on a misconception involving point collocation and the rayleigh hypothesis it is shown that the rayleigh hypothesis does not govern convergence of the simple point collocation approach to the numerical solution of scattering by a sinusoidal grating a previously developed numerical technique interval arithmetic is employed to perform some decisive numerical experiments which not only support but guarantee the validity of this conclusion

oscillation behavior of second order nonlinear neutral differential equations with deviating arguments oscillation criteria are established for second order nonlinear neutral differential equations with deviating arguments of the form r t z t a 1z t f t x s t 0 t t 0 where a 0 and z t x t p t x t t our results improve and extend some known results in the literature some illustrating examples are also provided to show the importance of our results

verified integration of linear nth order odes using large steps the solution y x of an ivp for a linear ode with analytic coefficient functions is represented as a power series a high order taylor polynomial is used for an approximate numerical solution the taylor remainder series is rigorously estimated by some geometric series the method has been implemented and tested on a computer guaranteed enclosures are achieved by taking into account all roundoff errors

oscillation criteria for second order quasi linear neutral delay differential equations abstract in this paper some new oscillation criteria for second order neutral nonlinear differential equation r t x t p t x t m 1 x t p t x t q t f x t 0 are established new oscillation criteria are different from most known ones in the sense that they are based on a class of new functions t s l defined in the sequel

interval analysis theory and applications we give an overview on applications of interval arithmetic among others we discuss verification methods for linear systems of equations nonlinear systems the algebraic eigenvalue problem initial value problems for odes and boundary value problems for elliptic pdes of second order we also consider the item software in this field and give some historical remarks

some new oscillation criteria for second order quasi linear neutral delay differential equations in this paper we will establish some new oscillation criteria for second order quasi linear neutral delay differential equations of the form r t x t p t x email protected a 1 x t p t x email protected q t f x s t 0 the results obtained essentially improve and extend those of dzurina and stavroulakis j dzurina i p stavroulakis oscillation criteria for second order delay differential equations appl math comput 140 2003 445 453 sun and meng y g sun f w meng note on the paper of dzurina and stavroulakis appl math comput 174 2006 1634 1641

on the properties of sard kernels and multiple error estimates for bounded linear functionals of bivariate functions with application to non product cubature each sard kernels theorem supplies multiple error estimates for a bounded linear functional provided the underlying multivariate function is sufficiently smooth the error estimation due to sard generally concerns the l 1 norms of sard kernels and the supremum norms of different partial derivatives of a given function this article presents a verified method for computing sard error constants to assist the verified computation we examine relevant properties of sard kernels for general bounded linear functionals of bivariate functions the derived l 1 norms together with interval taylor arithmetic make the multiple error estimates possible we demonstrate the flexibility and superiority of sard kernels method by different numerical examples that concern non product cubature for bivariate functions

new oscillation criteria for second order nonlinear neutral delay differential equations in this paper several new oscillation criteria for the second order nonlinear neutral delay differential equation r t x t p t x s t m 1 x t p t x s t q t f x t t 0 t t 0 are established these oscillation criteria extend and improve some known results an interesting example illustrating the importance of our results is also provided

on oscillation of second order neutral type delay differential equations oscillation criteria are obtained by using the so called h method for the second order neutral type delay differential equations of the form r t j x t z t q t f x s t 0 t t 0 where z t x t p t x t t r p q t s c t 0 r and f email protected c r r the results of the paper contains several results obtained previously as special cases furthermore we are also able to fix an error in a recent paper related to the oscillation of second order nonneutral delay differential equations

oscillation theorems for second order nonlinear neutral differential equations the purpose of this paper is to study the oscillation of the second order neutral differential equations of the form a t z t c q t x b s t 0 where z t x t p t x t t we explore properties of given equations by examining those of associated first order delay equations new comparison theorems essentially simplify the examination of the equations studied as they allow us to deduce the oscillation of the second order delay differential equation by applying the oscillation criteria obtained to the first order delay equations the results obtained are easy to verify

principles of verified numerical integration we describe methods for the numerical calculation of integrals with verified error bounds the problems range from integration over an interval to integration of parameter dependent integrands over the whole d variate space it is argued why we use bounds for the integrands in the complex plane as a tool for bounding the error in our own integration software

oscillation criteria for second order nonlinear differential equations with damping abstract for a second order nonlinear differential equation with a damping term we obtain new oscillation criteria without an assumption that has been required for related results reported in the literature over the last two decades two approaches refining the standard integral averaging technique suggested in the paper can be used to derive a variety of simpler oscillation theorems for different classes of nonlinear differential equations and both sets of oscillation criteria established in the paper are of independent interest several examples are provided to illustrate the relevance of the new theorems

oscillation theorems for certain neutral functional differential equations in this paper we shall offer sufficient conditions for the oscillation of all solutions to neutral functional differential equations of mixed type of the form dndtn x t ax t bx t q t x t g p t x t h 0 1 here p and q are periodic functions

lattice green function in uniform magnetic fields the green function of a square tight binding model in magnetic fields is expressed by means of continued fractions for rational fields the expression is reduced to a more useful one the green function expressing the out going wave is numerically computed at arbitrary sites for the first time it is found that the absolute value of the green function takes maximum values on the reciprocal lattice of the magnetic brillouin zone

interval inclusion computation for the solutions of the burgers equation in this paper we study the interval computation for the solutions of the burgers equation for the initial boundary value problems of the burgers equation by using the technique of the green function a new kind of interval method is proposed both algorithm and computational examples are given convergence is proved from the results we see that this interval method can get a better solution with our corroboration

oscillation theorems for second order neutral differential equations the aim of this paper is to study the oscillation of the second order neutral differential equations e r t x t p t x t t q t x s t 0 the obtained results are based on the new comparison theorems that enable us to reduce the problem of the oscillation of the second order equation to the oscillation of the first order equation the obtained comparison principles essentially simplify the examination of the studied equations

oscillation criteria for second order delay differential equations the aim of this paper is to establish some new oscillation criteria for the second order retarded differential equation r t u t a 1u t p t u t t a 1u t t 0 the results obtained essentially improve known results in the literature

analytic properties of the recursion method in the presence of band gaps the recursion method gives a good approximation of the density of states in the single band case where the recursion coefficients converge when band gaps exist it is known that they exhibit asymptotic undamped oscillations in order to clarify these asymptotic behaviours an analytic study of the recursion method applicable to the band gap case is presented the asymptotic properties of the system of orthogonal polynomials to which this method is closely related are investigated by use of the concept of the exterior mapping function various relations which connect the means of the recursion coefficients to the support of a spectrum are obtained the period of their asymptotic oscillations is written as a function of the support the numerical results by turchi ducastelle and treglia 1982 can be well interpreted by the present theory the asymptotic properties of the truncated basis are also examined

on the green function of the almost mathieu operator the square tight binding model in a magnetic field leads to the almost mathieu operator which for rational fields reduces to a q q matrix depending on the components of the wavevector in the magnetic brillouin zone we calculate the corresponding green function without explicit knowledge of eigenvalues and eigenfunctions and obtain analytical expressions for the diagonal and the first off diagonal elements the results which are consistent with the zero magnetic field case can be used to calculate several quantities of physical interest e g the density of states over the entire spectrum impurity levels in a magnetic field

oscillation criteria for second order nonlinear neutral differential equations by refining the standard integral averaging technique we obtain new oscillation criteria for a class of second order nonlinear neutral differential equations of the form r t x t p t x email protected q t f x t x s t 0 assumptions in our theorems are less restrictive whereas the proofs are significantly simpler compared to those in the recent paper by shi and wang 18 and related contributions to the subject examples are provided to illustrate the relevance of new theorems