a family of fifth order runge kutta pairs the construction of a runge kutta pair of order 5 4 with the minimal number of stages requires the solution of a nonlinear system of 25 order conditions in 27 unknowns we define a new family of pairs which includes pairs using 6 function evaluations per integration step as well as pairs which additionally use the first function evaluation from the next step this is achieved by making use of kutta s simplifying assumption on the original system of the order conditions i e that all the internal nodes of a method contributing to the estimation of the endpoint solution provide at these nodes cost free second order approximations to the true solution of any differential equation in both cases the solution of the resulting system of nonlinear equations is completely classified and described in terms of five free parameters optimal runge kutta pairs with respect to minimized truncation error coefficients maximal phase lag order and various stability characteristics are presented these pairs were selected under the assumption that they are used in local extrapolation mode the propagated solution of a problem is the one provided by the fifth order formula of the pair numerical results obtained by testing the new pairs over a standard set of test problems suggest a significant improvement in efficiency when using a specific pair of the new family with minimized truncation error coefficients instead of some other existing pairs

an operator valued t1 theory for symmetric czos abstract we provide a natural bmo criterion for the l 2 boundedness of calderon zygmund operators with operator valued kernels satisfying a symmetric property our arguments involve both classical and quantum probability theory in the appendix we give a proof of the l 2 boundedness of the commutators r j b whenever b belongs to the bourgain s vector valued bmo space where r j is the j th riesz transform a common ingredient is the operator valued haar multiplier studied by blasco and pott

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

high order zero dissipative runge kutta nystro m methods a new runge kutta nystrom pair of orders eight and six is presented here its main advantage is that it is of zero dissipation so it possesses an interval of periodicity numerical results over a set of problems demonstrate the superiority of the method in problems with periodic solution

pseudo localization of singular integrals and noncommutative calderon zygmund theory abstract the weak type 1 1 boundedness of singular integrals acting on matrix valued functions has remained open since the 1980s mainly because the methods provided by the vector valued theory are not strong enough in fact we can also consider the action of generalized calderon zygmund operators on functions taking values in any other von neumann algebra our main tools for its solution are two first the lack of some classical inequalities in the noncommutative setting forces to have a deeper knowledge of how fast a singular integral decreases l 2 sense outside of the support of the function on which it acts this gives rise to a pseudo localization principle which is of independent interest even in the classical theory second we construct a noncommutative form of calderon zygmund decomposition by means of the recent theory of noncommutative martingales this is a corner stone in the theory as application we obtain the sharp asymptotic behavior of the constants for the strong l p inequalities also unknown up to now our methods settle some basics for a systematic study of a noncommutative calderon zygmund theory

families of explicit two step methods for integration of problems with oscillating solutions we study a new sixth algebraic order explicit numerov type family of methods using every free parameter of the family even its nodes we manage to derive two methods the first with phase lag of order 12 while the other method has one stage less this is a considerable improvement over the 10th order phase lag order methods found in the literature until now numerical experiments confirm the superiority of our new methods over older methods

an operator valued tb theorem we establish conditions similar to the tb theorem of david journe and semmes which guarantee the boundedness of an integral transformation t with l x valued kernel on lxp rn where 1 p and x is a banach space with the unconditionality property of martingale differences umd

a t 1 theorem for non integral operators let x be a space of homogeneous type and let l be a sectorial operator with bounded holomorphic functional calculus on l 2 x we assume that the semigroup e tl t 0 satisfies davies gaffney estimates associated with l are certain approximations of the identity we call an operator t a non integral operator if compositions involving t and these approximations satisfy certain weighted norm estimates the davies gaffney and the weighted norm estimates are together a substitute for the usual kernel estimates on t in calderon zygmund theory in this paper we show under the additional assumption that a vertical littlewood paley stein square function associated with l is bounded on l 2 x that a non integral operator t is bounded on l 2 x if and only if t 1 in bmo l x and t 1 in bmo l x here bmo l x and bmo l x denote the recently defined bmo x spaces associated with l that generalize the space bmo x of john and nirenberg generalizing a recent result due to f bernicot we show a second version of a t 1 theorem under weaker off diagonal estimates which gives a positive answer to a question raised by him as an application we prove l 2 x boundedness of a paraproduct operator associated with l we moreover study criterions for a t b theorem to be valid

revised trigonometrically fitted two step hybrid methods with equation dependent coefficients for highly oscillatory problems for the numerical integration of highly oscillatory problems revised trigonometrically fitted two step hybrid methods rtftsh with equation dependent coefficients are considered the local truncation errors stability and phase properties of the new method are analyzed a feature of the new type of the methods is that the errors in the internal stages are assumed to contribute to the accuracy of the update a new revised method rtftsh4 of algebraic order four and phase lag order four is derived numerical experiments are reported to show that the new method rtftsh4 is much more efficient and robust than the standard fourth order method stftsh4

runge kutta nystrom interpolants for the numerical integration of special second order periodic initial value problems abstract the theory of the phase lag analysis for runge kutta nystrom methods and runge kutta nystrom interpolants is developed in this paper also a new runge kutta nystrom method with interpolation properties is developed to integrate second order differential equations of the formu t f t u when they possess an oscillatory solution

n fold supersymmetry and quasi solvability associated with x2 laguerre polynomials we construct a new family of quasi solvable and n fold supersymmetric quantum systems where each hamiltonian preserves an exceptional polynomial subspace of codimension 2 we show that the family includes as a particular case the recently reported rational radial oscillator potential whose eigenfunctions are expressed in terms of the x2 laguerre polynomials of the second kind in addition we find that the two kinds of the x2 laguerre polynomials are ingeniously connected with each other by the n fold supercharge

an optimized runge kutta method for the solution of orbital problems we present a new explicit runge kutta method with algebraic order four minimum error of the fifth algebraic order the limit of the error is zero when the step size tends to zero infinite order of dispersion and eighth order of dissipation the efficiency of the newly constructed method is shown through the numerical results of a wide range of methods when these are applied to well known periodic orbital problems

runge kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics in this paper a new runge kutta method with minimal dispersion and dissipation error is developed the chebyshev pseudospectral method is utilized using spatial discretization and a new fourth order six stage runge kutta scheme is used for time advancing the proposed scheme is more efficient than the existing ones for acoustic computations

mixed collocation methods for y f x y inline equatio abstract the collocation methods introduced here are based on linear combinations of trigonometric functions and powers the motivation is to provide better approximations for oscillatory solutions of initial value problems for differential equations of the special form y f x y the resulting methods for two or more collocation points are implicit runge kutta nystrom methods with coefficients which depend on both the fitted angular frequency and the steplength algebraic and trigonometric order conditions are considered and the stability properties of some methods are examined particular mixed collocation methods and other methods for the same class of problems are compared by applying them to a variety of test problems

zero dissipative phase fitted hybrid methods for solving oscillatory second order ordinary differential equations in this paper zero dissipative phase fitted two step hybrid methods are developed for the integration of second order periodic initial value problems the phase fitted hybrid methods are constructed using similar approaches introduced by papadopoulos et al 1 this new methods are based on the existing explicit hybrid methods of order four and six numerical illustrations indicate that the new methods are much more efficient than the existing methods

symplectic partitioned runge kutta methods with the phase lag property abstract in this work specially tuned symplectic partitioned runge kutta sprk methods with minimum phase lag and phase fitted have been considered the general framework for constructing sprk methods with minimum phase lag is given

minimum storage runge kutta schemes for computational acoustics abstract in this paper an implementation of some runge kutta schemes that requires 2n storage where n is the number of degrees of freedom of the system is proposed it is shown that some runge kutta schemes for wave propagation proposed by hu hussaini and manthey 1 can be written by using our implementation thus we provide an alternative to the 2n storage implementation proposed by stanescu and habashi 2 which follows the approach of williamson 3 carpenter and kennedy 4 and others in addition new runge kutta schemes with small dissipation and dispersion errors that can be written also by using our implementation are given finally the results of some numerical experiments are presented to compare the behaviour of these methods

a trigonometrically fitted explicit hybrid method for the numerical integration of orbital problems abstract in this paper we present a kind of trigonometrically fitted explicit two step hybrid method which achieves algebraic order six the new method is zero dissipative phase fitted and almost p stable numerical experiments from its application to well known periodic orbital problems show that the new method is more competitive when comparing with the codes proposed by franco j m franco a class of explicit two step hybrid methods for second order ivps j comput appl math 187 2006 41 57 anastassi and simos z a anastassi t e simos a trigonometrically fitted runge kutta method for the numerical solution of orbital problems new astron 10 2005 301 309

a two step method for the numerical solution of the radial schrodinger equation abstract a two step method is developed for the numerical solution of the radial schrodinger equation numerical results obtained for the integration of resonance problem and eigenvalue problem show that this new method is better than other similar methods

neural networks with multidimensional transfer functions we present a new type of neural network nn where the data for the input layer are the value x spl epsiv r the vector y spl epsiv r sup m associated to an initial value problem ivp with y x f y x and a steplength h then the stages of a runge kutta rk method with trainable coefficients are used as hidden layers for the integration of the ivp using f as transfer function we take as output two estimations y y spl circ of ivp at the point x h training the rk method at some test problems and counting the cost of the method under the coefficients used we may achieve coefficients that help the method to perform better at a wider class of problems

an adapted explicit hybrid method of numerov type for the numerical integration of perturbed oscillators in 1971 scheifele has designed a method for the integration of perturbed oscillators by refining the classical method of power series in this paper we examine how we can extend the algorithm of scheifele for the construction of a fourth order hybrid method without explicit first derivatives an application to some popular test problems such as the satellite problem shows the properties of the developed algorithm

optimal runge kutta methods for first order pseudospectral operators new runge kutta methods for method of lines solution of systems of ordinary differential equations arising from discretizations of spatial derivatives in hyperbolic equations by chebyshev or modified chebyshev methods are introduced these runge kutta methods optimize the time step necessary for stable solutions while holding dispersion and dissipation fixed it is found that maximizing dispersion minimizes dissipation and vice versa optimal methods with respect to large stability intervals on the imaginary axis and with respect to the eigenvalue spectra of the underlying pseudospectral discretizations are developed in the latter case stability regions are optimized to include the outliers of the spatial operators performance on a model problem in computational aeroacoustics is evaluated the optimized schemes have two more function evaluations per timestep than the standard fourth order runge kutta method but allow timesteps up to 1 7 times larger moreover dissipation and dispersion are reduced

explicit runge kutta methods for initial value problems with oscillating solutions new pairs of embedded runge kutta methods specially adapted to the numerical solution of first order systems of differential equations which are assumed to possess oscillating solutions are obtained these pairs have been derived taking into account not only the usual properties of accuracy stability and reliability of the local error estimator to adjust the stepsize of the underlying formulas but also the dispersion and dissipation orders of the advancing formula as defined by van der houwen and sommeijer 1989 three nine stage embedded pairs of runge kutta methods with algebraic orders 7 and 5 and higher orders of dispersion and or dissipation are selected among the members of a family of pairs depending on several free parameters some numerical results are presented to show the efficiency of the new methods

high order closed newton cotes trigonometrically fitted formulae for the numerical solution of the schrodinger equation abstract in this paper we investigate the connection between closed newton cotes formulae trigonometrically fitted methods symplectic integrators and efficient integration of the schrodinger equation the study of multistep symplectic integrators is very poor although in the last decades several one step symplectic integrators have been produced based on symplectic geometry see the relevant literature and the references here in this paper we study the closed newton cotes formulae and we write them as symplectic multilayer structures based on the closed newton cotes formulae we also develop trigonometrically fitted symplectic methods an error analysis for the one dimensional schrodinger equation of the new developed methods and a comparison with previous developed methods is also given we apply the new symplectic schemes to the well known radial schrodinger equation in order to investigate the efficiency of the proposed method to these type of problems

a filon type asymptotic approach to solving highly oscillatory second order initial value problems abstract in this paper we analyze and design filon type asymptotic methods for solving highly oscillatory second order initial value problems q t mq t f t q t q t where m is a non singular and diagonalizable matrix having large eigenvalues and m 1 we first derive filon type asymptotic methods for highly oscillatory linear systems and then extend the methods to highly oscillatory nonlinear systems the integrators are a blend of the existing filon type methods and asymptotic methods based on the variation of constants formula it is shown that the new integrators are convergent and the accuracy of the new integrators is improved as m grows large this point is significant and powerful for effectively dealing with highly oscillatory systems numerical experiments are carried out and the numerical results show the advantage and efficiency of our new integrators as compared with existing numerical methods for highly oscillatory problems from the scientific literature

block runge kutta methods for periodic initial value problems abstract block runge kutta methods with minimal phase lag for first order periodic initialvalue problems are developed it should be noted that the new methods are based on the runge kutta methods of algebraic order three and on a new error estimate introduced in this paper the numerical results indicate that these new methods are efficient for the numerical solution of differential equations with periodic solutions using variable step size

efficient energy preserving integrators for oscillatory hamiltonian systems in this paper we focus our attention on deriving and analyzing an efficient energy preserving formula for the system of nonlinear oscillatory or highly oscillatory second order differential equations q t mq t f q t where m is a symmetric positive semi definite matrix with m 1 and f q q u q is the negative gradient of a real valued function u q this system is a hamiltonian system with the hamiltonian h p q 1 2 p t p 1 2 q t mq u q the energy preserving formula exactly preserves the hamiltonian we analyze in detail the properties of the energy preserving formula and propose new efficient energy preserving integrators in the sense of numerical implementation the convergence analysis of the fixed point iteration is presented for the implicit integrators proposed in this paper it is shown that the convergence of implicit average vector field methods is dependent on m whereas the convergence of the new energy preserving integrators is independent of m the fermi pasta ulam problem and the sine gordon equation are carried out numerically to show the competence and efficiency of the novel integrators in comparison with the well known average vector field methods in the scientific literature

scheifele two step methods for perturbed oscillators two step methods specially adapted to the numerical integration of perturbed oscillators are obtained the formulation of the methods is based on a refinement of classical taylor expansions due to scheifele g scheifele on the numerical integration of perturbed linear oscillating systems z angew math phys 22 1971 186 210 the key property is that those algorithms are able to integrate exactly harmonic oscillators with frequency w the methods depend on a parameter email protected where h is the stepsize based on the b2 series theory of coleman j p coleman order conditions for a class of two step methods for y f x y ima j numer anal 23 2003 197 220 we derive the order conditions of this new type of method the linear stability and phase properties are examined the theory is illustrated with some fourth and fifth order explicit schemes numerical results carried out on an assortment of test problems such as the integration of the orbital motion of earth satellites show the relevance of the theory

a p stable complete in phase obrechkoff trigonometric fitted method for periodic initial value problems in this paper we derive a p stable trigonometric fitted obrechkoff method with phase lag frequency distortion infinity it is easy to see from numerical results presented that the new method is much more accurate than previous methods

a two stage fourth order almost p stable method for oscillatory problems abstract we propose a two stage fourth order almost p stable method for y f t y based on a pair of discretizations in which the displacement is computed using the trapezoidal rule with end point correction and the velocity is computed using suitably modified simpson s rule

a mono implicit runge kutta nystrom modification of the numerov method we present two two parameter families of fourth order mono implicit runge kutta nystrom methods each member of these families can be considered as a modification of the numerov method we analyze the stability and periodicity properties of these methods it is shown that i within one of these families there exist a stable even l stable and p stable methods and ii in both families there exist methods with a phase lag of order six

on modified runge kutta trees and methods modifled runge kutta mrk methods can have interesting properties as their coe cients may depend on the step length by a simple perturbation of very few coe cients we may produce various function fltted methods and avoid the overload of evaluating all the coe cients in every step it is known that for runge kutta methods each order condition corresponds to a rooted tree when we expand this theory to the case of mrk methods some of the rooted trees produce additional trees called mrk rooted trees and so additional conditions of order in this work we present the relative theory including a theorem for the generating function of these additional mrk trees and explain the procedure to determine the extra algebraic equations of condition generated for a major subcategory of these methods moreover e cient symbolic codes are provided for the enumeration of the trees and the generation of the additional order conditions finally phase lag and phase fltted properties are analyzed for this case and speciflc phase fltted pairs of orders 8 6 and 6 5 are presented and tested

on the numerical approximation of trajectories in the phase plane in this paper our work contributes to study the problem which consists in determining discrete trajectories which have relatively simple expressions by using implicit runge kutta methods and relatively simple computations

continuous runge kutta nystrom methods for initial value problems with periodic solutions abstract in the present work we are concerned with the derivation of continuous runge kutta nystrom methods for the numerical treatment of second order ordinary differential equations with nystrom methods for the numerical treatment of second order ordinary differential equations with periodic solutions numerical methods used for solving such problems are better to have the characteristics of high phase lag order first we analyse the construction algorithm for a high phase lag order scaled extension of an explicit runge kutta nystrom method using this procedure we manage to construct a phase lag order 14 continuous extension of a popular nine stages 8 6 order erkn pair in the literature only phase lag order 12 continuous extension of nine stage 8 6 erkn pairs can be found so the proposed scaling method has the higher until now dispersion order numerical tests for the proposed methods are done over various test problems

a runge kutta fehlberg method with phase lag of order infinity for initial value problems with oscillating solution abstract a runge kutta method is developed for the numerical solution of initial value problems with oscillating solution based on the runge kutta fehlberg 2 3 method a runge kutta method with phase lag of order infinity is developed based on these methods we produce a new embedded runge kutta fehlberg 2 3 method with phase lag of order infinity this method is called as runge kutta fehlberg phase fitted method rkfpf the numerical results indicate that this new method is much more efficient compared with other well known runge kutta methods for the numerical solution of differential equations with oscillating solution using variable step size

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

explicit runge kutta nystro m methods with reduced phase errors for computing oscillating solutions we construct explicit runge kutta nystrom methods for the integration of first and second order differential equations having an oscillatory solution special attention is paid to the phase errors or dispersion of the dominant components in the numerical oscillations when these methods are applied to a linear homogeneous test model rk n methods are constructed which are dispersive of orders up to 10 whereas the algebraic order of accuracy is only 2 or 3 application of these methods to equations describing free and weakly forced oscillations and to semidiscretized hyperbolic equations reveals that the phase errors can significantly be reduced

a global tb theorem for compactness and boundedness of calderon zygmund operators abstract we prove a tb theorem that characterizes all calderon zygmund operators that extend compactly on l p r n 1 p the result whose proof does not require the property of accretivity can be used to prove compactness of the double layer potential operator on a wide class of domains the study also provides conditions for boundedness of singular integral operators by means of non accretive testing functions

high accuracy solution of bi directional wave propagation in continuum mechanics solution of partial differential equations by numerical method is strongly affected due to numerical errors which are caused mainly by deviation of numerical dispersion relation from the physical dispersion relation to quantify and control such errors and obtain high accuracy solutions we consider a class of problems which involve second derivative of unknowns with respect to time here we analyse numerical metrics such as the numerical group velocity numerical phase speed and the numerical amplification factor for different methods in solving the model bi directional wave equation bdwe such equations can be solved directly for example by runge kutta nystrom rkn method alternatively the governing equation can be converted to a set of first order in time equations and then using four stage fourth order runge kutta rk4 method for time integration spatial discretisation considered are the classical second and fourth order central difference schemes along with lele s central compact scheme for evaluating second derivatives in another version we have used lele s scheme for evaluating first derivatives twice to obtain the second derivative as bdwe represents non dissipative non dispersive dynamics we also consider the canonical problem of linearised rotating shallow water equation lrswe in a new formulation involving second order derivative in time which represents dispersive waves along with a stationary mode the computations of lrswe with rk4 and rkn methods for temporal discretisation and lele s compact schemes for spatial discretisation are compared with computations performed with rk4 method for time discretisation and staggered compact scheme scs for spatial discretisation by treating it as a set of three equations as reported in rajpoot et al 2012 1

high phase lag order runge kutta and nystrom pairs we exploit the freedom in the selection of the free parameters of one family of eighth algebraic order runge kutta rk pairs and of three families of fourth sixth and eighth order rk nystrom rkn pairs with the purpose of obtaining specific pairs of the highest possible phase lag order which are also characterized by minimized principal truncation error coefficients we present a method for the analytic derivation of the dissipation order conditions for rk methods and the phase lag and dissipation order conditions for nystrom methods the rk pairs we study here are based on a one parameter generalization of some older families of pairs an algorithm and specific optimized 8 6 nystrom pairs are also provided for a class of initial value problems whose solution is known to be described by free oscillations or free oscillations of low frequency with forced oscillations of high frequency superimposed over long integration intervals these new pairs seem to offer some advantages with respect to some older pairs the latter are of the same algebraic orders as the new ones but are characterized by the minimal phase lag order according to their algebraic order and number of stages

runge kutta interpolants with minimal phase lag abstract we develop the basic theory for the construction of explicit scaled runge kutta methods srk for the integration of first order differential equations having an oscillatory solution based on this theory and on the theory of runge kutta methods with minimal phase lag we construct an embedded runge kutta 10 12 method and an explicit scaled runge kutta method numerical results indicate the efficiency of the new algorithms

an explicit almost p stable two step method with phase lag of order infinity for the numerical integration of second order pacific initial value problems an explicit almost p stable two step method with phase lag of order infinity is developed for the numerical integration of second order periodic initial value problems the method has algebraic order four extensive numerical testing indicates that the new method is generally more accurate than other two step methods

frequency determination and step length control for exponentially fitted runge kutta methods an exponentially tted runge kutta efrk fth order method with six stages is constructed which exactly integrates rst order di8erential initial value problems whose solutions are linear combinations of functions of the form exp x exp x r or ir by combining this efrk method with an equivalent classical embedded 4 5 runge kutta method a technique is developed for the estimation of the occurring values error and step length control is carried out by using the richardson extrapolation procedure some numerical experiments show the e ciency of the introduced methods c 2001 elsevier science b v all rights reserved

a new hybrid imbedded variable step procedure for the numerical integration of the schrodinger equation a new imbedded variable step procedure is developed for the numerical integration of the radial schrodinger equation the new imbedded method is based on p stable methods of exponential order eight ten 12 and 14 numerical results indicate that the new procedure is more efficient than similar variable step procedures

the operator valued marcinkiewicz multiplier theorem and maximal regularity given a closed linear operator on a umd space we characterize maximal regularity of the non homogeneous problem

a t1 theorem for integral transformations with operator valued kernel we consider generalized calderon zygmund operators whose kernel k x y takes values in l x continuous linear operators on the banach space x and satis es variants of the classical standard estimates involving rboundedness which has recently become a crucial notion in connection with operator valued singular integrals boundedness criteria in the spirit of the t1 theorem of david and journe are proved for such operators on the bochner spaces lp rn x where 1 p and x is a umd space for some results x is also required to have pisier s property in the special case t1 t 1 0 we obtain an essentially complete analogue of the scalar valued theorem we also provide su cient conditions for the general case but they are stronger in general than the necessary t1 t 1 bmo type conditions although they reduce to them in the classical situation a counterexample is given to show that the natural necessary conditions are not su cient in general

embedded methods for the numerical solution of the schrodinger equation abstract new methods for the approximate numerical integration of the one dimensional schrodinger equation are developed in this paper complete phase lag analysis of the new methods is included these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for a one dimensional schrodinger equation show the validity of the developed theory

diagonally implicit runge kutta nystro m methods for oscillatory problems implicit runge kutta nystrom rkn methods are constructed for the integration of second order differential equations possessing an oscillatory solution based on a linear homogeneous test model we analyse the phase errors or dispersion introduced by these methods and derive so called dispersion relations diagonally implicit rkn methods of relatively low algebraic order are constructed which have a high order of dispersion up to 10 application of these methods to a number of test examples linear as well as nonlinear yields a greatly reduced phase error when compared with conventional dirkn methods

phase fitted and amplification fitted two step hybrid methods for y f x y this paper provides a theoretical framework for a new type of phase fitted and amplification fitted two step hybrid ftsh methods which is introduced by the author in h van de vyver a phase fitted and amplification fitted explicit two step hybrid method for second order periodic initial value problems internat j modern phys c 17 2006 663 675 the methods constitute a modification of dissipative two step hybrid methods in the sense that two free parameters are added to eliminate the phase lag and the amplification error the methods are useful only when a good estimate of the frequency of the problem is known in advance the parameters depend on the product of the estimated frequency and the stepsize the algebraic order zero stability stability and phase properties are examined the theory is illustrated with sixth order explicit ftsh methods numerical results carried out on an assortment of test problems show the relevance of the theory

kato s square root problem in banach spaces abstract let l be an elliptic differential operator with bounded measurable coefficients acting in bochner spaces l p r n x of x valued functions on r n we characterize kato s square root estimates l u p u p and the h functional calculus of l in terms of r boundedness properties of the resolvent of l when x is a banach function lattice with the umd property or a noncommutative l p space to do so we develop various vector valued analogues of classical objects in harmonic analysis including a maximal function for bochner spaces in the special case x c we get a new approach to the l p theory of square roots of elliptic operators as well as an l p version of carleson s inequality

the performance of phase lag enhanced explicit runge kutta nystrom pairs on n body problems explicit runge kutta nystrom methods with enhanced phase lag order are intended for long integrations of initial value ordinary differential equations describing free oscillations or free oscillations of high frequency and forced oscillations of low frequency numerical comparisons by others of rkn4 3 rkn6 4 and rkn8 6 pairs has established that the pairs with enhanced phase lag order are more efficient on the intended problems than general purpose pairs we investigate if these gains in efficiency extend to n body problems used to model the orbital dynamics of the solar system the emphasis in our comparisons is on the rkn8 6 pairs because we are interested in long accurate integrations we have included the rkn4 3 and rkn6 4 pairs principally to gain insight about how the gains in efficiency depend on the order our main finding is that the gains for the rkn8 6 pair extend to the system of major planets except at severe accuracy requirements and to the system of regular satellites of these planets in addition we found for kepler s two body problem that the gains can be sensitive to small changes in eccentricity

adaptive two step methods for semidiscretized wave equations adaptive two step direct integration methods are constructed for the integration of second order semidiscrete evolution equations possessing oscillatory solutions the methods are based on a class of adaptive multistep methods for a semilinear test model whose frequency is known they are constructed following the notion of diagonally implicit rk methods by using efficient rational approximations to cos v v 0 our interest is centered on the dispersion or phase errors of the dominant components in the numerical oscillations when these methods are applied to a linear homogeneous test model two step methods which have high order of dispersion up to 12 whereas the algebraic order is relatively low 2 or 4 are derived applications of these methods to linear as well as non linear test models and to semidiscretized hyperbolic equations reveal a good behaviour with regard to error propagation when they are compared with other conventional methods

interpolation between vector valued hardy spaces abstract let 0 p and let h p h x denote the space of x valued harmonic functions on the half space with boundary values almost everywhere and poisson maximal function in l p r n and h p x the closure of the x valued analytic polynomials on the disc under the norm given by sup 0 r f r p it is shown that if 0 p 0 p 1 1 p 1 p 0 p 1 then h p0 h x 0 h p1 h x 1 0 h p h x 0 with the restriction p 1 we prove h p0 h x 0 h p1 h x 1 0 p h h p x 0 p a counterexample for the case p 1 is given for the case of real interpolation it is also proved that h p0 x 0 h p1 x 1 0 is in general smaller than h p x 0 finally bmo x is also considered as the end point for interpolation

optimized runge kutta pairs for problems with oscillating solutions three types of methods for integrating periodic initial value problems are presented these methods are i phase fitted ii zero dissipation iii both zero dissipative and phase fitted some particular modifications of well known explicit runge kutta pairs of orders five and four are constructed numerical experiments show the efficiency of the new pairs in a wide range of oscillatory problems

efficient iterations for gauss methods on second order problems we consider some important aspects about the implementation of high order implicit formulas specially the gauss methods for solving second order differential systems having high frequencies and small amplitudes superimposed the choice of an appropriate iterative scheme is discussed in detail important topics about the predictors initial guesses are analyzed and a variable order strategy to select the best predictor at each integration step is supplied a few numerical experiments on some standard test problems confirm the theory presented