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

an adaptive splitting approach for the quenching solution of reaction diffusion equations over nonuniform grids the numerical solution of a nonlinear degenerate reaction diffusion equation of the quenching type is investigated while spatial derivatives are discretized over symmetric nonuniform meshes a peaceman rachford splitting method is employed to advance solutions of the semidiscretized system the temporal step is determined adaptively through a suitable arc length monitor function a criterion is derived to ensure that the numerical solution acquired preserves correctly the positivity and monotonicity of the analytical solution weak stability is proven in a von neumann sense via the norm computational examples are presented to illustrate our results

a splitting moving mesh method for reaction diffusion equations of quenching type this paper studies the numerical solution of multi dimensional nonlinear degenerate reaction diffusion differential equations with a singular force term over a rectangular domain the equations may generate strong quenching singularities our work focuses on a variable temporal step peaceman rachford splitting method with an adaptive moving mesh in space the temporal and spatial adaptation is implemented based on arc length type of estimations of the time derivative of the solution since the time derivative of the solution approaches infinity when the quenching occurs the multi dimensional problem is split into a few one dimensional problems and the splitting procedure can also be parallelized so that the computational time is significantly reduced the physical monotonicity of the solution and stability of this variable step moving mesh scheme are analyzed for the time away from the quenching as stability analysis may not be valid when it is very close to the quenching thus an exact linear problem is introduced to justify the stability near the quenching time finally we provide some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method for the quenching problem or other problems with point singularities we will also show the significant reduction in computational time required with parallel implementation of the algorithm on a computer with multi cpu

stability of explicit arkn methods for perturbed oscillators the stability of explicit arkn methods for the numerical integration of perturbed oscillators is analyzed stability regions for the second order homogeneous linear test model are obtained these stability regions represent a generalization of the stability intervals for the classical rkn methods

an integral formula adapted to different boundary conditions for arbitrarily high dimensional nonlinear klein gordon equations with its applications in this paper we are concerned with the initial boundary value problem of arbitrarily high dimensional klein gordon equations posed on a bounded domain d for d 1 and equipped with the requirement of boundary conditions we derive and analyze an integral formula which is proved to be adapted to different boundary conditions for general klein gordon equations in arbitrarily high dimensional spaces the formula gives a closed form solution to arbitrarily high dimensional homogeneous linear klein gordon equations which is totally different from the well known d alembert poisson and kirchhoff formulas some applications are included as well

trigonometrically fitted three derivative runge kutta methods for solving oscillatory initial value problems abstract trigonometrically fitted three derivative runge kutta tfthdrk methods for solving numerically oscillatory initial value problems are proposed and developed tfthdrk methods improve three derivative runge kutta thdrk methods numer algor 74 247 265 2017 and integrate exactly the problem whose solutions can be expressed as the linear combinations of functions from the set of exp i w t exp i w t or equivalently the set cos wt sin wt where w approximate the main frequency of the problem the order conditions are deduced by the theory of rooted trees and b series and two new explicit special tfthdrk methods with order five and seven respectively are constructed linear stability of tfthdrk methods is examined numerical results show the superiority of the new methods over other methods from the scientific literature

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

fundamental solutions for second order subelliptic operators estimation de la solution fondamentale d un operateur aux derivees partielles lineaire du second ordre sur une variete compacte avec une mesure reguliere

remarks on quenching on considere le probleme aux limites parabolique non lineaire suivant u t u f u t 0 x u 0 t 0 n u 0 x u 0 x 0 x on demontre un phenomene de trempe pour des non linearites non convexes dans un espace de dimension arbitraire

holder estimates on cr manifolds with a diagonalizable levi form a for all e 0 the operator cl maps lip s po into lip s 2 m e po here lip s p denotes the space of square integrable forms of degree p q with 1 g q otheoperators o l o l q and0 map lip s p into lip s l m e po c for all e o the operators 8b8zo 8z8ho 1 q a aa cl 8 8 8 q 8 and 8 0 ab map lip s p into lip s p hebre aiag web asm acts on forms of type p q with 1 q 0 e zf f is a square integrable function on a with f e lip p then s f e lip s e p for all e 0 where s is the orthogonal projection on square integrable cr functions

exact solutions of the schrodinger equation d dx2 x2 λx2 1 gx2 ψ x eψ x the authors prove the existence of a class of exact eigenvalues and eigenfunctions of the schrodinger equation for the potential x2 x2 1 gx2 when certain algebraic relations between and g hold some of the properties of these solutions are discussed it is shown that in a certain sense they may be regarded as sturmians for the schrodinger equation with the potential x2 g g2x2

fast calculation of energy and mass preserving solutions of schrodinger poisson systems on unbounded domains this paper deals with the numerical solution of the time dependent schrodinger poisson system in the spherically symmetric case since the problem is posed on an unbounded domain one has to introduce artificial boundary conditions to confine the computational domain the main topic of this work is the construction of a so called discrete transparent boundary condition tbc for a crank nicolson type predictor corrector scheme for solving the schrodinger poisson system this scheme has the property of mass and energy conservation exactly on the discrete level we propose different strategies for the discrete tbc and present an efficient implementation finally a numerical example illustrate the findings and shows the comparison results between the different approaches

subunit balls for symbols of pseudodifferential operators abstract in this work we shall study a definition of subunit ball for non negative symbols of sub elliptic pseudodifferential operators extending in phase space the one given by stein nagel and wainger in the differential operator case using microlocal methods introduced by fefferman and phong we prove that these balls can be straightened by means of a canonical transformation to contain and be contained in boxes of certain sizes which we give in terms of the size of the symbol after microlocalizing the symbol in section 3 we define classes of subunit symbols and study some of their basic properties then we define the subunit ball in the last section the main structure theorems in the n n dimensional elliptic case and in the 1 1 and 2 2 dimensional nonelliptic nondegenerate cases are stated and proved

a priori estimates for second order operators with symplectic characteristic manifold we prove fefferman s sak principle for a class of classical pseudodifferential operators on with symplectic characteristic manifold

a 5 3 pair of explicit arkn methods for the numerical integration of perturbed oscillators a new embedded pair of explicit rkn methods specially adapted to the numerical integration of perturbed oscillators is obtained this pair depends on a parameter v h 0 h is the integration step and is the dominant frequency and it has four stages and algebraic orders five and three the numerical experiments carried out show that the new pair is very competitive when it is compared with high quality codes proposed in the scientific literature

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

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

an adaptive grid method for degenerate semilinear quenching problems abstract this paper studies the structure and implementation of a specially formulated adaptive grid method for computing the numerical solution of a class of degenerate semilinear quenching problems modified arc length procedures are used in both the space and time for robust adjustments of the spatial and temporal discretizations throughout the computation a milne like device is introduced in the error estimation and control of the computational error the numerical solution generated converges monotonically to the physical solution of the underlying problem up to the quenching time numerical examples are given for demonstrating our conclusions

a numerical study of three moving grid methods for one dimensional partial differential equations which are based on the method of lines in recent years several sophisticated packages based on the method of lines mol have been developed for the automatic numerical integration of time dependent problems in partial differential equations pdes notably for problems in one space dimension these packages greatly benefit from the very successful developments of automatic stiff ordinary differential equation solvers however from the pde point of view they integrate only in a semiautomatic way in the sense that they automatically adjust the time step sizes but use just a fixed space grid chosen a priori for the entire calculation for solutions possessing sharp spatial transitions that move e g travelling wave fronts or emerging boundary and interior layers a grid held fixed for the entire calculation is computationally inefficient since for a good solution this grid often must contain a very large number of nodes in such cases methods which attempt automatically to adjust the sizes of both the space and the time steps are likely to be more successful in efficiently resolving critical regions of high spatial and temporal activity methods and codes that operate this way belong to the realm of adaptive or moving grid methods following the mol approach this paper is devoted to an evaluation and comparison mainly based on extensive numerical tests of three moving grid methods for 1d problems viz the finite element method of miller and co workers the method published by petzold and a method based on ideas adopted from dorii and drury our examination of these three methods is aimed at assessing which is the most suitable from the point of view of retaining the acknowledged features of reliability robustness and efficiency of the conventional mol approach therefore considerable attention is paid to the temporal performance of the methods

a semi adaptive compact splitting method for the numerical solution of 2 dimensional quenching problems abstract this article studies a semi adaptive compact peaceman rachford splitting method for solving two dimensional nonlinear reaction diffusion equations with singular source terms while an adaption is utilized in the temporal direction uniform grids are considered in the space it is shown that the compact scheme is stable and convergent when its dimensional courant numbers are within the frame of a window determined by the given spatial domain though such a window implies a considerable restriction on the decomposed compact scheme the new computational strategy acquired is highly efficient and reliable for a variety of applications numerical examples are given to illustrate our conclusions and indicate that the method constructed is effective in determining key singularity characteristics such as the quenching time and critical domain

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

numerical solution of degenerate stochastic kawarada equations via a semi discretized approach abstract the numerical solution of a highly nonlinear two dimensional degenerate stochastic kawarada equation is investigated a semi discretized approximation in space is comprised on arbitrary nonuniform grids exponential splitting strategies are then applied to advance solutions of the semi discretized scheme over adaptive grids in time it is shown that key quenching solution features including the positivity and monotonicity are well preserved under modest restrictions the numerical stability of the underlying splitting method is also maintained without any additional restriction computational experiments are provided to not only illustrate our results but also provide further insights into the global nonlinear convergence of the numerical solution

solving degenerate reaction diffusion equations via variable step peaceman rachford splitting this paper studies the numerical solution of two dimensional nonlinear degenerate reaction diffusion differential equations with singular forcing terms over rectangular domains the equations considered may generate strong quenching singularities this investigation focuses on a variable time step peaceman rachford splitting method for the aforementioned problem the time adaptation is implemented based on arc length estimations of the first time derivative of the solution the two dimensional problem is split into several one dimensional problems so that the computational cost is significantly reduced the monotonicity and localized linear stability of the variable step scheme are investigated we give some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method over existing methods for the quenching problem it is also shown that the numerical solution obtained preserves important properties of the physical solution of the given problem

estimates for the bergman and szego kernels in mathbf c 2 the purpose of this paper some of whose conclusions were announced in nrsw is to study the bergman and szegd projection operators on pseudoconvex domains q of finite type in c2 the results we obtain are of three kinds i the size estimates of the bergman and szeg6 kernels and their derivatives ii the cancellation properties of those kernels expressed in terms of the actions of these operators on suitable bump functions these properties have an interest in their own right but are also crucial in iii below iii the sharp mapping properties of these operators on functions spaces such as lp the nonisotropic sobolev spaces and the nonisotropic lipschitz spaces which are naturally attached to the geometry of the boundary this also leads ultimately to sharp results in the regularity properties of solutions of the equation du f in q and 9bu f on the boundary let us describe these matters in more detail our starting points are certain basic geometric constructs associated to q a metric p defined in terms of the vector fields x1 and x2 which are the real and imaginary parts of the tangential levi vector field on d 2 and a function a p 8 a polynomial in 8 for p e d 2 which represents the higher levi invariant attached to d 9 the significance of these to the geometry of d q can be understood from the following facts the ball b p 8 c d q centered at p of radius 8 in the metric p when viewed in the appropriate coordinates has width 8 in the complex tangential directions but its width is a p 8 in the complementary direction thus the volume of the ball is 82a p 8 among other crucial properties

a nonlinear splitting algorithm for systems of partial differential equations with self diffusion systems of reactiondiffusion equations are commonly used in biological models of food chains the populations and their complicated interactions present numerous challenges in theory and in numerical approximation in particular self diffusion is a nonlinear term that models overcrowding of a particular species the nonlinearity complicates attempts to construct efficient and accurate numerical approximations of the underlying systems of equations in this paper a new nonlinear splitting algorithm is designed for a partial differential equation that incorporates self diffusion we present a general model that incorporates self diffusion and develop a numerical approximation the numerical analysis of the approximation provides criteria for stability and convergence numerical examples are used to illustrate the theoretical results

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

exponential splitting for n dimensional paraxial helmholtz equation with high wavenumbers this paper explores applications of the exponential splitting method for approximating highly oscillatory solutions of the n dimensional paraxial helmholtz equation an eikonal transformation is introduced for oscillation free platforms and matrix operator decompositions it is found that the sequential parallel and combined exponential splitting formulas possess not only anticipated algorithmic simplicity and efficiency but also the accuracy and asymptotic stability required for highly oscillatory wave computations

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

an energy preserving and symmetric scheme for nonlinear hamiltonian wave equations in this chapter we derive and analyse an energy preserving and symmetric scheme for nonlinear hamiltonian wave equations which can exactly preserve the energy of the underlying hamiltonian wave equations to this end we first define and discuss the bounded operator argument functions on the underlying domain we then introduce an operator variation of constants formula based on which we present an energy preserving scheme for nonlinear hamiltonian wave equations the scheme preserves the energy of the original continuous hamiltonian system exactly in comparison with the existing work on this topic such as the well known average vector field avf formula for hamiltonian ordinary differential equations the energy preserving scheme avoids the semi discretisation of spatial derivatives and exactly preserves the hamiltonian of the original continuous hamiltonian wave equation this point is very significant in comparison with the avf formula since the avf formula can preserve only the energy of hamiltonian ordinary differential equations hence the main theme of this chapter is to establish a scheme which can exactly preserve the energy of the nonlinear hamiltonian wave equation the chapter is also accompanied by some examples

blow up at the boundary for degenerate semilinear parabolic equations this paper treats a superlinear parabolic equation degenerate in the time derivative it is shown that the solution may blow up in finite time moreover it is proved that for a large class of initial data blow up occurs at the boundary of the domain when the nonlinearity is no worse than quadratic various estimates are obtained which determine the asymptotic behaviour near the blow up the mathematical analysis is then extended to equations with other degeneracies

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 trigonometrically fitted two step hybrid methods for oscillatory problems abstract in this paper we propose a general class of trigonometrically fitted two step hybrid tftsh methods for solving numerically oscillatory second order initial value problems the tftsh methods integrate exactly the differential system whose solutions can be expressed as the linear combinations of functions from the set exp i w t exp i w t or equivalently the set cos w t sin w t where w represents an approximation of the main frequency of the problem by introducing the generalized b2 series the necessary and sufficient conditions for tftsh methods of up to arbitrarily high order p are derived we also investigate the symmetry of tftsh methods and analyze the symmetric conditions of tftsh methods based on the order conditions and symmetric conditions a diagonally implicit two stage symmetric tftsh method with order four is constructed some numerical experiments are provided to confirm the theoretical expectations

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

finite difference scheme to solve schrodinger equations comparisons are made among several three node finite difference schemes fds s for solving time independent schr odinger equations it is shown that the mickens fds is although exact in some special cases generally two orders lower than the numerov fds an alternative fds the combined numerov mickens fds is introduced it has advantages of both the numerov and the mickens fds s numerical comparison among these fds s is presented

on a class of hyperbolic equations in weyl hormander calculus abstract in this paper we study the cauchy problem for hyperbolic equations in the setting of hormander s m g classes we provide regularity estimates existence and uniqueness in the scale of sobolev spaces h m g adapted to the weyl hormander calculus we also obtain estimates for some parabolic evolution equations

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

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

a complete characterization of extinction versus positivity of solutions to a parabolic problem of p laplacian type in graphs abstract this work is concerned with the extinction and positivity properties of solutions to the discrete p laplacian parabolic equation u t p u f u p 1 with a nonlinear absorption on a weighted graph we provide a complete characterization for f to see when we have an extinctive solution or a positive solution in addition the extinction time is estimated for the extinctive solutions and the decay rate is given to the positive solutions finally we give several numerical examples which illustrate the main results

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

on the positivity monotonicity and stability of a semi adaptive lod method for solving three dimensional degenerate kawarada equations abstract this paper concerns the numerical solution of three dimensional degenerate kawarada equations these partial differential equations possess highly nonlinear source terms and exhibit strong quenching singularities which pose severe challenges to the design and analysis of highly reliable schemes arbitrary fixed nonuniform spatial grids which are not necessarily symmetric are considered throughout this study the numerical solution is advanced through a semi adaptive local one dimensional lod integrator the temporal adaptation is achieved via a suitable arc length monitoring mechanism criteria for preserving the positivity and monotonicity are investigated and acquired the numerical stability of the splitting method is proven in the von neumann sense under the spectral norm extended stability expectations are proposed and investigated

trigonometrically fitted multi step rkn methods for second order oscillatory initial value problems abstract trigonometrically fitted multi step runge kutta nystrom tfmsrkn methods for solving numerically oscillatory special second order initial value problems are introduced tfmsrkn methods integrate exactly the differential system whose solutions can be expressed as the linear combinations of functions from the set exp i w t exp i w t or equivalently the set cos wt sin wt where w represents an approximation of the main frequency of the problem the corresponding order conditions are given and two explicit four stage tfmsrkn methods with order five are constructed stability of the new methods is examined and the regions of stability are depicted numerical results show that our new methods are more efficient in comparison with other well known high quality methods proposed in the scientific literature

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

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 complete characterization of nonlinear absorption for the evolution p laplacian equations to have positive or extinctive solutions this paper is concerned with long time behaviors of solutions to the initial boundary value problem of the evolution p laplacian equations with nonlinear absorption a long time behavior of solutions to these equations has been studied so far for particular types of nonlinear absorption term the purpose of this paper is to give a complete characterization of the nonlinear absorption term according to the parameter p in p laplacian operator and the growth of the nonlinear absorption term near the origin in order to determine whether the solution to the equations is extinctive or positive in addition we also give upper bounds for extinction times of solutions

controlling the error growth in long term numerical integration of perturbed oscillations in one or several frequencies in this paper we present a simple technique that allows us to limit the error growth in the long term numerical integration of perturbed multi dimensional oscillators while using highly efficient and accurate special multi step codes after laying down the theoretical basis of their behaviour we then illustrate them with some numerical examples including a case with non resonant frequencies seldom found in the related literature

new explicit adapted numerov methods for second order oscillatory differential equations this paper is concerned with a novel class of explicit adapted numerov an methods for solving initial value problems of second order oscillatory differential equations of the form y email protected 2y f x y the scheme of the methods incorporate the special structure of the problem carried by the term w 2y so that they can integrate the linear equation y email protected 2y 0 without truncation error a new type of simplifying assumption is proposed on the linear operators related to the quadratures that lead to the an scheme a practical an method of order five and three an methods of order six are derived the phase and stability properties of the new methods are analyzed the results of numerical experiments show the robustness and competence of the new methods compared with some highly efficient codes in the recent literature

a new family of runge kutta type methods for the numerical integration of perturbed oscillators our task in this paper is to present a new family of methods of the runge kutta type for the numerical integration of perturbed oscillators the key property is that those algorithms are able to integrate exactly without truncation error harmonic oscillators and that for perturbed problems the local error contains the perturbation parameter as a factor some numerical examples show the excellent behaviour when they compete with runge kutta nystrom type methods

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

multistep numerical methods based on the scheifele g functions with application to satellite dynamics in 1971 scheifele designed a method for the integration of perturbed oscillators by refining the classical method of power series this method consists of defining a certain sequence of entire functions and finding the sought solution in the form of a linear combination of them using coefficients obtained through recurrence formulas in spite of the good behavior of the scheifele method it is only practical for use in a few particular cases in which the force function is very simple given the complexity of the preliminary calculations needed to obtain the recurrence formulas this problem is resolved in this article by converting the g functions method into a multistep scheme the key property is that those algorithms are able to integrate without truncation error harmonic oscillators involving various frequencies and that for perturbed problems the local error contains the small perturbation parameter as a factor therefore the methods are useful for finding highly accurate solutions to problems whose solutions are close to quasi periodic ones an application is performed in investigating the behavior of the proposed schemes for the integration of the orbital motion of earth satellites

numerical solutions to singular reaction diffusion equation over elliptical domains solid fuel ignition models for which the dynamics of the temperature is independent of the single species mass fraction attempt to follow the dynamics of an explosive event such models may take the form of singular degenerate reaction diffusion equations of the quenching type that is the temporal derivative blows up in finite time while the solution remains bounded theoretical and numerical investigations have proved difficult for even the simplest of geometries and mathematical degeneracies quenching domains are known to exist for piecewise smooth boundaries but often lack theoretical estimates rectangular geometries have been primarily studied here this acquired knowledge is utilized to determine new theoretical estimates to quenching domains for arbitrary piecewise smooth connected geometries elliptical domains are of primary interest and a peaceman rachford splitting algorithm is then developed that employs temporal adaptation and nonuniform grids rigorous numerical analysis ensures numerical solution monotonicity positivity and linear stability of the proposed algorithm simulation experiments are provided to illustrate the accomplishments

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

numerical quenching for the semilinear heat equation with a singular absorption in this paper we study the numerical approximation for the heat equation with a singular absorption we prove that the numerical quenching rate coincides with the continuous one we also see that the quenching time and the quenching set converge to the continuous one in fact under some restriction on the initial data the numerical quenching coincides with the continuous one finally we give some numerical results to illustrate our analysis

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

a class of explicit two step hybrid methods for second order ivps a class of explicit two step hybrid methods for the numerical solution of second order ivps is presented these methods require a reduced number of stages per step in comparison with other hybrid methods proposed in the scientific literature new explicit hybrid methods which reach up to order five and six with only three and four stages per step respectively and which have optimized the error constants are constructed the numerical experiments carried out show the efficiency of our explicit hybrid methods when they are compared with classical runge kutta nystrom methods and other explicit hybrid codes proposed in the scientific literature

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

generalization of the stormer method for perturbed oscillators without explicit first derivatives abstract in this work we construct a numerical method specially designed for perturbed oscillators without explicit first derivatives the new method which generalizes the classical stormer and cowell codes for second order equations is useful when the calculation of the first derivative is not needed because this derivative is not computed some numerical examples of the artificial satellite problem show the good behaviour of the methods when they compete against classical multistep and runge kutta nystrom codes

an explicit sixth order method with phase lag of order eight for y f t y abstract recently thomas 6 gave a two step sixth order method with phase lag 1 101478 67 h8 for the numerical integration of periodic initial value problems y t y y t 0 y 0 y t 0 y 0 however thomas method is implicit it possesses an interval of periodicity of size 2 77 and for nonlinear problems her method requires 6i 1 function evaluations for i modified newton iterations for the solution of the resulting implicit equations at each step in the present paper we present a new two step sixth order method which also has phase lag of order eight but with a smaller constant given by 1 3 628 800 h8 in contrast with thomas method our method is explicit possesses a larger interval of periodicity of size 4 63 and it is decidedly more economical since it involves only six function evaluations per step numerical experiments confirm the superiority of our present method over thomas method