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

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

fast fourier galerkin methods for solving singular boundary integral equations numerical integration and precondition we develop a fast fully discrete fourier galerkin method for solving a class of singular boundary integral equations we prove that the number of multiplications used in generating the compressed matrix is o nlog 3n and the solution of the proposed method preserves the optimal convergence order o n t where n is the order of the fourier basis functions used in the method and t denotes the degree of regularity of the exact solution moreover we propose a preconditioning which ensures the numerical stability when solving the preconditioned linear system numerical examples are presented to confirm the theoretical estimates and to demonstrate the approximation accuracy and computational efficiency of the proposed algorithm

parameter dependence of solutions of differential equations on spaces of distributions and the splitting of short exact sequences we show that a linear partial differential operator with constant coefficients p d is surjective on the space of e valued ultra distributions over an arbitrary convex set if e is a nuclear frechet space with property dn in particular this holds if e is isomorphic to the space of tempered distributions s or to the space of germs of holomorphic functions over a one point set h 0 this result has an interpretation in terms of solving the scalar equation p d u f such that the solution u depends on parameter whenever the right hand side f also depends on the parameter in the same way a suitable analogue for surjective convolution operators over rd is obtained as well to get the above results we develop a splitting theory for short exact sequences of the form 0 x y z 0 where z is a frechet schwartz space and x y are pls spaces like the spaces of distributions or real analytic functions or their subspaces in particular an extension of the dn splitting theorem of vogt and wagner is obtained

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

fast computation of a class of highly oscillatory integrals in this paper the problem of numerical evaluation of a class of highly oscillatory integrals a b f x x a b x k 1 n x k k e i x dx where 1 k 1 a k b and 1 has been discussed based on analytic continuation the integrals are transformed into the problems of integrating the forms on 0 with the integrand does not oscillate and decays exponentially fast which can be efficiently computed by using general gauss laguerre quadrature rule theoretical results and numerical experiments perform that the method is very efficient in obtaining very high precision approximations if is sufficiently large

hadamard type operators on temperate distributions abstract we study hadamard operators on s r d and give a complete characterization they have the form l s s t where means the multiplicative convolution and t o h r d the space of distributions which are rapidly decreasing in infinity and at the coordinate hyperplanes we use the theory of convolution operators on the space y r d of exponentially decreasing c functions on r d see 7 17 and the exponential transformation to characterize the hadamard operators on s q q the positive quadrant and this result we use as a building block for our main result

efficient calculation and asymptotic expansions of many different oscillatory infinite integrals abstract this paper introduces and analyzes quadrature rules and asymptotic expansions of a few highly oscillatory infinite integrals we first derive a series of useful asymptotic expansions in inverse powers of the frequency parameter which clarify the large behavior of these integrals then based on the resulting asymptotic expansions two different interpolatory quadrature rules are given one is the so called filon type methods based on standard hermite interpolation of the non oscillatory and non singular part of the integrands at equidistant nodes the other is the filon clenshaw curtis type method fcc by using special hermite interpolation at n 1 clenshaw curtis points and the fast computation of modified moments the interpolation coefficients needed in the fcc method can be computed by a numerically stable algorithm in o nlog n operations based on fast fourier transform fft the required modified moments can be accurately and efficiently calculated by some recurrence relation formulae moreover for these quadrature rules their error analyses in inverse powers of the frequency are provided the presented methods share the advantageous property that the accuracy improves greatly for fixed n as increases numerical examples show the accuracy and efficiency of the proposed methods

on the computation of fourier transforms of singular functions abstract in this paper we consider a method for the computation of finite fourier transforms of functions having endpoint singularities of algebraic or logarithmic type this method may be considered as a modification of clenshaw curtis quadrature the nonoscillatory and nonsingular part of the integrand is replaced by a truncated chebyshev series approximation and the integral is then evaluated using recurrence relations the numerical stability of these recurrence relations is investigated

hadamard type operators on spaces of real analytic functions in several variables abstract we consider multipliers on the space of real analytic functions of several variables a r d open i e linear continuous operators for which all monomials are eigenvectors if zero belongs to these operators are just multipliers on the sequences of taylor coefficients at zero in particular euler differential operators of arbitrary order are multipliers we represent all multipliers via a kind of multiplicative convolution with analytic functionals and characterize the corresponding sequences of eigenvalues as moments of suitable analytic functionals moreover we represent multipliers via suitable holomorphic functions with laurent coefficients equal to the eigenvalues of the operator we identify in some standard cases what topology should be put on the suitable space of analytic functionals in order that the above mentioned isomorphism becomes a topological one when the space of multipliers inherits the topology of uniform convergence on bounded sets from the space of all endomorphisms on a we also characterize in the same cases when the discovered topology coincides with the classical topology of bounded convergence on the space of analytic functionals we provide several examples of multipliers and show surjectivity results for multipliers on a if r d

numerical integration of oscillatory airy integrals with singularities on an infinite interval abstract this work is devoted to the quadrature rules and asymptotic expansions for two classes of highly oscillatory airy integrals on an infinite interval we first derive two important asymptotic expansions in inverse powers of the frequency then based on structure characteristics of the two asymptotic expansions in inverse powers of the frequency both the so called filon type method and the more efficient clenshaw curtis filon type method are introduced and analyzed the required moments in the former can be explicitly expressed by the meijer g functions the latter can be implemented in o n log n operations based on fast fourier transform fft and fast computation of the modified moments here we can construct two useful recurrence relations for computing the required modified moments accurately with the help of the airy s equation and some properties of the chebyshev polynomials particularly we also provide their error analyses in inverse powers of the frequency furthermore the presented error analysis shows the advantageous property that the accuracy improves greatly as increases numerical examples are provided to illustrate the efficiency and accuracy of the proposed methods

computation of integrals with oscillatory singular factors of algebraic and logarithmic type in this paper we present the clenshaw curtis filon methods and the higher order methods for computing many classes of oscillatory integrals with algebraic or logarithmic singularities at the two endpoints of the interval of integration the methods first require an interpolant of the nonoscillatory and nonsingular parts of the integrands in n 1 clenshaw curtis points then the required modified moments can be accurately and efficiently computed by constructing some recurrence relations moreover for these quadrature rules their absolute errors in inverse powers of the frequency are given the presented methods share the advantageous property that the accuracy improves greatly for fixed n as increases numerical examples show the accuracy and efficiency of the proposed methods

evaluation of cauchy principal value integrals of oscillatory kind abstract this paper studies the evaluation of cauchy principal value c p v integrals of oscillatory kind a b f x x e i x d x where a b the singularity of the c p v integral is transferred to an individual oscillatory integral that can be analytically calculated the remaining integral which is non singular can be well calculated with a stable and fast quadrature method namely the improved levin method numerical examples confirm the advantages of the proposed method

on the evaluation of cauchy principal value integrals of oscillatory functions the problem of the numerical evaluation of cauchy principal value integrals of oscillatory functions 1 1e i w xf x x tdx where 1 t 1 has been discussed based on analytic continuation if f is analytic in a sufficiently large complex region g containing 1 1 the integrals can be transformed into the problems of integrating two integrals on 0 with the integrand that does not oscillate and that decays exponentially fast which can be efficiently computed by using the gauss laguerre quadrature rule the validity of the method has been demonstrated in the provision of two numerical experiments and their results

method for numerical integration of rapidly oscillating functions in diffraction theory the quadrature of general highly oscillatory integrals is a relatively complicated computational problem that occurs in a wide range of practical applications e g in physics chemistry and image analysis it is often necessary to use a high number of nodal points with classical quadrature formulas in order to achieve a required accuracy of numerical integration of rapidly oscillating functions however an increase in integration points leads to a larger computational time our work describes and analyses a method for numerical integration of rapidly oscillating functions which enables to obtain the required accuracy with a smaller number of nodal points than classical integration rules and with a relatively simple integration formula the proposed method is generally suitable for integration of rapidly oscillating functions in various applications the method is demonstrated in examples of calculation of the diffraction integral in optics where the integrand is often a rapidly oscillatory function the results can be readily adapted to similar problems from other fields copyright 2009 john wiley sons ltd

efficient integration for a class of highly oscillatory integrals abstract this paper presents some quadrature methods for a class of highly oscillatory integrals whose integrands may have singularities at the two endpoints of the interval one is a filon type method based on the asymptotic expansion the other is a clenshaw curtis filon type method which is based on a special hermite interpolation polynomial and can be evaluated efficiently in o n log n operations where n 1 is the number of clenshaw curtis points in the interval of integration in addition we derive the corresponding error bound in inverse powers of the frequency for the clenshaw curtis filon type method for the class of highly oscillatory integrals the efficiency and the validity of these methods are testified by both the numerical experiments and the theoretical results

some results on surjectivity of augmented differential operators abstract considering a problem of bonet and domanski 2006 1 problem 9 1 we prove that for a polynomial p on r 2 surjectivity of the differential operator p d on d x implies surjectivity of the augmented operator p d on d x r where p x 1 x 2 x 3 p x 1 x 2 moreover we give a sufficient geometrical condition on an open subset x of r d such that an analogous implication is true for arbitrary dimension d in case of p being homogeneous semi elliptic or of principal type

delaminating quadrature method for multi dimensional highly oscillatory integrals a delaminating quadrature method for multi dimensional highly oscillatory integrals is put forward in comparison with the levin method the new method has the benefits of being numerically stable and having a high computational speed numerical experiments testified the advantages of this method

on the fundamental solution of the operator of dynamic linear thermoelasticity abstract the fundamental matrix of the system of partial differential operators which governs the diffusion of heat and the strains in elastic media is investigated qualitatively singular support and explicit representation formulae in the form of simple definite integrals resp series are given

computation of oscillatory integrals with an exponential kernel and jacobi type singularities abstract in this communication two rules are considered for computing oscillatory integrals with an exponential kernel and jacobi type singularities an asymptotic rule is presented with the aid of integration by parts an asymptotic filon type quadrature rule is constructed on the basis of a two point taylor interpolation polynomial error analysis on the proposed rules in inverse powers of large frequency is provided numerical examples are provided to illustrate that the two rules are very efficient in obtaining very high precision approximations if is sufficiently large

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

letter to the editor on the filon and levin methods for highly oscillatory integral this paper shows that for any suitably smooth function f x and arbitrarily selected interpolation nodes c1 c2 cv in a b the filon method and the levin method for with the polynomial interpolation approach are identical when g x is a linear function based on this result a new efficient levin quadrature for is presented

composition operators with closed range for one dimensional smooth symbols abstract we give a sufficient condition for a smooth not necessarily injective function r r ensuring that the composition operator c c r c r f f has closed range this generalizes in a special case the result of kenessey and wengenroth which gave a description of composition operators c with closed range for smooth injective symbols r r d

inheritance of surjectivity for partial differential operators on spaces of real analytic functions abstract for an open set r n let a be the space of real analytic functions on improving our previous results we prove a new quantitative characterization of the linear partial differential operators p d which are surjective on a this implies that p d is surjective on a r n if p d is surjective on a for some further inheritance properties for the surjectivity of p d on a are also obtained

interpolation of holomorphic functions and surjectivity of taylor coefficient multipliers abstract we prove criteria for global analytic solvability for some euler type partial differential equations a class of linear partial differential equations with variable polynomial coefficients this is based on a very general mellin type theory developed in the present paper which is valid for analytic functionals in particular distributions of compact support substituting the classical mellin transform as a consequence we get a characterization of moment sequences of analytic functionals by interpolation of holomorphic functions satisfying some restrictive growth conditions this allows to study the so called hadamard type multipliers on real analytic functions of several variables i e operators for which monomials are eigenvectors we characterize multiplier sequences i e sequences of eigenvalues of multipliers by interpolation and describe the image of some multipliers especially the image of the principal examples of hadamard multipliers euler type partial differential operators we also get some results showing that the image remains unchanged after some perturbation of the original hadamard type operator

the geometry of analytic varieties satisfying the local phragmen lindelof condition and a geometric characterization of the partial differential operators that are surjective on ℝ the local phragmen lindelof condition for analytic subvarieties of c n at real points plays a crucial role in complex analysis and in the theory of constant coefficient partial differential operators as hormander has shown here necessary geometric conditions for this phragmen lindelof condition are derived they are shown to be sufficient in the case of curves in arbitrary dimension and of surfaces in c 3 the latter result leads to a geometric characterization of those constant coefficient partial differential operators which are surjective on the space of all real analytic functions on r 4

differentiation formulas of some hypergeometric functions with respect to all parameters in this work we present two methods to derive some differentiation formulas of the generalized hypergeometric function m f n a 1 a m b 1 b n z including the most commonly used gauss hypergeometric function 2 f 1 z and kummer confluent hypergeometric function 1 f 1 z as special cases with respect to all parameters we first briefly describe the direct derivative method for the convergent power series of hypergeometric functions secondly we mainly focus on the differential equation method which is based on differentiating the generalized hypergeometric differential equation with respect to parameters particularly by using the differential equation method some general analytical expressions of any sth derivatives with respect to single parameter can be deduced by induction in s moreover we can obtain all the mixed derivatives of higher order very conveniently finally examples are given to illustrate the usefulness of these derivatives in mathematics physics and other related fields numerical examples for computing those singular oscillatory integrals presented in kang et al 2013 and kang and ling in press in turn verify that the approximation value of the required derivatives can be of great precision and show the correctness of differentiation formulas obtained by the proposed methods

phragmen lindelof principles on algebraic varieties from several results in recent years starting with h6rmander s characterization of the constant coefficient partial differential equations p d u f that have a real analytic solution u for every real analytic function f it has become clear that certain properties of the partial differential operator p d are equivalent to estimates of phragmen lindelof type for plurisubharmonic functions on the algebraic variety

on the explicit calculation of fundamental solutions the notion of a fundamental solution in the sequel abbreviated as fs gradually became clearer during the 19th and 20th century it is only in the setting of laurent schwartz theory of distributions that fss can be defined in general and can be applied via the convolution of distributions to the solution of linear partial differential equations with constant coefficients in part the relevant concepts were worked out by john horvath cf 20 25 in this survey paper we first review some important steps in the history of fss second we explain why the singular support of the fss of homogeneous operators p is just the dual hypersurface of the zero set of p if p is of principal type this is of fundamental importance in the third part where we present some recent results in the calculation of fss of homogeneous cubic and quartic operators in three dimensions finally we discuss what is known for the system of crystal optics where similarly as in dynamic anisotropic elasticity many questions are still open

on the calculation of highly oscillatory integrals with an algebraic singularity abstract in this paper we explore the evaluation of highly oscillatory integrals containing an algebraic singularity based on the change of variable t x r where r is a positive real number and the analytical continuation robust numerical results demonstrate the accuracy and effectiveness of the proposed approach

hadamard multipliers on spaces of real analytic functions abstract we consider multipliers on spaces of real analytic functions of one variable i e maps for which monomials are eigenvectors we characterize sequences of complex numbers which are sequences of eigenvalues for some multiplier we characterize invertible multipliers in particular we find which euler differential operators of infinite order have global analytic solutions on the real line we present a number of examples where our theory applies in some cases we give algorithms for solving the respective equations perturbation results for solvability are presented

meshless methods for two dimensional oscillatory fredholm integral equations abstract in this paper a meshless solution procedure incorporating delaminating quadrature method for two dimensional highly oscillatory fredholm integral equation is put forward the proposed method is an extension of our earlier findings of meshless methods for two dimensional oscillatory fredholm integral equation having kernel function free of stationary phase point s siraj ul islam et al 2015 the current method deals not only with the kernels having no stationary phase point s but also with the kernels having stationary phase point s in the context of highly oscillatory integral equations the new method is numerically stable and computationally fast numerical experiments are presented to testify the robustness and efficiency of the proposed method

hyperfunction fundamental solutions of surjective convolution operators on real analytic functions let a r the surjectivity of the convolution operator t on real analytic functions is characterized by two equivalent conditions 1 t admits hyperfunctional elementary solutions e and e which are analytic on an angular neighbourhood of c respectively c for some c 0 2 the fourier tranform of satisfies a slowly decreasing condition and the following alternative there is 0 such that im z re z or im z o z for any z with z 0 was obtained for a 0 by napalkov rudakov and meyer

on numerical computation of integrals with integrands of the form f x sin w xr on 0 1 with existing numerical integration methods and algorithms it is difficult in general to obtain accurate approximations to integrals of the form 0 1f x sin wx r dxor 0 1f x cos wx r dx r 0 where f is a sufficiently smooth function on 0 1 gautschi has developed software as scripts in matlab for computing these integrals for the special case r w 1 in this paper an algorithm as a mathematica program is developed for computing these integrals to arbitrary precision for any given values of the parameters in a certain range numerical examples are given of testing the performance of the algorithm program

filon clenshaw curtis rules for highly oscillatory integrals with algebraic singularities and stationary points in this paper we propose and analyze composite filon clenshaw curtis quadrature rules for integrals of the form i k a b f g int a b f x exp mathrm i kg x mathrm d x where k geq 0 f may have integrable singularities and g may have stationary points our composite rule is defined on a mesh with m subintervals and requires mn 1 evaluations of f it satisfies an error estimate of the form c n k r m n 1 r where r is determined by the strength of any singularity in f and the order of any stationary points in g and c n is a constant which is independent of k and m but depends on n the regularity requirements on f and g are explicit in the error estimates for fixed k the rate of convergence of the rule as m rightarrow infty is the same as would be obtained if f was smooth moreover the quadrature error decays at least as fast as k rightarrow infty as does the original integral i k a b f g for the case of nonlinear oscill

on the evaluation of highly oscillatory integrals by analytic continuation we consider the integration of one dimensional highly oscillatory functions based on analytic continuation rapidly converging quadrature rules are derived for a general class of oscillatory integrals with an analytic integrand the accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency and for the case of an increasing number of points and fixed frequency these results are then used to obtain quadrature rules for more general oscillatory integrals i e for functions that exhibit some smoothness but that are not analytic the approach described in this paper is related to the steepest descent method but it does not employ asymptotic expansions it can be used for small or moderate frequencies as well as for very high frequencies the approach is compared with the oscillatory integration techniques recently developed by iserles and norsett

efficient methods for highly oscillatory integrals with weakly singular and hypersingular kernels abstract in this paper we present a special clenshaw curtis filon ccf type scheme for approximation of highly oscillatory integrals with weak and hypersingular kernels the non oscillatory and nonsingular part of the integrand is replaced by a special hermite interpolation polynomial error bounds with respect to the frequency k and the number of the clenshaw curtis points n are considered the overall computational complexity for the scheme is o nlog n numerical experiments support the theoretical analysis

implementing the complex integral method with the transformed clenshaw curtis quadrature abstract gauss laguerre quadrature plays an important role in implementing the numerical steepest decent method for computing highly oscillatory integrals however it consumes too much time when the analytic region of the integrand is narrow in this paper we analyze the convergence rate of the transformed clenshaw curtis quadrature and show that this method also shares the property that the higher the oscillation the better the calculation moreover it is efficient to compute highly oscillatory integral with a nearly singularity numerical tests are performed to verify our given results

on graded meshes for weakly singular volterra integral equations with oscillatory trigonometric kernels we compare the collocation methods on graded meshes with that on uniform meshes for the solution of the weakly singular volterra integral equation of the second kind with oscillatory trigonometric kernels due to in general unbounded derivatives at the left endpoint of the interval of integration we should approximate the solution of the integral equation by collocation methods on graded meshes however we show that this non smooth behavior of the solution has little effect on the approximate solution when the kernels of integral equation are highly oscillatory numerical examples are given to confirm the proposed results

a note on a recent study of oscillatory integration rules abstract in a recent comparison of methods of highly oscillatory numerical integration the authors appear to have overlooked some works which seem to compete effectively with several of those described it is the intention briefly to illuminate this with a number of examples reworked in so doing it is found that three commonly used fortran compilers give curiously differing results on a number of occasions

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

a user friendly method for computing indefinite integrals of oscillatory functions for indefinite integrals q f x 1 x f t e i t d t x 1 1 torii and the first author hasegawa and torii 1987 developed a quadrature method of clenshaw curtis c c type its improvement was made and combined with sidi s m w transformation by sidi and the first author hasegawa and sidi 1996 to compute infinite oscillatory integrals the improved method per se however has not been elucidated in its attractive features which here we reveal with new results and its detailed algorithm a comparison with a method of c c type for definite integrals q f 1 due to dominguez et al 2011 suggests that a smaller number of computations is required in our method this is achieved by exploiting recurrence and normalization relations and their associated linear system we show their convergence and stability properties and give a verified truncation error bound for a result computed from the linear system with finite dimension for f z analytic on and inside an ellipse in the complex plane z the error of the approximation to q f x of the improved method is shown to be bounded uniformly numerical examples illustrate the stability and performance of the method

efficient filon type methods for int_a bf x rm e rm i omega g x rm d x based on the transformation y g x some new efficient filon type methods for integration of highly oscillatory function int a bf x rm e rm i omega g x rm d x with an irregular oscillator are presented one is a moment free filon type method for the case that g x has no stationary points in a b the others are based on the filon type method or the asymptotic method together with filon type method for the case that g x has stationary points the effectiveness and accuracy are tested by numerical examples

numerical evaluation of a class of highly oscillatory integrals involving airy functions in this paper we consider the implementation of the clenshaw curtis filon type method for a class of highly oscillatory integrals 0 1 x 1 x f x ai x dx where ai x is an airy function and 1 1 by replacing f by its chebyshev interpolation polynomial at the clenshaw curtis points so that the modified moments can be computed by recursive formula based on special functions an efficient and stable method for this integral is presented error analysis for the presented method is given moreover the method shares the property that the larger the the higher the precision theoretical results and numerical examples show that the method is very efficient in obtaining very high precision approximations if is sufficiently large

surjectivity of euler operators on temperate distributions abstract euler operators are partial differential operators of the form p where p is a polynomial and j x j x j we show that every non trivial euler operator is surjective on the space of temperate distributions on r d this is in sharp contrast to the behaviour of such operators when acting on spaces of differentiable or analytic functions

the singular terms in the fundamental matrix of crystal optics we derive an explicit formula for the singular part of the fundamental matrix of crystal optics it consists of a singularity remaining fixed at the origin x 0 of delta terms located on the positively curved parts of the wave surface the well known fresnel surface and of a cauchy principal value distribution on the negatively curved part of the wave surface

on conical refraction in hexagonal and cubic media the occurrence of conical refraction in hexagonal and in cubic elastic media is investigated by the method of localizations this is accomplished by generalizing the atiyah bott garding theory from the scalar case to hyperbolic square systems of linear partial differential systems with constant coefficients

oscillation free solutions to volterra integral and integro differential equations with periodic force terms volterra equations governed by oscillatory functions arise frequently in applied fields to construct efficient algorithms for solving these equations oscillatory properties of their solutions should be studied in this paper oscillatory orders of solutions to a class of highly oscillatory volterra integral equations are presented then some modified solutions are given with the help of asymptotic expansions of highly oscillatory integrals finally theoretical analysis verifies these modified solutions enjoy less oscillation than the original one and extensions to highly oscillatory volterra integro differential equations are considered

computation of integrals with oscillatory and singular integrands using chebyshev expansions we present a general method for computing oscillatory integrals of the form 1 1f x g x e i w xdx where f is sufficiently smooth on 1 1 w is a positive parameter and g is a product of singular factors of algebraic or logarithmic type based on a chebyshev expansion of f and the properties of chebyshev polynomials the proposed method for such integrals is constructed with the help of the expansion of the oscillatory factor e i w x furthermore due to numerically stable recurrence relations for the modified moments the devised scheme can be employed to compute oscillatory integrals with algebraic or logarithmic singularities at the end or interior points of the interval of integration numerical examples are provided to confirm our analysis

euler type partial differential operators on real analytic functions abstract we describe all euler partial differential operators which act on the space of real analytic functions and we identify them among the taylor multipliers on these spaces partial differential operators of the form f a d f d d 1 1 d d d d j f x q j 1 x j f x j x q j 0 x j f x where q j 1 q j 0 a j b j c are called generalized euler differential operators whenever all d j are conjugate to the classical euler differential f t t f t we find criteria when a linear differential operator with analytic coefficients on the space of real analytic functions is a generalized euler differential operator it turns out that this happens for a wide variety of linear operators with variable coefficients using our earlier results on solvability of classical euler operators of finite order we then study the question of surjectivity or big image for generalized euler partial differential operators with analytic coefficients i e global solvability of the considered equations in spaces of real analytic functions