation of canonical commutation relations commutation properties of two dimensional momentum operators with gauge potentials are investigated a notion of local quantization of magnetic flux is introduced to characterize physically the strong commutativity of the momentum operators in terms of the notion a necessary and sufficient condition is given for the position and the momentum operators to be equivalent to the schrodinger representation of the canonical commutation relations

gauge theory on a non simply connected domain and representations of canonical commutation relations a quantum system of a particle interacting with a non abelian gauge field on the non simply connected domain m r2 an nn 1 is considered where an n 1 n are fixed isolated points in r2 the gauge potential a of the gauge field is a p p anti hermitian matrix valued 1 form on m and may be strongly singular at the points an n 1 n if a is flat then the physical momentum and the position operators pj qj 2j 1 of the particle satisfy the canonical commutation relations ccr with two degrees of freedom on a suitable dense domain of the hilbert space l2 r2 cp a necessary and sufficient condition for this representation to be the schrodinger 2 system is given in terms of the wilson loops of the rectangles not intersecting an n 1 n this also gives a characterization for the representation to be non schrodinger it is proven that for a class of gauge potentials which is not necessarily flat pj is essentially self adjoint moreover an example which gives a class of non schrodinger repres

the numerical solution of volterra functional differential equations of neutral type in this paper methods of adams moulton type for neutral functional differential equations are investigated a general convergence theorem is presented and the order of convergence of these methods is examined some numerical examples are given

stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations we prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically stable and unstable periodic solutions of the linear delay differential equation dot y t a t y t b t y t tau f t by a periodic boundary value approach this equation arises e g in the study of the numerical stability of collocation methods for computing periodic solutions of nonlinear delay equations we obtain convergence results for the standard collocation algorithm and for two variants in particular estimates of the difference between the collocation solution and the true solution are derived for the standard collocation scheme the convergence results are unconditional that is they do not require mesh ratio restrictions numerical results that support the theoretical findings are also given

an extension of numerical stability criteria for linear neutral multidelay integro differential equations in this paper the underlying general linear methods glms are adapted to linear neutral multidelay integro differential equations nmides in order to obtain stability criteria of the extended glms the corresponding results in paper of zhang and vandewalle 2008 are generalized based on the concepts of a stability and a stability of the underlying glms a serial of asymptotic stability criteria of the extended glms are obtained

nonlinear stability of runge kutta methods for neutral delay integro differential equations the present paper is concerned with the numerical solution to initial value problems of neutral delay integro differential equations ndides the sufficient conditions for the stability and asymptotic stability of runge kutta methods for a class r a b 1 b 2 c of nonlinear systems of ndides are derived a numerical test that confirms the theoretical results is given in the end

on the pauli operator for the aharonov bohm effect with two solenoids we consider a spin 1 2 charged particle in the plane under the influence of two idealized aharonov bohm fluxes we show that the pauli operator as a differential operator is defined by appropriate boundary conditions at the two vortices further we explicitly construct a basis in the deficiency subspaces of the symmetric operator obtained by restricting the domain to functions with supports separated from the vortices this construction makes it possible to apply the krein s formula to the pauli operator

canonical commutation relations the weierstrass zeta function and infinite dimensional hilbert space representations of the quantum group uq sl2 a two dimensional quantum system of a charged particle interacting with a vector potential determined by the weierstrass zeta function is considered the position and the physical momentum operators give a representation of the canonical commutation relations with two degrees of freedom if the charge of the particle is not an integer the case corresponding to the aharonov bohm effect then the representation is inequivalent to the schrodinger representation it is shown that the inequivalent representation induces infinite dimensional hilbert space representations of the quantum group uq sl some properties of these representations of uq sl are investigated

runge kutta theory for volterra integrodifferential equations the present paper develops the theory of general runge kutta methods for volterra integrodifferential equations the local order is characterized in terms of the coefficients of the method we investigate the global convergence of mixed and extended runge kutta methods and give results on asymptotic error expansions in a further section we construct examples of methods up to order 4

error estimates with explicit constants for sinc approximation sinc quadrature and sinc indefinite integration error estimates with explicit constants are given for approximations of functions definite integrals and indefinite integrals by means of the sinc approximation although in the literature various error estimates have already been given for these approximations those estimates were basically for examining the rates of convergence and several constants were left unevaluated giving more explicit estimates i e evaluating these constants is of great practical importance since by this means we can reinforce the useful formulas with the concept of verified numerical computations in this paper we reveal the explicit form of all constants in a computable form under the same assumptions of the existing theorems the function to be approximated is analytic in a suitable region we also improve some formulas themselves to decrease their computational costs numerical examples that confirm the theory are also given

runge kutta theory for volterra integral equations of the second kind the present paper develops the theory of general runge kutta methods for volterra integral equations of the second kind the order conditions are derived by using the theory of p series which for our problem reduces to the theory of v series these results are then applied to two special classes of runge kutta methods introduced by pouzet and by bel tyukov

convergence analysis of fitted numerical method for a singularly perturbed nonlinear volterra integro differential equation with delay abstract in this paper the initial value problem for a quasilinear singularly perturbed delay volterra integro differential equation was considered by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with the weight and remainder term in integral form a fitted difference scheme is constructed and analysed it is shown that the method displays first order uniform convergence in perturbation parameter some numerical results are given to confirm the theoretical analysis

essential self adjointness for semi bounded magnetic schrodinger operators on non compact manifolds abstract we prove essential self adjointness for semi bounded below magnetic schrodinger operators on complete riemannian manifolds with a given positive smooth measure which is fixed independently of the metric some singularities of the scalar potential are allowed this is an extension of the povzner wienholtz simader theorem the proof uses the scheme of wienholtz but requires a refined invariant integration by parts technique as well as the use of a family of cut off functions which are constructed by a non trivial smoothing procedure due to karcher

on the stability of runge kutta methods for delay integral equations we present a class of runge kutta methods for the numerical solution of a class of delay integral equations dies described by two different kernels and with a fixed delay the stability properties of these methods are investigated with respect to a test equation with linear kernels depending on complex parameters the results are then applied to collocation methods in particular we obtain that any collocation method for dies resulting from ana stable collocation method for odes with a stepsize which is submultiple of the delay preserves the asymptotic stability properties of the analytic solutions

stability of runge kutta methods for delay integro differential equations we study stability of runge kutta rk methods for delay integro differential equations with a constant delay on the basis of the linear equation du dt lu t mu t k t t u d where l m k are constant complex matrices in particular we show that the same result as in the case k 0 koto bit 34 1994 262 267 holds for this test equation i e every a stable rk method preserves the delay independent stability of the exact solution whenever a step size of the form h m is used where m is a positive integer

de sinc methods have almost the same convergence property as se sinc methods even for a family of functions fitting the se sinc methods in this paper the theoretical convergence rate of the trapezoidal rule combined with the double exponential de transformation is given for a class of functions for which the single exponential se transformation is suitable it is well known that the de transformation enables the rule to achieve a much higher rate of convergence than the se transformation and the convergence rate has been analyzed and justified theoretically under a proper assumption here it should be emphasized that the assumption is more severe than the one for the se transformation and there actually exist some examples such that the trapezoidal rule with the se transformation achieves its usual rate whereas the rule with de does not such cases have been observed numerically but no theoretical analysis has been given thus far this paper reveals the theoretical rate of convergence in such cases and it turns out that the de s rate is almost the same as but slightly lower than that of the se by using the analysis technique developed here the theoretical convergence rate of the sinc approximation with the de transformation is also given for a class of functions for which the se transformation is suitable the result is quite similar to above the convergence rate in the de case is slightly lower than in the se case numerical examples which support those two theoretical results are also given

numerical modelling in biosciences using delay differential equations our principal purposes here are i to consider from the perspective of applied mathematics models of phenomena in the biosciences that are based on delay differential equations and for which numerical approaches are a major tool in understanding their dynamics ii to review the application of numerical techniques to investigate these models we show that there are prima facie reasons for using such models i they have a richer mathematical framework compared with ordinary differential equations for the analysis of biosystem dynamics ii they display better consistency with the nature of certain biological processes and predictive results we analyze both the qualitative and quantitative role that delays play in basic time lag models proposed in population dynamics epidemiology physiology immunology neural networks and cell kinetics we then indicate suitable computational techniques for the numerical treatment of mathematical problems emerging in the biosciences comparing them with those implemented by the bio modellers

aharonov bohm effect with δ type interaction a quantum particle interacting with a thin solenoid and a magnetic flux is described by a five parameter family of hamilton operators obtained via the method of self adjoint extensions one of the parameters the value of the flux corresponds to the aharonov bohm effect the other four parameters correspond to the strength of a singular potential barrier the spectrum and eigenstates are computed and the scattering problem is solved

numerical stability analysis and computation of hopf bifurcation points for delay differential equations we present a numerical technique for the stability analysis and the computation of branches of hopf bifurcation points in nonlinear systems of delay differential equations with several constant delays the stability analysis of a steady state solution is done by a numerical implementation of the argument principle which allows to compute the number of eigenvalues with positive real part of the characteristic matrix the technique is also used to detect bifurcations of higher singularity hopf and fold bifurcations during the continuation of a branch of hopf points this allows to trace new branches of hopf points and fold points

numerical bifurcation analysis of immunological models with time delays in recent years a large number of mathematical models that are described by delay differential equations ddes have appeared in the life sciences to analyze the models dynamics numerical methods are necessary since analytical studies can only give limited results in turn the availability of efficient numerical methods and software packages encourages the use of time delays in mathematical modelling which may lead to more realistic models we outline recently developed numerical methods for bifurcation analysis of ddes and illustrate the use of these methods in the analysis of a mathematical model of human hepatitis b virus infection

two formulas for numerical indefinite integration we derive two formulas for approximating the indefinite integral over a finite interval the approximation error is 0 c cv uniformly where m is the number of integrand evaluations the integrand is required to be analytic in the interior of the integration interval but may be singular at the endpoints some sample calculations indicate that the actual convergence rate accords with the error bound derived

a remark concerning canonical commutation relations an elementary quantum mechanical example of canonical commutation relations not equivalent to the schrodinger representation is discussed it demonstrates that the nelson phenomenon occurs in a natural situation

stochastic parameter estimation in nonlinear time delayed vibratory systems with distributed delay abstract the stochastic estimation of parameters and states in linear and nonlinear time delayed vibratory systems with distributed delay is explored the approach consists of first employing a continuous time approximation to approximate the delayed integro differential system with a large set of ordinary differential equations having stochastic excitations then the problem of state and parameter estimation in the resulting stochastic ordinary differential system is represented as an optimal filtering problem using a state augmentation technique by adapting the extended kalman bucy filter to the augmented filtering problem the unknown parameters of the time delayed system are estimated from noise corrupted possibly incomplete measurements of the states similarly the upper bound of the distributed delay can also be estimated by the proposed technique as an illustrative example to a practical problem in vibrations the parameter delay upper bound and state estimation from noise corrupted measurements in a distributed force model widely used for modeling machine tool vibrations in the turning operation is investigated

a finite difference method for a singularly perturbed delay integro differential equation we consider the singularly perturbed initial value problem for a linear first order volterra integro differential equation with delay our purpose is to construct and analyse a numerical method with uniform convergence in the perturbation parameter the numerical solution of this problem is discretized using implicit difference rules for differential part and the composite numerical quadrature rules for integral part on a layer adapted mesh error estimations for the approximate solution are established numerical examples supporting the theory are presented

representation theoretic aspects of two dimensional quantum systems in singular vector potentials canonical commutation relations quantum algebras and reduction to lattice quantum systems some representation theoretic aspects of a two dimensional quantum system of a charged particle in a vector potential a which may be singular on an infinite discrete subset d of r2 are investigated for each vector v in a set v d r2 0 the projection pv of the physical momentum operator p p a to the direction of v is defined by pv v p as an operator acting in l2 r2 where p idx idy x y r2 with dx resp dy being the generalized partial differential operator in the variable x resp y and r is a parameter denoting the charge of the particle it is proven that pv is essentially self adjoint and an explicit formula is derived for the strongly continuous one parameter unitary group eitpv t r generated by the self adjoint operator pv the closure of pv i e the magnetic translation to the direction of the vector v the magnetic translations along curves in r2 d are also considered conjugately to pv and pw w v d a self adjoint multiplication operator qv w is introduced which is a

stability properties of second order delay integro differential equations a basic theorem on the behavior of solutions of scalar linear second order delay integro differential equations is established as a consequence of this theorem a stability criterion is obtained

general linear methods for volterra integro differential equations with memory a new class of numerical methods for volterra integro differential equations with memory is developed the methods are based on the combination of general linear methods with compound quadrature rules sufficient conditions that guarantee global and asymptotic stability of the solution of the differential equation and its numerical approximation are established numerical examples illustrate the convergence and effectiveness of the numerical methods

stability of linear multistep methods for delay integro differential equations this paper is concerned with the numerical solution of delay integro differential equations the adaptation of linear multistep methods is considered the emphasis is on the linear stability of numerical methods it is shown that every a stable strongly 0 stable linear multistep method of pouzet type can preserve the delay independent stability of the underlying linear systems in addition some delay dependent stability conditions for the stability of numerical methods are also given

the double exponential sinc collocation method for singular sturm liouville problems sturm liouville problems are abundant in the numerical treatment of scientific and engineering problems in the present contribution we present an efficient and highly accurate method for computing eigenvalues of singular sturm liouville boundary value problems the proposed method uses the double exponential formula coupled with sinc collocation method this method produces a symmetric positive definite generalized eigenvalue system and has exponential convergence rate numerical examples are presented and comparisons with single exponential sinc collocation method clearly illustrate the advantage of using the double exponential formula

numerical indefinite integration by double exponential sinc method we present a numerical method for approximating an indefinite integral by the double exponential sinc method the approximation error of the proposed method with n integrand function evaluations is o exp c 1 n log c 2 n for a reasonably wide class of integrands including those with endpoint singularities the proposed method compares favorably with the existing formulas based on the ordinary sinc method computational results show the accordance of the actual convergence rates with the theoretical estimate

function classes for double exponential integration formulas the double exponential de formulas for numerical integration are known to be highly efficient more efficient than the single exponential se formulas in many cases function classes suited to the se formulas have already been investigated in the literature through rigorous mathematical analysis whereas this is not the case with the de formulas this paper identifies function classes suited to the de formulas in a way compatible with the existing theoretical results for the se formulas the de formulas are good for more restricted classes of functions but more efficient for such functions two concrete examples demonstrate the subtlety in the behavior of the de formulas that is revealed by our theoretical analysis

deficiency indices and singular boundary conditions in quantum mechanics we consider schrodinger operators h in l2 rn n n with countably infinitely many local singularities of the potential which are separated from each other by a positive distance it is proved that due to locality each singularity yields a separate contribution to the deficiency index of h in the special case where the singularities are pointlike and the potential exhibits certain symmetries near these points we give an explicit construction of self adjoint boundary conditions

near optimality of the sinc approximation near optimality of the sinc approximation is established in a variety of spaces of functions analytic in a strip region about the real axis each space being characterized by the decay rate of their elements functions in the neighborhood of the infinity

summary of sinc numerical methods sinc approximation methods excel for problems whose solutions may have singularities or infinite domains or boundary layers this article summarizes results obtained to date on sinc numerical methods of computation sinc methods provide procedures for function approximation over bounded or unbounded regions encompassing interpolation approximation of derivatives approximate definite and indefinite integration solving initial value ordinary differential equation problems approximation and inversion of fourier and laplace transforms approximation of hilbert transforms and approximation of indefinite convolutions the approximate solution of partial differential equations and the approximate solution of integral equations methods for constructing conformal maps and methods for analytic continuation indeed sinc are ubiquitous for approximating every operation of calculus

error estimates with explicit constants for the sinc approximation over infinite intervals abstract the sinc approximation is a function approximation formula that attains exponential convergence for rapidly decaying functions defined on the whole real axis even for other functions the sinc approximation works accurately when combined with a proper variable transformation the convergence rate has been analyzed for typical cases including finite semi infinite and infinite intervals recently for verified numerical computations a more explicit computable error bound has been given in the case of a finite interval in this paper such explicit error bounds are derived for other cases

numerical bifurcation analysis of delay differential equations numerical methods for the bifurcation analysis of delay differential equations ddes have only recently received much attention partially because the theory of ddes smoothness boundedness stability of solutions is more complicated and less established than the corresponding theory of ordinary differential equations as a consequence no established software packages exist at present for the bifurcation analysis of ddes we outline existing numerical methods for the computation and stability analysis of steady state solutions and periodic solutions of systems of ddes with several constant delays

stability of θ methods for delay integro differential equations stability of methods for delay integro differential equations dides is studied on the basis of the linear equation du dt u t u t k t t u d where k are complex numbers and is a constant delay it is shown that every a stable method possesses a similar stability property to p stability i e the method preserves the delay independent stability of the exact solution under the condition that k is real and h is an integer where h is a step size it is also shown that the method does not possess the same property if h is not an integer as a result no method can possess a similar stability property to gp stability with respect to dides

double exponential formulas for numerical indefinite integration in this paper we derive a formula for indefinite integration of analytic functions over 1 s where 1 s 1 by means of the double exponential transformation and the sinc method the integrand must be analytic on 1 x 1 but may have a singularity at the end points x 1 the error of the formula behaves approximately as exp c1n logc2n where n is the number of function evaluations of the integrand this error term shows a much faster convergence to zero when n becomes large than that of the known formula by haber also we derive efficient double exponential formulas for numerical evaluation of indefinite integrals over 0 s 0 s and over s s several numerical examples indicate high efficiency of the formulas

recent developments of the sinc numerical methods abstract this paper gives a survey of recent developments of the sinc numerical methods a variety of sinc numerical methods have been developed by stenger and his school for a certain class of problems the sinc numerical methods have the convergence rates of o exp n with some 0 where n is the number of nodes or bases used in the methods recently it has turned out that the sinc numerical methods can achieve convergence rates of o exp n log n with some 0 for a smaller but still practically meaningful class of problems and that these convergence rates are best possible the present paper demonstrates these facts for two sinc numerical methods the sinc approximation and the sinc collocation method for two point boundary value problems

dissipativity and asymptotic stability of nonlinear neutral delay integro differential equations this paper is concerned with the dissipativity and asymptotic stability of the theoretical solutions of a class of nonlinear neutral delay integro differential equations ndides we first give a generalization of the halanay inequality which plays an important role in the study of dissipativity and stability of differential equations then we apply the generalization of the halanay inequality to ndides and the dissipativity and the asymptotic stability results of the theoretical solution of ndides are obtained from a numerical point of view it is important to study the potential of numerical methods in preserving the qualitative behavior of the analytical solutions therefore the results presented in this paper provide the theoretical foundation for analyzing the dissipativity and the asymptotic stability of the numerical methods when they are applied to these systems

the double exponential transformation in numerical analysis the double exponential transformation was first proposed by takahasi and mori in 1974 for the efficient evaluation of integrals of an analytic function with end point singularity afterwards this transformation was improved for the evaluation of oscillatory functions like fourier integrals recently it turned out that the double exponential transformation is useful not only for numerical integration but also for various kinds of sinc numerical methods the purpose of the present paper is to review the double exponential transformation in numerical integration and in a variety of sinc numerical methods

convergence analysis of an arnoldi order reduced runge kutta method for integro differential equations of pantograph type in this paper we consider a numerical method for linear functional integro differential equations with pantograph delays to deal with the pantograph delays we introduce a geometrically increasing mesh and propose a new kind of runge kutta methods based on the arnoldi order reduced technique the convergence analysis of the new method is presented and the method is proved to be at least first order the numerical experiments attached show that the new method can achieve almost the same accuracy as the corresponding runge kutta method for some properly chosen krylov subspaces

stability and rupture of bifurcation bridges in semiconductor lasers subject to optical feedback the bifurcation diagram of a single mode semiconductor laser subject to a delayed optical feedback is examined by using numerical continuation methods for this we show how to cope with the special symmetry properties of the equations as the feedback strength is increased branches of modes and antimodes appear and we have found that pairs of modes and antimodes are connected by closed branches of periodic solutions bifurcation bridges such connections seem generically present as new pairs of modes and antimodes appear we subsequently investigate the behavior of the first connection as a function of the linewidth enhancement factor and the feedback phase our results extend and confirm existing results and hypotheses reported in the literature for large values of the linewidth enhancement factor alpha 5 6 bridges break through homoclinic orbits changing the feedback phase unfolds the bifurcation diagram of the modes and antimodes allowing different types of connections between modes

magnetic translation groups in an n dimensional torus and their representations a charged particle in a uniform magnetic field in a two dimensional torus has a discrete noncommutative translation symmetry instead of a continuous commutative translation symmetry we study topology and symmetry of a particle in a magnetic field in a torus of arbitrary dimensions the magnetic translation group mtg is defined as a group of translations that leave the gauge field invariant we show that the mtg in an n dimensional torus is isomorphic to a central extension of a cyclic group z 1 z 2l tm by u 1 with 2l m n we construct and classify irreducible unitary representations of the mtg in a three torus and apply the representation theory to three examples we briefly describe a representation theory for a general n torus the mtg in an n torus can be regarded as a generalization of the so called noncommutative torus

point interactions in two dimensions basic properties approximations and applications to solid state physics on etudie rigoureusement l operateur de schrodinger formellement donne par h y dans l 2 r 2 et on montre comment l approcher en termes d hamiltoniens a courte portee transformes d echelles on considere des questions semblables pour l operateur h j 1 n j y j n n on etudie les proprietes spectrales du modele a un electron d un cristal a 2 dimensions avec des interactions ponctuelles

function classes for successful de sinc approximations the de sinc formulas resulting from a combination of the sinc approximation formula with the double exponential de transformation provide a highly efficient method for function approximation in many cases they are more efficient than the se sinc formulas which are the sinc approximation formulas combined with the single exponential se transformations function classes suited to the se sinc formulas have already been investigated in the literature through rigorous mathematical analysis whereas this is not the case with the de sinc formulas this paper identifies function classes suited to the de sinc formulas in a way compatible with the existing theoretical results for the se sinc formulas furthermore we identify alternative function classes for the de sinc formulas as well as for the se sinc formulas which are more useful in applications in the sense that the conditions imposed on the functions are easier to verify

convergence of one leg methods for neutral delay integro differential equations the error analysis of one leg methods for a class of nonlinear neutral delay integro differential equations ndides is given it is proved that an a stable one leg method with an appropriate quadrature rule applied to ndides is convergent of order at least min p q 12 if the one leg method is consistent of order p2 in the classical sense for odes and the error order of the quadrature rule is o hq 1 numerical examples further confirm the theoretical results

classical theory of runge kutta methods for volterra functional differential equations abstract for solving volterra functional differential equations vfdes a class of discrete runge kutta methods based on canonical interpolation is studied which was first presented by the same author in a previous published paper classical stability and convergence theories for this class of methods are established the methods studied and the theories established in this paper can be directly applied to non stiff non linear initial value problems in delay differential equations ddes integro differential equations ides delay integro differential equations dides and vfdes of other type which appear in practice and can be used as a necessary basis for the study of splitting methods for complex nonlinear stiff vfdes

dissipativity of runge kutta methods for neutral delay integro differential equations this paper is concerned with the numerical dissipativity of a class of nonlinear neutral delay integro differential equations the dissipativity results are obtained for algebraically stable runge kutta methods when they are applied to above problems

convergence of runge kutta methods for neutral delay integro differential equations the present paper is concerned with the convergence of numerical methods to solve initial value problems of neutral delay integro differential equations ndides the error estimation of runge kutta methods for a class of nonlinear systems of ndides are derived some numerical experiments are given which confirm the theoretical results

optimality of the double exponential formula functional analysis approach in the light of the functional analysis theory we establish the optimality of the double exponential formula the argument consists of the following three ingredients 1 introduction of a number of spaces of functions analytic in a strip region about the real axis each space being characterized by the decay rate of their elements functions in the neighborhood of the infinity 2 proof of the near optimality of the trapezoidal formula in each space introduced in 1 by showing the near equality between an upper estimate for the error norm of the trapezoidal formula and a lower estimate for the minimum error norm of quadratures 3 nonexistence theorem for the spaces the characterizing decay rate of which is more rapid than the double exponential

de sinc methods have almost the same convergence property as se sinc methods even for a family of functions fitting the se sinc methods part ii indefinite integration in this paper the theoretical convergence rate of the sinc indefinite integration combined with the double exponential de transformation is given for a class of functions for which the single exponential se transformation is suitable although the de transformation is considered as an enhanced version of the se transformation for sinc related methods the function space for which the de transformation is suitable is smaller than that for se and therefore there exist some examples such that the de transformation is not better than the se transformation even in such cases however some numerical observations in the literature suggest that there is almost no difference in the convergence rates of se and de in fact recently the observations have been theoretically explained for two explicit approximation formulas the sinc quadrature and the sinc approximation the conclusion is that in such cases the de s rate is slightly lower but almost the same as that of the se the contribution of this study is the derivation of the same conclusion for the sinc indefinite integration numerical examples that support the theoretical result are also provided

collocation methods for the computation of periodic solutions of delay differential equations in this paper we investigate collocation methods for the computation of periodic solutions of autonomous delay differential equations ddes periodic solutions are found by solving a periodic two point boundary value problem which is an infinite dimensional problem for ddes in contrast to the case of ordinary differential equations we investigate three collocation methods based on piecewise polynomials we discuss computational issues and show numerical orders of convergence using an extensive number of tests we compare our numerical results with known theoretical convergence results for initial value problems for ddes in particular we show how superconvergence at the mesh points can be lost or recovered depending on the dde model under consideration and on the choice of collocation discretization we end with a brief discussion of adaptive mesh selection

an analysis of delay dependent stability for ordinary and partial differential equations with fixed and distributed delays this paper is concerned with the study of the stability of ordinary and partial differential equations with both fixed and distributed delays and with the study of the stability of discretizations of such differential equations we start with a delay dependent asymptotic stability analysis of scalar ordinary differential equations with real coefficients we study the exact stability region of the continuous problem as a function of the parameters of the model next it is proved that a time discretization based on the trapezium rule can preserve the asymptotic stability for the considered set of test problems in the second part of the paper we study delay partial differential equations the stability region of the fully continuous problem is analyzed first then a semidiscretization in space is applied it is shown that the spatial discretization leads to a reduction of the stability region when the standard second order central difference operator is employed to approximate the diffusion operator finally we consider the delay dependent stability of the fully discrete problem where the partial differential equation is discretized both in space and in time some numerical examples and further discussions are given

retarded differential equations retarded differential equations rdes are differential equations having retarded arguments they arise in many realistic models of problems in science engineering and medicine where there is a time lag or after effect numerical techniques for such problems may be regarded as extensions of dense output methods for ordinary differential equations odes but scalar rdes are inherently infinite dimensional with a richer structure than their ode counterparts we give background material develop a theoretical foundation for the basic numerics and give some results not previously published