the equimaterial approach for the numerical solution of the one dimensional transient diffusion equation with zero flux boundary conditions abstract a new approach is proposed for the grid motion for the numerical solution of a general transient diffusion equation in one spatial dimension with zero flux boundary conditions the new criterion for grid motion is that the solute amount contained in each discretization section should be a pre described fraction of the total solute amount at each time step this requirement is not explicitly enforced to the solution technique but it is implicitly included in the equation through the appropriate variable transformation the results showed that although the technique leads to the required grid motion the numerical results are of pure quality due to the appearance of singularities during the variable transformation procedure nevertheless it is shown that by appropriate numerical handling of the solution at the singularity region the technique can lead to accurate results and potentially can replace the existing moving grid algorithms at least for the particular problem at hand

conservative modified cranknicolson and time splitting wavelet methods for modeling boseeinstein condensates in delta potentials this paper explores two wavelet based energy conserving algorithms for the grosspitaevskii equation with delta potentials in boseeinstein condensates named modified cranknicolson wavelet method and time splitting wavelet method respectively both proposed methods can preserve the intrinsic properties of original problems as much as possible meanwhile the rigorous error estimates and some conservative properties are investigated they are proved to preserve the charge conservation exactly the global energy conservation laws can be preserved under several conditions in practical computations to avoid a large drift in energy values caused by discontinuous potential well an improved discrete delta function is implemented numerical experiments for attractive and repulsive cases are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis

on the efficiency of newton and broyden numerical methods in the investigation of the regular polygon problem of n 1 bodies numerical methods of finding the roots of a system of non linear algebraic equations are treated in this paper this paper attempts to give an answer to the selection of the most efficient method in a complex problem of celestial dynamics the so called ring problem of n 1 bodies we apply newton and broyden s method to these problems and we investigate by means of their use the planar equilibrium points the five equilibrium zones which are symbolized by a 1 a 2 b c 2 and c 1 by order of appearance from the center o to the periphery of the imaginary circle on which the primaries lie t j kalvouridis a planar case of the n 1 body problem the ring problem astrophys space sci 260 3 1999 309 325 and the attracting regions of the system the efficiency of these methods is studied through a comparative process the obtained results are demonstrated in figures and are discussed

symplectic integrators for the numerical solution of the schrodinger equation the solution of the one dimensional time independent schrodinger equation is considered by symplectic integrators the schrodinger equation is first transformed into a hamiltonian canonical equation the concept of asymptotic symplecticness is introduced and asymptotically symplectic methods of order up to 3 are developed numerical results are obtained for the one dimensional harmonic oscillator the hydrogen atom and the one dimensional double well anharmonic oscillator

an algebraic method to solve the radial schrodinger equation we propose a method of numerical integration of differential equations of the type x 2y f x y 0 by approximating its solution with solutions of equations of the type x 2y ax 2 bx c y 0 this approximation is performed by segmentary approximation on an interval we apply the method to obtain approximate solutions of the radial schrodinger equation on a given interval and test it for two different potentials we conclude that our method gives a similar accuracy than the taylor method of higher order

bounds for variable degree rational l approximations to the matrix exponential abstract in this work we derive new alternatives for efficient computation of the matrix exponential which is useful when solving linear initial value problems vibratory systems or after semidiscretization of pdes we focus especially on the two classes of normal and nonnegative matrices and we present intervals of applications for rational l approximations of various degrees for these types of matrices in the lines of 7 8 our method relies on remez algorithm for rational approximation while the innovation here is the choice of the starting set of non symmetrical chebyshev points only one remez iteration is then usually enough to quickly approach the actual l approximant

high algebraic high phase lag order embedded numerov type methods for oscillatory problems an embedded numerov type method of algebraic order eight with phase lag orders 10 20 for the numerical integration of second order periodic initial value problems is developed in this paper the method has 10 stages an error estimation based on varying the phase lag order is introduced and an embedded technique is described each method of the embedded pair has a large interval of periodicity theoretical evidence and numerical and theoretical results obtained for several well known problems show the efficiency of the new methods

newton cotes formulae for long time integration the connection between closed newton cotes differential methods and symplectic integrators is considered in this paper several one step symplectic integrators have been developed based on symplectic geometry however multistep symplectic integrators have seldom been investigated zhu et al j chem phys 104 1996 2275 converted open newton cotes differential methods into a multilayer symplectic structure also chiou and wu j chem phys 107 1997 6894 have written on the construction of multistep symplectic integrators based on the open newton cotes integration methods in this work we examine the closed newton cotes formulae and we write them as symplectic multilayer structures we apply the symplectic schemes in order to solve hamilton s equations of motion which are linear in position and momentum we observe that the hamiltonian energy of the system remains almost constant as integration proceeds

a generator of hybrid symmetric four step methods for the numerical solution of the schrodinger equation in this paper a generator of hybrid explicit four step methods with minimal phase lag is developed the methods are of sixth algebraic order and have large intervals of periodicity the coefficients of the methods are determined in order to have minimal phase lag the efficiency of the new methods is showed by their application to the schrodinger equation and by their comparison with other well known methods

runge kutta nystrom triples abstract the embedded runge kutta nystrom process for the numerical solution of the special second order initial value problem can be extended by the addition of dense formulae for both y and y which yield solutions at points within the normal step intervals as for runge kutta methods two modes of implementation are possible and it is shown that these are equivalent under certain conditions the cases of dense formulae applied to rkn4 3 and rkn6 4 families are considered and test results are presented

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

explicit high order methods for the numerical integration of periodic initial value problems two explicit two step hybrid methods of order 7 and 8 for the numerical integration of second order periodic initial value problems are developed in this paper the first of them has a large interval of periodicity and the other a minimal phase lag each of them has seven stages per iteration numerical and theoretical results obtained for several well known problems show the efficiency of the new methods

a family of high order multistep methods with vanished phase lag and its derivatives for the numerical solution of the schrodinger equation many simulation algorithms chemical reaction systems differential systems arising from the modelling of transient behaviour in the process industries etc contain the numerical solution of systems of differential equations for the efficient solution of the above mentioned problems linear multistep methods or runge kutta single step methods are used for the simulation of chemical procedures the radial schrodinger equation is used frequently in the present paper we will study a class of linear multistep methods more specifically the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial schrodinger equation and related problems this algorithm belongs in the category of the multistep methods in order to produce an efficient multistep method the phase lag property and its derivatives are used hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions the reason of their efficiency as the analysis proved is that the phase lag and its derivatives are eliminated another reason of the efficiency of the new obtained methods is that they have high algebraic order

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

a method for computing phase shifts for scattering abstract the method developed by papageorgiou and raptis for computing the eigenvalues is now applied to scattering problems in particular a new approach is developed for the computation of the phase shifts of the one dimensional schrodinger equation the algorithm is very simple to code and very efficient for computational purposes the method is also general and can be applied to any kind of potential functions coulomb type etc

modified runge kutta methods for the numerical solution of odes with oscillating solutions three new modified runge kutta methods with minimal phase lag are developed for the numerical solution of ordinary differential equations with oscillating solutions these methods are based on the well known runge kutta method of dormand and prince rk5 4 t see 1 of order five numerical and theoretical results show that this new approach is more efficient compared with the well known classical fifth order runge kutta dormand and prince method

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

harmonic reflection in quasicircles and well posedness of a riemann hilbert problem on quasidisks abstract a complex harmonic function of finite dirichlet energy on a jordan domain has boundary values in a certain conformally invariant sense by a construction of h osborn we call the set of such boundary values the douglas osborn space one may then attempt to solve the dirichlet problem on the complement for these boundary values this defines a reflection of harmonic functions we show that quasicircles are precisely those jordan curves for which this reflection is defined and bounded we then use a limiting cauchy integral along level curves of green s function to show that the plemelj sokhotski jump formula holds on quasicircles with boundary data in the douglas osborn space this enables us to prove the well posedness of a riemann hilbert problem with boundary data in the douglas osborn space on quasicircles

dissipative high phase lag order numerov type methods for the numerical solution of the schrodinger equation a generator of families of explicit hybrid methods with minimal phase lag is developed in this paper the methods of the generator have algebraic order six the main characteristic of the new methods is that they are dissipative i e they are not symmetric and they have not an interval of periodicity the generator is of dissipation order eight numerical results indicate that these new methods are more efficient than older ones i e the property of the phase lag is more crucial than the nonempty interval of periodicity for the construction of the numerical methods for the numerical solution of the schrodinger type equations

representation of l q μ ˆ x by vector valued function spaces abstract in this paper we use results from the theory of tensor products of banach spaces to establish the isometry of the space of so called strongly p integrable functions in a banach space x to the space of integral x valued operators on l p and the complete projective tensor product of l p with an arbitrary banach space x although there are similar results in recent papers for instance in q bu p k lin radon nikodym property for the projective tensor product of kothe function spaces j math anal appl 293 2004 149 159 and j diestel j h fourie j swart the projective tensor product ii the radon nikodym property rev r acad cienc ser a mat 100 2006 75 100 our contribution in the present paper is to remove all restrictions attached to the banach space x through similar techniques the result is then also considered in the context of the complete projective tensor product of a banach lattice x satisfying some conditions with an arbitrary banach space y

a family of numerov type exponentially fitted predictor corrector methods for the numerical integration of the radial schro dinger equation a family of predictor corrector exponential numerov type methods is developed for the numerical integration of the one dimensional schrodinger equation the formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions the new methods are very simple and integrate more exponential functions than both the well known fourth order numerov type exponentially fitted methods and the sixth algebraic order runge kutta type methods numerical results also indicate that the new methods are much more accurate than the other exponentially fitted methods mentioned above

a fourth algebraic order trigonometrically fitted predictor corrector scheme for ivps with oscillating solutions a scheme of trigonometrically fitted predictor corrector p c adams bashforth moulton methods is constructed in this paper our new p c method is based on the third order adams bashforth scheme as predictor and on the fourth order adams moulton scheme as corrector we tested the efficiency of our newly developed scheme against well known methods with excellent results the numerical experimentation showed that our method is considerably more efficient compared to well known methods used for the numerical solution of initial value problems with oscillating solutions

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

multiderivative methods of eighth algebraic order with minimal phase lag for the numerical solution of the radial schrodinger equation multiderivative methods with minimal phase lag are introduced in this paper for the numerical solution of the one dimensional schrodinger equation the methods are called multiderivative since they use derivatives of order two four or six numerical application of the newly introduced method to the resonance problem of the one dimensional schrodinger equation shows its efficiency compared with other similar well known methods of the literature

stage reduction on p stable numerov type methods of eighth order we present an implicit hybrid two step method for the solution of second order initial value problem it costs only six function evaluations per step and attains eighth algebraic order the method satisfy the p stability property requiring one stage less we conclude dealing with implementation issues for the methods of this type and give some first pleasant results from numerical tests

explicit two step methods with minimal phase lag for the numerical integration of special second order initial value problems and their application to the one dimensional schro dinger equation abstract two step sixth order methods with phase lag of order eight ten and twelve are developed for the numerical integration of the special second order initial value sproblem an application to the one dimensional schrodinger equation on the resonance problem indicates that these new methods are generally more accurate than methods developed by chawla rao and neta this journal 1986 1987

an explicit four step phase fitted method for the numerical integration of second order initial value problems abstract an explicit four step method with phase lag of infinite order is developed for the numerical integration of second order initial value problems extensive numerical testing indicates that this new method can be generally more efficient than other four step methods

a discrete eigenfunctions method for computing mixed hyperbolic problems based on an implicit difference scheme in this paper numerical solutions of mixed hyperbolic problems are computed using a discrete eigenfunctions method combined with an implicit difference scheme this new numerical technique preserves the qualitative properties of the analytic solution due to the sturm liouville structure of the underlying discrete linear boundary value problem and has computational stability advantages vs other methods illustrative examples are included

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

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

new finite difference formulas for numerical differentiation conventional numerical differentiation formulas based on interpolating polynomials operators and lozenge diagrams can be simplified to one of the finite difference approximations based on taylor series and closed form expressions of these finite difference formulas have already been presented in this paper we present new finite difference formulas which are more accurate than the available ones especially for the oscillating functions having frequency components near the nyquist frequency closed form expressions of the new formulas are given for arbitrary order a comparison of the previously available three types of approximations is given with the presented formulas a computer program written in mathematica based on new formulas is given in the appendix for numerical differentiation of a function at a specified mesh point

an extended numerov type method for the numerical solution of the schrodinger equation abstract a new method which is based on the new property of phase lag for computing eigenvalues of schrodinger equations with potentials is developed in this paper we investigate two cases 1 i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wavefunctions tend to zero for x 2 ii the general case for positive and negative eigenvalues and for the well known cases of the morse potential and of the woods saxon or optical potential numerical and theoretical results show that this new method is more efficient compared with previously derived methods

bessel and neumann fitted methods for the numerical solution of the schrodinger equation abstract the bessel and neumann fitted methods for the numerical solution of the schrodinger equation is the subject of this paper an eighth algebraic order method for the numerical solution of the schrodinger equation is developed in this paper the new method has free parameters which are defined such that the method is fitted to spherical bessel and neumann functions a variable step procedure is obtained based on the newly developed method and the method of simos 1 the results produced based on the numerical solution of the radial schrodinger equation and of coupled differential equations arising from the schrodinger equation indicate that this new approach is more efficient than other well known methods

shooting methods for the schrodinger equation a simple stable shooting method for the schrodinger equation is described and is shown to work with various integrators including the numerov one quantum mechanical techniques are extended to apply to problems for which e is not a traditional eigenvalue and are shown to permit the extraction of useful information even from an unstable shooting method

fourth order symplectic integration abstract in this paper we present an explicit fourth order method for the integration of hamilton s equations this method preserves the property that the time evolution of such a system yields a canonical transformation from the initial conditions to the final state that is the integration step is an explicit symplectic map although the result is first derived for a specific type of hamiltonian it is shown to be quite general in particular the results can be applied to any lie group

a family of trigonometrically fitted partitioned runge kutta symplectic methods abstract we are presenting a family of trigonometrically fitted partitioned runge kutta symplectic methods of fourth order with six stages the solution of the one dimensional time independent schrodinger equation is considered by trigonometrically fitted symplectic integrators the schrodinger equation is first transformed into a hamiltonian canonical equation numerical results are obtained for the one dimensional harmonic oscillator and the exponential potential

a p stable exponentially fitted method for the numerical integration of the schrodinger equation a p stable exponentially fitted method is developed in this paper for the numerical integration of the schrodinger equation an application to the resonance problem of the radial schrodinger equation indicates that the new method is generally more efficient than the previously developed exponentially fitted methods of the same kind

radon nikodym property for the projective tensor product of kothe function spaces abstract let e be an order continuous kothe function space or an order continuous banach lattice and x be a dual banach space then e x the projective tensor product of e and x has the radon nikodym property if and only if both e and x do

a sixth order exponentially fitted method for the numerical solution of the radial abstract a new sixth order method of runge kutta type is developed for the numerical integration of the single channel radial schrodinger equation the formula derived contains certain free parameters which allows it to be fitted automatically to exponential functions extensive numerical testing on the resonance problem and on the bound states problem indicates that this new method is generally more efficient than other previously developed finite difference methods

two step fourth order p stable methods with phase lag of order six for y t y abstract we present a new family of two step fourth order methods which when applied to the test equation y 2y 0 are at once p stable and have a phase lag of order h6 h h h is the step size

a generator of high order embedded p stable methods for the numerical solution of the schro dinger equation a generator of new embedded p stable methods of order 2n 2 where n is the number of layers used by the embedded methods for the approximate numerical integration of the one dimensional schrodinger equation is developed in this paper these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for one dimensional differential equations of the schrodinger type show the validity of the developed theory

a new method for the numerical solution of fourth order bvp s with oscillating solutions abstract a new finite difference scheme with minimal phase lag for the numerical solution of fourth order differential equations is developed in this paper numerical and theoretical results show that this new approach is more efficient compared with previously derived methods

explicit exponentially fitted methods for the numerical solution of the schrodinger equation we present here some accurate explicit methods for the numerical integration of the radial schrodinger equation the formulae considered contain free parameters which are defined in order to integrate exponential functions numerical results also indicate that the new methods are much more accurate than other explicit numerov type wellknown methods

banach space sequences and projective tensor products abstract in this paper we use results from the theory of tensor products of banach spaces to establish the isometry of the space of 1 p summing sequences also known as strongly p summable sequences in a banach space x the space of nuclear x valued operators on l p and the complete projective tensor product of l p with x through similar techniques from the theory of tensor products the isometry of the sequence space l p x recently introduced in a paper by bu quaestiones math 2002 to appear the space of nuclear x valued operators on l p 0 1 with a suitable equivalent norm and the complete projective tensor product of l p 0 1 with x is established moreover we find conditions for the space of p q summing multipliers to have the gak property generalized ak property use multiplier sequences to characterize banach space valued bounded linear operators on the vector sequence space of absolutely p summable sequences in a banach space and present short proofs for results on p summing multipliers

a family of four step exponential fitted methods for the numerical integration of the radial schrodinger equation abstract a family of exponential four step methods is developed for the numerical integration of the one dimensional schrodinger equation the formula developed contains certain free parameters which allows it to be fitted automatically to exponential functions the new methods integrate more exponential functions and are very simple compared with the well known sixth algebraic order runge kutta type methods numerical results indicate that the new methods are much more accurate than other exponentially fitted methods

an accurate finite difference method for the numerical solution of the schro dinger equation abstract an accurate finite difference approach for computing eigenvalues of schrodinger equations is developed in this paper we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wave functions tend to zero for x we investigate the well known potential of the onedimensional anharmonic oscillator the symmetric double well potential the razavy potential and the doubly anharmonic oscillator potential ii the general case for positive and negative eigenvalues and for the well known cases of the morse potential and woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient than previously derived methods

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

p q summing sequences abstract a sequence x j in a banach space x is p q summing if for any weakly q summable sequence x j in the dual space we get a p summable sequence of scalars x j x j we consider the spaces formed by these sequences relating them to the theory of p q summing operators we give a characterization of the case p 1 in terms of integral operators and show how these spaces are relevant for a general question on banach spaces and their duals in connection with grothendieck theorem

flowing maps to minimal surfaces we introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow but is more effective at finding minimal surfaces in the genus 0 case our flow is just the harmonic map flow and it tries to find branched minimal 2 spheres as in sacks uhlenbeck 1981 and struwe 1985 etc in the genus 1 case we show that our flow is exactly equivalent to that considered by ding li liu 2006 in general we recover the result of schoen yau 1979 and sacks uhlenbeck 1982 that an incompressible map from a surface can be adjusted to a branched minimal immersion with the same action on pi 1 and this minimal immersion will be homotopic to the original map in the case that pi 2 0

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

explicit multi symplectic extended leap frog methods for hamiltonian wave equations in this paper we study the integration of hamiltonian wave equations whose solutions have oscillatory behaviors in time and or space we are mainly concerned with the research for multi symplectic extended runge kutta nystrom erkn discretizations and the corresponding discrete conservation laws we first show that the discretizations to the hamiltonian wave equations using two symplectic erkn methods in space and time respectively lead to an explicit multi symplectic integrator eleap frogi then we derive another multi symplectic discretization using a symplectic erkn method in time and a symplectic partitioned runge kutta method which is equivalent to the well known stormer verlet method in space eleap frogii these two new multi symplectic schemes are extensions of the leap frog method the numerical stability and dispersive properties of the new schemes are analyzed numerical experiments with comparisons are presented where the two new explicit multi symplectic methods and the leap frog method are applied to the linear wave equation and the sine gordon equation the numerical results confirm the superior performance and some significant advantages of our new integrators in the sense of structure preservation

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 exponentially fitted and trigonometrically fitted method for the numerical solution of periodic initial value problems abstract an explicit symmetric multistep method is presented in this paper the new method is exponentially fitted and trigonometrically fitted and is of algebraic order eight the effectiveness of the exponential fitting is proved by the application of the new method and the classical one with constant coefficients to well known periodic problems

a new variable step method for the numerical integration of the one dimensional schro dinger equation the numerical integration of the one dimensional schrodinger equation is considered a new variable step method is presented within the canonical functions scheme characterized namely by i the replacement of the schrodinger equation by the equivalent integral equation ii the representation of the wavefunction by a superposition of simple integrals iii the definition of taylor series like difference equations at each interval an estimate of the step size h is determined in order to obtain the desired local accuracy for the practical case of a given potential function with polynomial interpolations the step size h is directly given by a simple formula depending on i the potential function in the interval ii the initial values of the wavefunction at the origin of the interval this variable step method is applied to the morse potential function used for the recent raptis and cash method comput phys commun 36 113 1985 for this application the present method presents some advantages over the previous one

some new variable step methods with minimal phase lag for the numerical integration of special second order initial value problem abstract two new variable step methods with minimal phase lag are developed for the numerical integration of the special second order initial value problem an application to the one dimensional schrodinger equation on the phase shift problem indicates that these new methods are generally more accurate than other previously developed finite difference methods especially in the case of the high energies

explicit numerov type methods with reduced number of stages we present in this paper a new approach for the derivation of hybrid explicit numerov type methods the new methodology does not require the intermediate use of high accuracy inter polatory nodes since we only need the taylor expansion of the internal points as a consequence a sixth order method is produced at a cost of only four stages per step instead of six stages needed for the methods which have appeared in the literature until now numerical results over some well known problems in physics and mechanics indicate the superiority of the new method 2003 elsevier science ltd all rights reserved

a finite difference method for the numerical solution of the schro dinger equation a new approach which is based on a new property of phase lag for computing eigenvalues of schrodinger equations with potentials is developed in this paper we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wave functions tend to zero for x ii the general case of the morse potential and of the woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient compared to previously derived methods

a fourth order bessel fitting method for the numerical solution of the schro dinger equation abstract a new fourth order method is developed for the numerical integration of the one dimensional radial schrodinger equation this method integrates bessel and neumann functions exactly it is shown that for large r this new formula is much more accurate and rapid than the bessel fitting method of second order which is developed by raptis and cash 1987 the benefit of using this new approach is demonstrated by considering some numerical examples based on the lenard jones potential

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

high order embedded runge kutta formulae abstract the criteria to be satisfied by embedded runge kutta pairs of formulae are reviewed two new formulae of orders 6 and 8 are presented together with tests on their efficiency relative to other high order formulae in current use

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

a runge kutta nystro m method for the numerical integration of special second order periodic initial value problems abstract a new runge kutta nystrom method is developed to integrate second order differential equations of the form u t f t u when they posses an oscillatory solution through an appropriate definition of the parameters of the method a fourth algebraic order is obtained and the phase lag is reduced significantly

explicit runge kutta methods for starting integration of lane emden problem abstract traditionally when constructing explicit runge kutta methods we demand the satisfaction of the trivial simplifying assumption thus f 1 f x 0 y 0 is always used as the first stage of these methods when applied to the initial value problem ivp y x f x y y x 0 y 0 here we examine the case with f 1 f x 0 c 1 h y 0 h the step and c1 0 we derive the order conditions for arbitrary order and construct a 5th order method at the standard cost of six stages per step this method is found to outperform other classical runge kutta pairs with orders 5 4 when applied to problems with singularity at the beginning e g lane emden problem

a smooth approximation based on exponential spline solutions for nonlinear fourth order two point boundary value problems methods of order two four and six based on exponential spline functions consisting of a polynomial part of degree three and an exponential part is developed to find approximation of linear and nonlinear fourth order two point boundary value problems it is shown that the free parameter k of the exponential part can be used to raise the order of accuracy of the new scheme convergence analysis of these methods is shown along with numerical examples each for linear and nonlinear are included to illustrate the practical usefulness of our methods

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

an explicit hybrid method of numerov type for second order periodic initial value problems abstract we consider a two parameter family of explicit hybrid methods of numerov type for the numerical integration of second order intial value problems when these methods are applied to the linear equation y 2 y 0 0 we determine the parameters so that the phase lag frequency distortion of the method is minimal the resulting method has algebraic order 4 and a small frequency distortion of size 1 3628800 v 8 v h h being the step size and in addition it possesses an interval of periodicity of size 4 63 which is larger than the interval of periodicity corresponding to the explicit method of chawla and rao 1986 the application of this method to equations describing free and weakly forced oscillations reveals its superiority over other methods

a variable mesh c splage method of accuracy o k2hl 1 khl hl3 for 1d nonlinear parabolic equations a new high accuracy two level implicit cubic spline formulation for the solution of nonlinear parabolic equation u x email protected x t u u x u t is presented the strategy of the cubic spline alternating group explicit c splage method is developed the proposed c splage method requires only 3 spatial grid points and applicable to singular problems the convergence analysis of the proposed c splage method is given the obtained c splage method is compared to the corresponding successive over relaxation sor method both in terms of stability and performance

on the stability of mesh equidistribution strategies for time dependent partial differential equations abstract we study the stability of several mesh equidistribution schemes for time dependent partial differential equations in one space dimension the schemes move a finite difference or finite element mesh so that a given quantity is uniform over the domain we consider mesh moving methods that are based on solving a system of ordinary differential equations for the mesh velocities and show that some of these methods are unstable with respect to an equidistributing mesh when the partial differential system is dissipative using linear perturbation techniques we are able to develop simple criteria for determining the stability of a particular method and show how to construct stable differential systems for the mesh velocities several examples illustrating stable and unstable mesh motions are presented

a new finite difference scheme with minimal phase lag for the numerical solution of the schrodinger equation a new finite difference approach for the numerical solution of the eigenvalue problem of the schrodinger equations is developed in this paper the new approach is based on the maximization of the algebraic order and the minimization of the phase lag we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wavefunctions tend to zero for x ii the general case for positive and negative eigenvalues and for the well known cases of the morse potential and woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient when compared with previously derived methods

on computing eigenvalues of the schrodinger equation for symmetrical potentials the problem of the numerical determination of eigenvalues of the one dimensional schrodinger equation with symmetrical potential v x is considered the problem is reduced to the definition of an eigenvalue function g e well defined for the given potential v it is theoretically proved that the even parity eigenvalues are given by g e 0 and the odd parity eigenvalues are given by g e infinity the method is canonical in the sense that it is independent of the eigenfunction yet along with the eigenvalue it allows the determination of the eigenfunction initial values this method is applied to the potentials v x2 v x6 bx2 v x2 lambda x2 1 gx2 where exact eigenvalues are available the numerical results compared with the exact ones and with those of previous confirmed methods show that the present method is accurate and efficient

a family of four step exponentially fitted predictor corrector methods for the numerical integration of the schro dinger equation a family of predictor corrector exponential four step methods is developed for the numerical integration of the one dimensional schrodinger equation the formula developed contains certain free parameters which allows it to be fitted automatically to exponential functions the new methods integrate more exponential functions and are very simple compared with the well known sixth algebraic order runge kutta type methods numerical results indicate that the new method is much more accurate than other exponentially fitted methods

a four step method for the numerical solution of the schro dinger equation abstract a new four step exponentially fitted method is developed in this paper the expressions for the coefficients of the method are found such as to ensure the optimal approximation to the eigenvalue schrodinger equation i e equivalent to positive energy

a noumerov type method with minimal phase lag for the integration of second order periodic initial value abstract in a recent paper 2 we gave a noumerov type method with minimal phase lag for the integration of second order initial value problems y f t y y t0 y0 y t0 y 0 however the method given there is implicit we show here the interesting result that if the noumerov type methods of 2 are made explicit with the help of the classical second order method then there exists a selection of the free parameter for which the resulting method has a considerably small frequency distortion of size 1 40320 h 6 and also a slightly larger interval of periodicity of size 2 75 than the phase lag of size 1 12096 h 6 and interval of periodicity of size 2 71 for the implicit method 2 more interestingly it turns out that noumerov made explicit of chawla 3 also has less frequency distortion than the implicit noumerov method we shall assume a familiarity with the notation and discussion given in 2

trigonometrically fitted predictor corrector methods for ivps with oscillating solutions in this paper we develop a trigonometrically fitted predictor corrector p c scheme which is based on the well known two step second order adams bashforth method as predictor and on the third order adams moulton method as corrector numerical experiments show that the new trigonometrically fitted p c method is substantially more efficient than widely used methods for the numerical solution of initial value problems ivps with oscillating solutions

a new implementation of en method en method proposed by eirola and nevalinna can be more efficient than gmres in some cases but be not the case in general which has been studied by vuik and van der vorst comparison of the efficiency of the two methods in practice shows that gmres is preferable a new implementation of en method with significant saving both in the amount of work and memory requirements is introduced in the present paper numerical experiments indicate that this new version is competitive

numerical methods for solving radial schro dinger equations abstract an algorithm previously introduced by brown et al 1963 for solving radial schrodinger equations is revisited and implemented in a more accurate way the method is firstly applied to equations where potentials are present which are finite at the origin and which have an asymptotic behaviour v r 0 as r typical examples of potentials belonging to that class are the woods saxon and the morse potential furthermore the method is also used for coulomb like potentials such as the hulthen and the hellmann potential a comparison between the approximated numerical values and other available numerical and exact bound state energies is made