on stable numerical differentiation based on a regularized volterra equation two different approaches for numeri cal differentiation are considered the first approach consists of solving a regularized volterra equation while the second approach is based on solving a disretized version of the regularized volterra equation numerical experiments show that these methods are efficient and compete fa vorably with the variational regularization method for stable calculating the derivatives of noisy functions

interior regularity estimates for elliptic systems of difference equations we prove interior regularity estimates for a large class of difference approximations for elliptic systems of partial differential equations these estimates are analogous to those for the systems of differential equations from these regularity estimates we obtain estimates on the convergence of the solutions of the difference equations a theory of pseudo difference operators with order is used to prove the regularity results we also comment on factors to be considered in choosing a difference scheme for computations

locating invariant tori for a family of two dimensional hamiltonian mappings departamento de estatistica matemat apl e computaoinstituto de geociencias e ciencias exatas unesp univ estadual paulista av 24a 1515bela vista cep 13506 900 rio claro sp

finite dimensional approximation settings for infinite dimensional moore penrose inverses after a brief survey of basic results about finite dimensional approximation settings such as mathematical frameworks of various projection methods including the least squares method the dual least squares method and the galerkin method for infinite dimensional moore penrose inverses this paper proceeds to a detailed study from the following aspects for projection methods we investigate convergence and weak convergence of their approximation setting to develop a unified theory on projection methods for infinite dimensional moore penrose inverses this investigation yields a fundamental convergence theorem theorem 2 2 from which the criterion for convergence the criterion for weak convergence and the generalized dual least squares method are derived we also derive general results on the least squares method by which two flaws in groestch s results are corrected for nonprojection methods whose approximation setting is a more general framework we investigate weak perfect convergence of their approximation setting and provide a necessary and sufficient condition of such convergence holding theorem 3 2 several examples are proposed as counterexamples to illustrate the differences between some important concepts or as concrete algorithms to show how the present work can help to analyze their behavior

stability instability and error of the force based quasicontinuum approximation due to their algorithmic simplicity and high accuracy force based model coupling techniques are popular tools in computational physics for example the force based quasicontinuum qcf approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model in this paper we present a detailed stability and error analysis of this method our optimal order error estimates provide a theoretical justification for the high accuracy of the qcf approximation they clearly demonstrate that the computational efficiency of continuum modeling can be utilized without a significant loss of accuracy if defects are captured in the atomistic region the main challenge we need to overcome is the fact that the linearized qcf operator is typically not positive definite moreover we prove that no uniform inf sup stability condition holds for discrete versions of the w1 p w1 q duality pairing with 1 p 1 q 1 if 1 p however we were able to establish an inf sup stability condition for a discrete version of the w1 w1 1 duality pairing which leads to optimal order error estimates in a discrete w1 norm

fine structure of sticky sets in mushroom billiards we consider sticky sets in the phase space of circular mushroom billiards which are referred to in physics literature as mupos marginally unstable periodic orbits an exact description of the set of parameters widths of the mushrooms stems without or with a finite set of mupos is given it is shown that there exist a continuum of mupoless parameters we also estimate from above a number of mupos which exist for widths of the mushroom s stem which correspond to rational numbers and present a simple approach for finding mushroom billiards without mupos

stability of a force based hybrid method with planar sharp interface we study a force based hybrid method that couples an atomistic model with a cauchy born elasticity model with sharp transition interface we identify stability conditions that guarantee the convergence of the hybrid scheme to the solution of the atomistic model with second order accuracy as the ratio between lattice parameter and the characteristic length scale of the deformation tends to zero convergence is established for hybrid schemes with planar sharp interface for systems without defects with general finite range atomistic potential and simple lattice structure the key ingredients of the proof are regularity and stability analysis of elliptic systems of difference equations we apply the results to atomistic to continuum scheme for a two dimensional triangular lattice with planar interface

quasi implicit and two level explicit finite difference procedures for solving the one dimensional advection equation a variety of explicit and implicit algorithms has been studied dealing with the solution of the one dimensional advection equation these schemes are based on the weighted finite difference approximations this solution approach proves to be very effective this procedure is carried out in many disciplines including hydrology oceanography and meteorology the main idea behind the finite difference methods for obtaining the solution of a given partial differential equation is to approximate the derivatives appearing in the equation by a set of values of the function at a selected number of points the most usual way to generate these approximations is through the use of taylor series the basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach it is worth noting that from the truncation error of the modified equivalent equation it is possible to eliminate the dominant error terms associated with the finite difference equations that contain weights thus leading to more accurate techniques quasi implicit and two level explicit finite difference techniques are considered in this article to approximately solve the one dimensional advection problem stability of the techniques are investigated using the von neumann stability analysis the results of a numerical experiment are presented and the accuracy and central processor unit cpu time needed are discussed and compared numerical tests show that the new third order or fourth order methods are more accurate than either of the second order techniques

power law decay and self similar distributions in stadium type billiards orbits of particles in hamiltonian systems may spend long times near invariant sets these orbits called sticky orbits can lead to self similar probability distributions and power law decay we study problems in stadium type billiards where the sticky invariant sets consist of orbits which are perpendicular to the straight boundaries of the billiard we consider the time dependence originating from various initial distributions of the angle of incidence for an ensemble of particles in the stadium billiard in an open variants of the stadium billiard in which most of the circular wall is removed allowing orbits to leave the billiard and in a quarter stadium billiard in which the stadium is bisected by horizontal and vertical walls with a porous vertical wall we find that in each of these cases the relaxing distributions are asymptotically self similar and that the particle populations exhibit algebraic decay with time power law decay exponents are determined for the various situations considered

stickiness in hamiltonian systems from sharply divided to hierarchical phase space we investigate the dynamics of chaotic trajectories in simple yet physically important hamiltonian systems with nonhierarchical borders between regular and chaotic regions with positive measures we show that the stickiness to the border of the regular regions in systems with such a sharply divided phase space occurs through one parameter families of marginally unstable periodic orbits and is characterized by an exponent ensuremath gamma 2 for the asymptotic power law decay of the distribution of recurrence times generic perturbations lead to systems with hierarchical phase space where the stickiness is apparently enhanced due to the presence of infinitely many regular islands and cantori in this case we show that the distribution of recurrence times can be composed of a sum of exponentials or a sum of power laws depending on the relative contribution of the primary and secondary structures of the hierarchy numerical verification of our main results are provided for area preserving maps mushroom billiards and the newly defined magnetic mushroom billiards

regularity estimates up to the boundary for elliptic systems of difference equations this paper proves regularity estimates up to the boundary for solutions of elliptic systems of finite difference equations the regularity estimates obtained for boundary fitted coordinate systems on domains with smooth boundary involve discrete sobolev norms and are proved using pseudodifference operators to treat systems with variable coefficients the elliptic systems of difference equations and the boundary conditions that are considered are very general in form it is proved that regularity of a regular elliptic system of difference equations is equivalent to the nonexistence of eigensolutions the regularity estimates obtained are analogous to those in the theory of elliptic systems of partial differential equations and to the results of gustafsson kreiss and sundstrom math comp 26 1972 pp 649 686 and others for hyperbolic difference equations

a meshless galerkin scheme for the approximate solution of nonlinear logarithmic boundary integral equations utilizing radial basis functions abstract this paper presents a computational scheme to solve nonlinear logarithmic singular boundary integral equations these types of integral equations arise from boundary value problems of laplace s equations with nonlinear robin boundary conditions the discrete galerkin method together with the inverse multiquadric radial basis functions established on scattered points is utilized to approximate the solution the discrete galerkin method for solving boundary integral equations results from the numerical integration of all integrals in the method the proposed scheme uses a special accurate quadrature formula via the nonuniform gauss legendre integration rule to compute logarithm like singular integrals appeared in the scheme since the numerical method developed in the current paper does not require any mesh generations on the boundary of the domain it is meshless and does not depend to the domain form we also investigate the error analysis of the proposed method illustrative examples show the reliability and efficiency of the new scheme and confirm the theoretical error estimates

meeting time distributions in bernoulli systems meeting time is defined as the time for which two orbits approach each other within the distance in phase space we show that the distribution of the meeting time is exponential in p1 pk bernoulli systems in the limit of 0 the distribution converges to exp where is the meeting time normalized by the average the exponent is shown to be kl 1pl 1 pl for the bernoulli systems

poincare recurrences and transient chaos in systems with leaks in order to simulate observational and experimental situations we consider a leak in the phase space of a chaotic dynamical system we obtain an expression for the escape rate of the survival probability by applying the theory of transient chaos this expression improves previous estimates based on the properties of the closed system and explains dependencies on the position and size of the leak and on the initial ensemble with a subtle choice of the initial ensemble we obtain an equivalence to the classical problem of poincare recurrences in closed systems which is treated in the same framework finally we show how our results apply to weakly chaotic systems and justify a split of the invariant saddle into hyperbolic and nonhyperbolic components related respectively to the intermediate exponential and asymptotic power law decays of the survival probability

weakly noisy chaotic scattering received 30 april 2013 published 23 september 2013 the effect of a weak source of noise on the chaotic scattering is relevant to situations of physical interest we investigate how a weak source of additive uncorrelated gaussian noise affects both the dynamics and thetopology of a paradigmatic chaotic scattering problem as the one taking place in the open nonhyperbolic regimeof the h enon heiles hamiltonian system we have found long transients for the time escape distributions forcritical values of the noise intensity for which the particles escape slower as compared with the noiseless case an analysis of the survival probability of the scattering function versus the gaussian noise intensity shows asmooth curve with one local maximum and with one local minimum which are related to those long transientsand with the basin structure in phase space on the other hand the computation of the exit basins in phase spaceshows a quadratic curve for which the basin boundaries lose their fractal like structure as noise turned on doi 10 1103 physreve 88 032914 pacs number s 05

a wavelet galerkin method for high order numerical differentiation numerical differentiation is a classical ill posed problem in this paper we propose a wavelet galerkin method for high order numerical differentiation by an appropriate choice of the regularization parameter an order optimal stability estimate of holder type is obtained some numerical examples show that the method is effective and stable

open mushrooms stickiness revisited we investigate mushroom billiards a class of dynamical systems with sharply divided phase space for typical values of the control parameter of the system an infinite number of marginally unstable periodic orbits mupos exist making the system sticky in the sense that unstable orbits approach regular regions in phase space and thus exhibit quasi regular behaviour for long periods of time the problem of finding these mupos is expressed as the well known problem of finding optimal rational approximations of a real number subject to some system specific constraints by introducing a generalized mushroom and using properties of continued fractions we describe for the first time a zero measure set of control parameter values 0 1 for which all mupos are destroyed and therefore the system is less sticky leading to a different power law exponent for the poincar recurrence time distribution statistics the open mushroom billiard with a hole is then considered in order to quantify the stickiness exhibited due to mupos and exact leading order expressions for the algebraic decay of the survival probability function are calculated for mushrooms with triangular and rectangular stems numerical simulations are also performed which confirm our predictions for both sticky and less sticky mushrooms

structure size and statistical properties of chaotic components in a mixed type hamiltonian system we perform a detailed study of the chaotic component in mixed type hamiltonian systems on the example of a family of billiards introduced by robnik in j phys a math gen 16 3971 1983 jphac50305 447010 1088 0305 4470 16 17 014 the phase space is divided into a grid of cells and a chaotic orbit is iterated a large number of times the structure of the chaotic component is discerned from the cells visited by the chaotic orbit the fractal dimension of the border of the chaotic component for various values of the billiard shape parameter is determined with the box counting method the cell filling dynamics is compared to a model of uncorrelated motion the so called random model robnik et al j phys a math gen 30 l803 1997 jphac50305 447010 1088 0305 4470 30 23 003 and deviations attributed to sticky objects in the phase space are found the statistics of the number of orbit visits to the cells is analyzed and found to be in agreement with the random model in the long run the stickiness of the various structures in the phase space is quantified in terms of the cell recurrence times the recurrence time distributions in a few selected cells as well as the mean and standard deviation of recurrence times for all cells are analyzed the standard deviation of cell recurrence time is found to be a good quantifier of stickiness on a global scale three methods for determining the measure of the chaotic component are compared and the measure is calculated for various values of the billiard shape parameter lastly the decay of correlations and the diffusion of momenta is analyzed

numerical solution of the three dimensional advection diffusion equation the study of advection diffusion equation continues to be an active field of research the subject has important applications to fluid dynamics as well as many other branches of science and engineering in this paper several different numerical techniques will be developed and compared for solving the three dimensional advection diffusion equation with constant coefficient these techniques are based on the two level fully explicit and fully implicit finite difference approximations the basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach developed from the 1974 work of warming and hyett this allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes another nice feature of the modified equivalent partial differential equation approach is that a high order of accuracy can be combined with excellent stability properties the new second order accurate methods are free of numerical diffusion the results of a numerical experiment are presented and the accuracy and central processor cpu time needed are discussed and compared

weak dissipative effects on trajectories from the edge of basins of attraction abstract the purpose of this work is to present convergence properties of regular and chaotic conservative trajectories under small dissipation it is known that when subjected to dissipation stable periodic points become sinks attracting the surrounding trajectories which belong to rational irrational tori while chaotic trajectories converge to a chaotic attractor if it exists using the standard map and a mixed plot we show that this simple scenario can be rather complicated and strongly depends on the dissipation intensity for small dissipations the huge amount of attractors of the phase space generates a complicated and intricate dynamics where trajectories are steered to their attractors based on the local non hyperbolicity measured by the lyapunov vectors dissipation creates holes or attracting channels on the torus from the conservative limit and allows trajectories to penetrate it these holes are regions of large local hyperbolicity and are related to sticky channels reported recently for stronger dissipation sinks tend to attract all trajectories prevailing over the chaotic attractor

a finite difference finite element technique with error estimate for space fractional tempered diffusion wave equation abstract an efficient numerical technique is proposed to solve one and two dimensional space fractional tempered fractional diffusion wave equations the space fractional is based on the riemann liouville fractional derivative at first the temporal direction is discretized using a second order accurate difference scheme then a classic galerkin finite element is employed to obtain a full discrete scheme furthermore for the time discrete and the full discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method finally two test problems have been illustrated to verify the efficiency and simplicity of the proposed technique

convergence of a force based hybrid method in three dimensions we study a force based hybrid method that couples an atomistic model with the cauchy born elasticity model we show that the proposed scheme converges to the solution of the atomistic model with second order accuracy since the ratio between lattice parameter and the characteristic length scale of the deformation tends to 0 convergence is established for the three dimensional system without defects with general finite range atomistic potential and simple lattice structure the proof is based on consistency and stability analysis general tools for stability analysis are developed in the framework opseudodifference operators in arbitrary dimensions 2012 wiley periodicals inc

cauchy born rule and the stability of crystalline solids static problems we study the connection between atomistic and continuum models for the elastic deformation of crystalline solids at zero temperature we prove under certain sharp stability conditions that the correct nonlinear elasticity model is given by the classical cauchy born rule in the sense that elastically deformed states of the atomistic model are closely approximated by solutions of the continuum model with stored energy functionals obtained from the cauchy born rule the analysis is carried out for both simple and complex lattices and for this purpose we develop the necessary tools for performing asymptotic analysis on such lattices our results are sharp and they also suggest criteria for the onset of instabilities of crystalline solids

escape of particles in a time dependent potential well we investigate the escape of an ensemble of noninteracting particles inside an infinite potential box that contains a time dependent potential well the dynamics of each particle is described by a two dimensional nonlinear area preserving mapping for the variables energy and time leading to a mixed phase space the chaotic sea in the phase space surrounds periodic islands and is limited by a set of invariant spanning curves when a hole is introduced in the energy axis the histogram of frequency for the escape of particles which we observe to be scaling invariant grows rapidly until it reaches a maximum and then decreases toward zero at sufficiently long times a plot of the survival probability of a particle in the dynamics as function of time is observed to be exponential for short times reaching a crossover time and turning to a slower decay regime due to sticky regions observed in the phase space

galerkin method with trigonometric basis on stable numerical differentiation abstract this paper considers the p p 1 2 3 order numerical differentiation on function y in 0 2 they are transformed into corresponding fredholm integral equation of the first kind computational schemes with analytic solution formulas are designed using galerkin method on trigonometric basis convergence and divergence are all analysed in corollaries 5 1 5 2 and a priori error estimate is uniformly obtained in theorem 6 1 7 1 7 2 therefore the algorithm achieves the optimal convergence rate o 2 2 1 1 2 or 1 with periodic sobolev source condition of order 2 p besides we indicate a noise independent a priori parameter choice when the function y possesses the form of k 0 p 1 a k t k k 1 n 1 b k cos k t k 1 n 2 c k sin k t b n 1 c n 2 0 in particular in numerical differentiations for functions above good filtering effect error approaches 0 is displayed with corresponding parameter choice in addition several numerical examples are given to show that even derivatives with discontinuity can be recovered well

force based atomistic continuum blending for multilattices we formulate the blended force based quasicontinuum method for multilattices and develop rigorous error estimates in terms of the approximation parameters choice of atomistic region blending region and continuum finite element mesh balancing the approximation parameters yields a convergent atomistic continuum multiscale method for multilattices with point defects including a rigorous convergence rate in terms of the computational cost the analysis is illustrated with numerical results for a stone wales defect in graphene