error sources in the standard numerical integration of the schrodinger equation an improved method abstract a numerical study of the standard numerov method for the solution of the schrodinger equation is done by looking at the errors of the solution and the limits of precision for this method comparison is made with the results of an improved method that traces back all sources of errors minimizing them and making the error of each of the sources of comparable magnitude to the others in order to optimize the computing time a reduction factor of more than 100 is found for the computing time given a certain accuracy on the other hand fixing a certain computational time a substantial increase in accuracy is gained the improved method proves particularly suitable for the study of weakly bound states potentials with discontinuities singular potentials or those mixing atomic and nuclear degrees of freedom

invariant subspaces of the dirichlet shift and pseudocontinuations in this paper we study extremal functions for invariant subspaces x of the dirichlet shift i e solutions v of the extremal problem sup if n 0 1 1f lid f e x f o here n is the smallest nonnegative integer such that the sup is positive it is known that such a function v generates x we show that the derivative z o has a pseudocontinuation to the exterior disc this pseudocontinuation is an analytic continuation exactly near those points of the unit circle where 0 is bounded away from zero we also show that the radial limit of the absolute value of an extremal function exists at every point of the unit circle some of our results are valid for all functions that are orthogonal to a nonzero invariant subspace

application of he s homotopy perturbation method for solving the cauchy reaction diffusion problem in this paper the solution of cauchy reaction diffusion problem is presented by means of the homotopy perturbation method reaction diffusion equations have special importance in engineering and sciences and constitute a good model for many systems in various fields application of homotopy perturbation method to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution

application of homotopy perturbation methods for solving systems of linear equations abstract in this paper homotopy perturbation methods hpms are applied to obtain the solution of linear systems and conditions are deduced to check the convergence of the homotopy series moreover we have adapted the richardson method the jacobi method and the gauss seidel method to choose the splitting matrix the numerical results indicate that the homotopy series converges much more rapidly than the direct methods for large sparse linear systems with a small spectrum radius

cyclicity and invariant subspaces in dirichlet spaces let be a positive finite measure on the unit circle and d the associated dirichlet space the generalized brown shields conjecture asserts that an outer function f d is cyclic if and only if c z f 0 where c is the capacity associated with d and z f is the zero set of f in this paper we prove that this conjecture is true for measures with countable support we also give in this case a complete and explicit characterization of invariant subspaces

kernel and capacity estimates in dirichlet spaces abstract we study the capacity in the sense of beurling deny associated with the dirichlet space d where is a finite positive borel measure on the unit circle first we obtain a sharp asymptotic estimate of the norm of the reproducing kernel of d it allows us to give an estimate of the capacity of points and arcs of the unit circle we also provide new conditions on closed sets to be polar our method is based on sharp estimates of norms of some outer functions which allow us to transfer these problems to an estimate of the reproducing kernel of an appropriate weighted sobolev space

homotopy perturbation method a new nonlinear analytical technique in this paper a new perturbation method is proposed in contrast to the traditional perturbation methods this technique does not require a small parameter in an equation in this method according to the homotopy technique a homotopy with an imbedding parameter email protected 0 1 is constructed and the imbedding parameter is considered as a small parameter so the method is called the homotopy perturbation method which can take the full advantages of the traditional perturbation methods and homotopy techniques to illustrate its effectiveness and its convenience a duffing equation with high order of nonlinearity is used the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter and is more accurate than the perturbation solutions

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

cantor sets and cyclicity in weighted dirichlet spaces we treat the problem of characterizing the cyclic vectors in the weighted dirichlet spaces extending some of our earlier results in the classical dirichlet space the absence of a carleson type formula for weighted dirichlet integrals necessitates the introduction of new techniques

exponentially fitted numerov methods the error in the estimate of the kth eigenvalue of a regular sturm liouville problem obtained by numerov s method with mesh length h is o k 6h 4 it is shown that the error can be reduced to o k 3h 4 by using one of the three versions of the exponentially fitted numerov method numerical examples demonstrate the usefulness of this approach even for low values of k

on the brown shields conjecture for cyclicity in the dirichlet space abstract let d be the dirichlet space namely the space of holomorphic functions on the unit disk whose derivative is square integrable we establish a new sufficient condition for a function f d to be cyclic i e for p f p a polynomial to be dense in d this allows us to prove a special case of the conjecture of brown and shields that a function is cyclic in d iff it is outer and its zero set defined appropriately is of capacity zero

homotopy perturbation technique abstract the homotopy perturbation technique does not depend upon a small parameter in the equation by the homotopy technique in topology a homotopy is constructed with an imbedding parameter p 0 1 which is considered as a small parameter some examples are given the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters

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

accurate computation of higher sturm liouville eigenvalues the computation of eigenvalues of regular sturm liouville problems is considered it is shown that a simple step dependent linear multistep method can be used to reduce the error of the orderk 4 h 2 of the centered finite difference estimate of thek th eigenvalue with uniform step lengthh to an error of orderkh 2 by an appropriate minimization of the local error term of the method one can obtain even more accurate results a comparison of the simple correction techniques of paine de hoog and anderssen and of andrew and paine is given numerical examples demonstrate the usefulness of this correction even for low values ofk

approximate analytic solution for critical parameters in a thermal explosion problem abstract in this paper a new approximate analytical solution based on the homotopy perturbation method hpm is applied to predict critical conditions for a thermal explosion problem apart from the conceptual simplicity and flexibility the method gives rise to an easy converging solution leading to the predictions of critical parameters for thermal explosion

a symmetric matrix method for schrodinger eigenstates it is shown how the one dimensional schrodinger eigenvalue equation can be transformed into a symmetric generalised matrix eigenproblem with a local truncation error of only delta 4 where delta is the step size as in the numerov algorithm standard sparse matrix library packages are available for the solution and it is demonstrated that groups of eigenvalues and their associated eigenvectors wavefunctions can be determined simultaneously with high precision and speed

symmetric matrix methods for schrodinger eigenvectors the one dimensional single particle schrodinger eigenvalue equation is represented by a generalized matrix eigenvalue equation of the form a psi ea psi where a is a symmetric matrix a is a symmetric positive definite matrix and psi is a column vector the local error in making the representation is proportional to delta 4 where delta is the step size several methods of solution which make use of the sparsity of the matrices are investigated

construction of asymptotic solutions to discrete bessel equations abstract we construct four finite difference models for the bessel differential equation they correspond respectively to the standard numerov mickens ramadhani and combined numerov mickens schemes the asymptotic behavior of the solutions to these difference equations is calculated and compared to the asymptotic solution of the bessel differential equation these results are then related to the problem of numerically integrating schrodinger type ordinary differential equations

numerical approximations for highly oscillatory bessel transforms and applications abstract this paper presents an efficient numerical method for approximating highly oscillatory bessel transforms based on analytic continuation we transform the integrals into the problems of integrating the forms on 0 with the integrand that does not oscillate and decays exponentially fast which can be efficiently computed by using gauss laguerre quadrature rule we then derive the error of the method depending on the frequency and the node number moreover we apply the scheme for studying the approximations of the solutions of two kinds of highly oscillatory integral equations preliminary numerical results show the efficiency and accuracy of numerical approximations

numerov method maximally adapted to the schro dinger equation abstract we construct and investigate the two step scheme which is maximally specialized for the schrodinger equation the new scheme is found to preserve all attractive features of the standard numerov method ease in programming flexibility in application speed in run etc with the important advantage of being more efficient for problems involving high values of energy its efficiency is actually found to be close to that of the piecewise perturbation methods a fact which strongly recommends it for further applications

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

level sets and composition operators on the dirichlet space abstract we consider composition operators in the dirichlet space of the unit disc in the plane various criteria on boundedness compactness and hilbert schmidt class membership are established some of these criteria are shown to be optimal

solving the 1 2 and 3 dimensional schrodinger equation for multiminima potentials using the numerov cooley method an extrapolation formula for energy eigenvalues as demonstrated with examples the numerov cooley algorithm can be applied to more complicated potentials especially multiminima potentials there are no stability problems in the classically forbidden regions an extrapolation formula for energy eigenvalues is deduced which gets an additional accuracy of 2 4 digits for energy eigenvalues this formula is checked as follows for one state of a double minimum potential which is used to approximate hydrogen bond potentials an analytical solution is given this solution is compared with numerical results also the accuracy of the eigenfunctions is checked computer dependent rounding errors cdc cyber 995 are estimated the eigenfunctions corresponding to 1 dimensional potentials v x x and v y y are used as basis functions for a perturbed 2 dimensional multiminima potential v x x v y y email protected x y for a 63 minima potential as an example the accuracy of the eigenvalues is 4 6 digits depending on the perturbation dv x y the computer time depending on the accuracy is tabulated for different potentials

cyclic vectors in the dirichlet space we study the hilbert space of analytic functions with finite dinchlet integral in the open unit disc we try to identify the functions whose polynomial multiples are dense in this space theorems 1 and 2 confirm a special case of the following conjecture if ij z i ig z l at all points and if g is cyclic thenjis cyclic theorems 3 5 give a sufficient condition t is an outer function with some smoothness and the boundary zero set is at most countable and a necessary condition the radial limit can vanish only for a set of loganthmic capacity zero for a function j to be cyclic introduction in this paper we shall study the hilbert space of analytic functions in the open unit disc a in the complex plane that have a finite dirichlet integral jj if t12 dx dy g z l for some cyclic g and all z question 4 asks if f must be cyclic whenever f and l f are both in the space no examples are known where either of these questions has a negative answer in 2 we begin the study of cyclic vectors in the dirichlet space d theorems 1 and 2 give a partial answer to question 3 above for this space this section also contains 2 propositions 10 11 and 4 questions 7 10 proposition 11 says that if f and g are bounded functions in d whose product is cyclic then bothf and g must be cyclic theorem 2 gives a partial converse we require that gl be dini continuous on received by the editors july 21 1983 1980 mathematics subject classification primary 30h05 secondary 46e15 46e20 47b37

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

regularity for generators of invariant subspaces of the dirichlet shift abstract let d denote the classical dirichlet space of analytic functions on the open unit disc whose derivative is square area integrable for a set e d we write d e f d lim r 1 f r e i t 0 q e where q e stands for except possibly for e i t in a set of logarithmic capacity 0 we show that if e is a carleson set then there is a function f d e that is also in the disc algebra and that generates d e in the sense that d e clos p f p is a polynomial we also show that if d is an extremal function i e p p 0 for every polynomial p then the limits of z exist for every e i t d as z approaches e i t from within any polynomially tangential approach region

an exponentially fitted eighth order method for the numerical solution of the schrodinger equation an eighth algebraic order exponentially fitted method is developed for the numerical integration of the schrodinger equation the formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions an comparative error analysis is also given numerical and theoretical results indicate that the new method is much more efficient than other classical and exponentially fitted methods

analysis of regularized long wave equation associated with a new fractional operator with mittag leffler type kernel abstract in this work we aim to present a new fractional extension of regularized long wave equation the regularized long wave equation is a very important mathematical model in physical sciences which unfolds the nature of shallow water waves and ion acoustic plasma waves the existence and uniqueness of the solution of the regularized long wave equation associated with atangana baleanu fractional derivative having mittag leffler type kernel is verified by implementing the fixed point theorem the numerical results are derived with the help of an iterative algorithm in order to show the effects of various parameters and variables on the displacement the numerical results are presented in graphical and tabular form

approximate analytical solution for seepage flow with fractional derivatives in porous media abstract in this paper a new and more exact model for seepage flow in porous media with fractional derivatives has been proposed which has modified the well known darcy law and overcome the continuity assumption of seepage flow a new kind of analytical method of nonlinear problem called the variational iteration method is described and used to give approximate solutions of the problem the results show that the proposed iteration method requiring no linearization or small perturbation is very effective and convenient

asymptotic properties of solutions to discrete coulomb equations abstract we construct two finite difference models for the coulomb differential equation which arises in the quantum mechanics analysis of the scattering of two charged point particles these difference equations correspond to the standard and mickens ramadhani schemes for the coulomb equation our major goal is to determine the first two terms in the asymptotic solutions and compare them to the corresponding solutions of the coulomb differential equation in particular the form of the anomalous phase term is examined

application of he s methods to nonlinear chemistry problems in this paper he s variational iteration method vim and homotopy perturbation method hpm are implemented for solving analytically two systems of ordinary differential equations that often appear in chemical applications in both methods the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions the solution procedure is of utter simplicity and the obtained solution is of high accuracy

approximate solution of nonlinear differential equations with convolution product nonlinearities in this paper a new iteration method is proposed to solve nonlinear problems special attention is paid to nonlinear differential equations with convolution product nonlinearities the results reveal the approximations obtained by the proposed method are uniformly valid for both small and large parameters in nonlinear problems furthermore the first order of approximations are more accurate than perturbation solutions at high order of approximation