existence and stability of nonlinear functional differential equations existence and stability results for a class of nonlinear functional differential equations which contains the generalised pantograph equation as a special case are obtained

numerical evaluation of highly oscillatory bessel transforms abstract in this paper we mainly focus on numerical evaluation of highly oscillatory bessel transforms based on the multiple integral the schemes for computing this class of the transform are presented by means of the bounds of special functions we give the corresponding errors of the schemes to demonstrate the efficiency and accuracy of the proposed methods some test examples are shown

some new retarded nonlinear integral inequalities and their applications in differential integral equations in this paper we discuss some new retarded nonlinear gronwall bellman pachpatte integral inequalities and give upper bound estimation of the unknown function by integral inequality technique our results generalize those of abdeldaim and yakout a abdeldaim m yakout on some new integral inequalities of gronwall bellman pachpatte type appl math comput 217 2011 7887 7899 to some inequalities with both retarded argument and nonlinear integrand an application to estimate the solutions of differential integral equations is also given

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

henry gronwall type retarded integral inequalities and their applications to fractional differential equations with delay abstract this paper presents retarded integral inequalities of henry gronwall type applying these inequalities we study certain properties of solutions to fractional differential equations with delay

a method for computing bessel function integrals abstract a method for numerical calculation of integrals containing bessel functions of integer or integer plus one half order is described the calculation involves first a one dimensional fourier sine or cosine transform followed by evaluation of the coefficient of the chebyshev series of the fourier transformed function in the case of the bessel function and evaluation of the legendre expansion coefficient in the case of the spherical bessel function a generalization of the method for the computation of an integral involving the bessel function of arbitrary real order v is presented as well

a generalized gronwall inequality and its application to a fractional differential equation this paper presents a generalized gronwall inequality with singularity using the inequality we study the dependence of the solution on the order and the initial condition of a fractional differential equation

on evaluation of bessel transforms with oscillatory and algebraic singular integrands in this paper we study efficient methods for computing the integrals of the form 0 1x a 1 x bf x j v wx dx where a b v w are the given constants and w 1 j v is the bessel function of the first kind and of order v f is a sufficiently smooth function on 0 1 firstly we express the moments in a closed form with the aid of special functions secondly we induce the filon type method based on the taylor interpolation polynomial at two endpoints and the hermite interpolation polynomial at clenshaw curtis points on evaluating the highly oscillatory bessel integrals with algebraic singularity theoretical results and numerical experiments perform that the methods are very efficient in obtaining very high precision approximations if w is sufficiently large

procedures for computing one and two dimensional integrals of functions with rapid irregular oscillations a collocation procedure for efficient integration of rapidly oscillatory functions is presented the integration problem is transformed into a certain o d e problem and this is solved by a collocation technique the method is also extended to two dimensional integration and some numerical results are appended showing the efficiency of the method in handling difficult cases of rapid irregular oscillations 1 the procedure for one dimensional integrals we consider integrals of the form a 1 1 1 f x e iq x dx where f is smooth and nonoscillatory and iq x i b a 1 two practical methods for evaluating rapidly oscillatory integrals are described in 1 the use of approximation as in filon s method 3 4 and the speedup method of longman 5 formally both methods are applicable to any integral of the form 1 1 but their best performance is for the case of a constant frequency q w in this note we present an efficient method which is applicable for cases of varying frequency q using only a small number of values of f and q in a b and the values q a and q b the proposed method follows the spirit of filon s method it is based upon the fact that if f were of the form 1 2 f x iq x p x p x l l1p x a x b then the integral could be evaluated directly as 1 3 j b iq x p x pt x eiq x dx f d p x eiq x dx b e qbp a eiqa equation 1 2 can be considered as a differential equation for p x and any solution of this equation can be used in 1 3 for evaluating i the general solution of this equation is 1 4 p x e iq x f t eiq t dt cj received march 24 1981 revised july 29 1981 1980 mathematics subject classification primary 65d30 secondary 65d32 1982 american mathematical society 0025 5718 8 1 ooo 0 1 23 03 25 531 this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 36 01 utc all use subject to http about jstor org terms

explicit bounds on certain integral inequalities abstract in this paper explicit bounds on retarded gronwall bellman and bihari like integral inequalities and their two independent variable generalizations are established some applications are also given to illustrate the usefulness of one of our results

oscillations of first order linear retarded differential equations the oscillatory behavior of differential equations with deviating arguments has been studied by many authors in particular several authors have studied the oscillations caused by the deviating arguments the study of such oscillations is a relatively new field and is very interesting in applications for some contributions in this area see the papers by arino gyori and jawhari 11 bowcock and yu 2 gyiiri 4 5 j hunt and yorke 6 kulenovic ladas and meimaridou 7 ladas s ladas sficas and stavroulakis 9 lo ladas and stavroulakis 111 sugie 141 tramov 15 and yan 16 we also refer to the recent book by ladde lakshmikantham and zhang 123 the purpose of this paper is to investigate the oscillatory behavior of the solutions of first order linear retarded differential equations with variable coefftcients and nonconstants delays consider the first order retarded differential equation

fast integration of rapidly oscillatory functions a collocation method for approximating integrals of rapidly oscillatory functions is presented the method is efficient for integrals involving bessel functions jv rx with large oscillation parameter r as well as for many other one and multi dimensional integrals of functions with rapid irregular oscillations

on property i for knots in this paper deals with the question of which knot surgeries on s 3 can yield 3 manifolds homeomorphic to or with the same fundamental group as the poincare homology 3 sphere

asymptotic expansions of bessel anger and weber transformations in this paper we explore higher order numerical quadrature for the integrations of systems containing bessel anger and weber functions the method is constructed by finding the approximate solution of the differential equation and truncating the asymptotic series numerical examples based on theoretical results are presented to demonstrate the efficiency of the method

on the evaluation of bessel transformations with the oscillators via asymptotic series of whittaker functions in this paper we are concerned with numerical evaluation of bessel transformations with the oscillators first we rewrite the integrals by whittaker functions then based on the asymptotic series of whittaker functions the integrals are transformed into the problems involving the complex exponential functions which can be efficiently computed by using the complex integration method at the same time we obtain the entirely different asymptotic expansions from the references in particular the asymptotic expansions can be denoted by the incomplete gamma function the error analysis shows that the decay of the error drastically improves as the frequency grows the effectiveness and accuracy of the methods are tested for large arguments of bessel functions

fourier or bessel transformations of highly oscillatory functions an algorithm for integral transformation of highly oscillatory functions is presented it is a generalization of the standard gaussian quadrature method but has the virtue that the usual tedious tests of convergence of the numerical results is not necessary the effectiveness and accuracy of the algorithm is tested for both large arguments and higher orders of the bessel function

the fast multipole method we study integral methods applied to the resolution of the maxwell equations where the linear system is solved using an iterative method which requires only matrix vector products the fast multipole method fmm is one of the most efficient methods used to perform matrix vector products and accelerate the resolution of the linear system a problem involving n degrees of freedom may be solved in cniternlogn floating operations where c is a constant depending on the implementation of the method in this article several techniques allowing one to reduce the constant c are analyzed this reduction implies a lower total cpu time and a larger range of application of the fmm in particular new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms several numerical tests are also described these confirm the efficiency and the theoretical complexity of the fmm

growth in an impulsive integral inequality abstract in this paper we study a finite sum integral inequality with a sequence of impulses making a sequential monotonization we give recursively defined functions to estimate solutions of the inequality piecewise in order to use the estimate to study boundedness and asymptotics of solutions for differential equations we regard the estimating function as a solution of a nonautonomous difference equation simplify the equation with composition of operators and reduce our discussion to qualitative analysis of the difference equation so that we give results on monotonicity boundedness and weighted boundedness for solutions of the inequality as applications we study boundedness and asymptotics of solutions for two impulsive differential equations

generalization of a retarded gronwall like inequality and its applications abstract this paper generalizes a lipovan s result of gronwall like inequalities j math anal appl 252 2000 389 to a new type of retarded inequalities which includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals from our result bihari s result and pinto s result can be deduced as some special cases our result can be applied to give the global existence and estimate solutions we also apply it to improve an estimation for almost periodicity of an invariant manifold

finite dehn surgery on knots let k be a knot with a closed tubular neighbourhood n k in a connected orientable closed 3 manifold w such that the exterior of k m w intn k is irreducible we consider the problem of which dehn surgeries on k or equivalently which dehn fillings on m can produce 3 manifolds with finite fundamental group for convenience a surgery is called a g surgery if the resultant 3 manifold has fundamental group g if g is cyclic or finite the surgery is also called a cyclic surgery or a finite surgery similar terminology will be used for dehn fillings the manifold obtained by the dehn filling on m along m with slope r is denoted by m r let r1 r2 denote the minimal geometric intersection number the distance between two slopes r1 and r2 on m according to t1 m belongs to one of the following three mutually exclusive categories i m is a seifert fibred space admitting no essential torus ii m is a hyperbolic manifold i e intm admits a complete hyperbolic metric of finite volume iii m contains an essential torus it is a remarkable result the so called cyclic surgery theorem cgls that if m is not seifert fibred then all cyclic surgery slopes of k have mutual distance no larger than 1 and consequently there are at most 3 cyclic surgeries on k in this paper we consider finite dehn surgery on k and we prove for instance that if m is not a manifold of type i and is not a union along a torus of the twisted i bundle over the klein bottle and a cabled space then there are at most 6 finite and cyclic surgeries on k of maximal mutual distance 5 henceforth we shall use finite cyclic to mean either finite or infinite cyclic it turns out to be convenient to consider the three cases mentioned above separately in case i it is well known that one can completely classify the finite cyclic surgeries on k considering the torus knots for instance one sees that there exist infinitely many knots whose exteriors are of type i each of which admit an infinity of distinct finite cyclic or non cyclic surgery slopes our contributions deal with the cases ii and iii for the former we obtain

on generalized of certain retarded nonlinear integral inequalities and its applications in retarded integro differential equations in this paper some new nonlinear retarded integral inequalities of gronwall bellman pachpatte type are investigated some applications are also presented to illustrate the usefulness of some of our results in estimation of solution of certain retarded integro differential equations with the initial conditions

estimates on solutions of some new nonlinear retarded volterra fredholm type integral inequalities abstract some new explicit bounds on solutions to a class of new nonlinear retarded volterra fredholm type integral inequalities are established which can be used as effective tools in the study of certain integral equations applications examples are also indicated

on some generalized retarded integral inequalities and the qualitative analysis of integral equations abstract in this paper we obtain estimates on solutions of some retarded integral inequalities which include retarded gronwall like inequalities and retarded volterra fredholm type inequalities some particular cases with weakly singular kernels are discussed we also apply our results to investigate the boundedness uniqueness and continuous dependence of solutions of some integral equations

on van der corput type lemmas for bessel and airy transforms and applications abstract this paper presents van der corput type lemmas for bessel and airy transforms with shifted jacobi weight functions these results on the asymptotic orders of the highly oscillatory integrals on the frequency are optimal furthermore from these estimates the convergence rates on filon type methods are easily derived

a generalization of retarded integral inequalities in two independent variables and their applications the object of this paper is to establish some nonlinear retarded integral inequalities in two independent variables which can be used as handy tools in the theory of partial differential and integral equations with time delays

existence of solutions to some retarded integrodifferential equations via the topological transversality theorem abstract we prove the existence of solutions to some retarded integrodifferential equations under some suitable conditions on the involved functions some applications of our results are also provided

on the implementation of the asymptotic filon type method for infinite integrals with oscillatory bessel kernels in this paper we consider the implementation of the asymptotic filon type method for the semi infinite highly oscillatory bessel integrals of the form 1 f x c v wx dx where c v wx denotes bessel function j v wx of the first kind y v wx of the second kind h v 1 wx and h v 2 wx of the third kind and modified bessel function k v wx of the second kind respectively f is a smooth function on 1 lim x f k x 0 k 0 1 2 and w is large by approximating f by a linear combination of negative integer powers so that the moments can be expressed by some special functions we complete the implementation of the method furthermore we give the error analysis of the method for computing the integrals the method is very efficient in obtaining very high precision approximations if w is sufficiently large numerical examples are provided to confirm our analysis

a fast algorithm for the electromagnetic scattering from a large cavity a fast algorithm is presented for solving electromagnetic scattering from a rectangular open cavity embedded in an infinite ground plane the medium inside the cavity is assumed to be vertically layered by introducing a transparent artificial boundary condition the problem in the open cavity is reduced to a bounded domain problem a simple finite difference method is then applied to solve the model helmholtz equation the fast algorithm is designed for solving the resulting discrete system in terms of the discrete fourier transform in the horizontal direction a gaussian elimination in the vertical direction and a preconditioning conjugate gradient method with a complex diagonal preconditioner for the indefinite interface system the existence and uniqueness of the finite difference solution are established for arbitrary wave numbers our numerical experiments for large numbers of mesh points up to 16 million unknowns and for large wave numbers e g between 100 and 200 wavelengths show that the algorithm is extremely efficient the cost for calculating the radar cross section which is of significant interest in practice is o m2 for an m times m mesh the proposed algorithm may be extended easily to solve discrete systems from other discretization methods of the model problem

stability and convergence of collocation schemes for retarded potential integral equations time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations solving such equations numerically is both complicated and computationally intensive and numerical methods often prove to be unstable collocation schemes are easier to implement than full finite element formulations but little appears to be known about their stability and convergence here we derive and analyze some new stable collocation schemes for the single layer equation for transient acoustic scattering and use spatial fourier and temporal laplace transform techniques to demonstrate that such stable schemes are second order convergent

some nonlinear gronwall bellman type retarded integral inequalities with power and their applications abstract the purpose of this paper is to establish some new retarded integral inequalities of gronwall bellman type which generalizes some known integral inequalities we use two lemmas to simplify the complex inequalities and give an upper bound estimation of the unknown function the inequalities given here can be used in the analysis of the qualitative properties of certain classes of differential equations and integral equations we apply our results to the boundedness of the solutions of integral equations and differential system

oscillation and stability of linear impulsive delay differential equations the main results of this paper are that the oscillation and the stability of a linear impulsive delay differential equation are equivalent respectively to the oscillation and the stability of a corresponding linear delay differential equation without impulses the results of this paper can be used to improve and derive some of the known results in the literature

a sparse discretization for integral equation formulations of high frequency scattering problems we consider two dimensional scattering problems formulated as an integral equation defined on the boundary of the scattering obstacle the oscillatory nature of high frequency scattering problems necessitates a large number of unknowns in classical boundary element methods in addition the corresponding discretization matrix of the integral equation is dense we formulate a boundary element method with basis functions that incorporate the asymptotic behavior of the solution at high frequencies the method exhibits the effectiveness of asymptotic methods at high frequencies with only few unknowns but retains accuracy for lower frequencies new in our approach is that we combine this hybrid method with very effective quadrature rules for oscillatory integrals as a result we obtain a sparse discretization matrix for the oscillatory problem moreover numerical experiments indicate that the accuracy of the solution actually increases with increasing frequency the sparse discretization applies to problems where the phase of the solution can be predicted a priori for example in the case of smooth and convex scatterers

asymptotic properties of solutions to some nonlinear integral equations of convolution type for a class of nonlinear integral equations of convolution type we give necessary and sufficient conditions for the boundedness of nonnegative solutions moreover conditions for the solution to converge asymptotically to a determined limit are obtained

integral inequalities for retarded volterra equations integral inequalities are very useful in the qualitative analysis of differential and integral equations starting with o lipovan a retarded gronwall like inequality and its applications j math anal appl 252 2000 389 401 several recent investigations see o lipovan a retarded integral inequality and its applications j math anal appl 285 2003 436 443 b g pachpatte explicit bounds on certain integral inequalities j math anal appl 267 2002 48 61 b g pachpatte on some retarded integral inequalities and applications j inequal pure appl math 3 2002 article 18 b g pachpatte on a certain retarded integral inequality and its applications j inequal pure appl math 5 2004 article 19 b g pachpatte on some new nonlinear retarded integral inequalities j inequal pure appl math 5 2004 article 80 were devoted to retarded integral inequalities in this paper we consider the case of retarded volterra integral equations we establish bounds on the solutions and by means of examples we show the usefulness of our results in investigating the asymptotic behaviour of the solutions

continuability of solutions of perturbed differential equations on considere les systemes d equations differentielles suivants e x f t x p x f t x g t x per x f t x h t ou x f g h sont des n vecteurs f t x g t x sont continues sur 0 r n et h t est continue sur 0 on demontre des theoremes sur la perturbation de la continuite des solutions de e a p et per

integral inequalities in two independent variables for retarded volterra equations in this paper we obtain some retarded volterra integral inequalities in two independent variables which can be used as tools in the theory of partial differential and integral equations with time delays

on retarded integral inequalities and their applications abstract the main results of j math anal appl 285 2003 436 443 are here generalized to the following retarded integral inequalities u m t c m m n m m n 0 t f s u n s w u s g s u n s d s and u m t c m m n m m n 0 t f s u n s w u s d s m m n 0 t g s u n s w u s d s where m n 0 are constants and t r 0 the results given here can be applied to the global existence of solutions to differential equations with time delay

some generalized integral inequalities and their applications in this paper we generalize some integral inequalities to more general situations we establish bounds on the solutions and by means of examples we show the usefulness of our results in investigating the global existence of the solutions of integral equations and differential equations

a retarded gronwall like inequality and its applications we prove a gronwall like inequality and present some of its applications to the qualitative study of retarded differential equations the problems of global continuation of the solutions and the existence of nonoscillatory solutions are considered by means of examples we show the usefulness of our results

the superconvergence of newton cotes rules for the hadamard finite part integral on an interval we study the general composite newton cotes rules for the computation of hadamard finite part integral with the second order singularity and focus on their pointwise superconvergence phenomenon i e when the singular point coincides with some a priori known point the convergence rate is higher than what is globally possible we show that the superconvergence rate of the composite newton cotes rules occurs at the zeros of a special function and prove the existence of the superconvergence points several numerical examples are provided to validate the theoretical analysis

a generalized retarded gronwall like inequality in two variables and applications to bvp abstract in this paper we establish a generalized retarded integral inequality of gronwall like type in two variables which includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals without assumption of monotonicity using our result we can solve both the integral inequality in w s cheung some new nonlinear inequalities and applications to boundary value problems nonlinear anal 64 2006 2112 2128 and the one in r p agarwal s deng w zhang generalization of a retarded gronwall like inequality and its applications appl math comput 165 2005 599 612 we apply our result to a boundary value problem of a partial differential equation for boundedness uniqueness and continuous dependence

efficient quadrature of highly oscillatory integrals using derivatives in this paper we explore quadrature methods for highly oscillatory integrals generalizing the method of stationary phase we expand such integrals into asymptotic series in inverse powers of the frequency the outcome is two families of methods one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of filon filon 1928 proc r soc edinb 49 38 47 1 both kinds of methods approximate the integral as a linear combination of function values and derivatives with coefficients that may depend on frequency we determine asymptotic properties of these methods proving perhaps counterintuitively that their performance drastically improves as frequency grows the paper is accompanied by numerical results that demonstrate the potential of this set of ideas 1 ref 4

some new nonlinear inequalities and applications to boundary value problems abstract in this paper we establish some new nonlinear integral inequalities of the gronwall bellman ou iang type in two variables these on the one hand generalizes and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of differential equations we illustrate this by applying our new results to certain boundary value problem

on certain integral inequalities in two independent variables for retarded equations in this paper we establish some new retarded integral inequalities in two independent variables which can be used as tools in the theory of integral equations with time delays applications are given to illustrate the usefulness of the inequalities

estimation of unknown function of a class of integral inequalities and applications in fractional integral equations in this paper we establish a class of iterated integral inequalities which includes a nonconstant term outside the integrals the upper bound of the embedded unknown function is estimated explicitly by adopting novel analysis techniques such as change of variable amplification method differential and integration the derived result can be applied in the study of qualitative properties of solutions of fractional integral equations

discrete non linear inequalities and applications to boundary value problems in this paper we generalize some existing discrete gronwall bellman ou iang type inequalities to more general situations these are in turn applied to study the boundedness uniqueness and continuous dependence of solutions of certain discrete boundary value problem for difference equations

letter to the editor on quadrature of bessel transformations a method for integral transformations of highly oscillatory functions bessel functions is presented it is based on the filon type method and the decay of the error can be increased as increases the effectiveness and accuracy of the quadrature is tested for both large arguments and higher orders of bessel functions in the case where the orders are nonnegative integers

on some new integral inequalities and their applications the aim of the present paper is to establish some new integral inequalities in two independent variables which provide explicit bounds on unknown functions the inequalities given here can be used as tools in the qualitative theory of certain partial differential equations

on retarded integral inequalities in two independent variables and their applications the object of this paper is to establish some nonlinear retarded integral inequalities in two independent variables which can be used as handy tools in the theory of partial differential and integral equations with time delays

some new delay integral inequalities and their applications the aim of the present paper is to establish some new delay integral inequalities which provide explicit bounds on unknown functions the inequalities given here can be used as tools in the qualitative theory of certain delay differential equations and delay integral equations

knot floer homology detects genus one fibred knots ozsv ath and szab o conjectured that knot floer homology detects fibred knots we propose a strategy to approach this conjecture based on contact topology and gabai s theory of sutured manifold decomposition we implement this strategy for genus one knots obtaining as a corollary that if rational surgery on a knot k gives the poincar e homology sphere sigma 2 3 5 then k is the left handed trefoil knot

a retarded integral inequality and its applications we prove a retarded nonlinear integral inequality and present some applications of it to the global existence of solutions to differential equations with time delay