pade numerical schemes for the sine gordon equation abstract the sine gordon equation turn up in several problems in science and engineering although it is integrable in practical applications its numerical solution is powerful and versatile four novel implicit finite difference methods based on q s pade approximations with q s th order in space have been developed and analyzed for this equation all share the same treatment for the nonlinearity and integration in time concretely 0 4 2 2 2 4 and 4 4 pade methods additionally the energy conserving strauss vazquez scheme has been considered in a 0 2 pade implementation these methods have been compared among them for both the kink antikink and breather solutions in terms of global error computational cost and energy conservation the 0 4 and 2 4 pade methods are the most cost effective ones for small and large global error respectively our results indicate that spatial order of accuracy is more relevant to effectiveness of a method than energy conservation even in very long time integrations

vibration analysis by discrete singular convolution this paper explores the utility of a discrete singular convolution algorithm for vibration analysis a number of different realizations of singular convolution kernels are selected to illustrate the present algorithm vibration analysis of strings rods beams diatomic molecules membranes waveguides and thin plates are utilized to test numerical accuracy and speed of convergence of the present approach numerical experiments indicate that the discrete singular convolution is a simple and reliable algorithm for vibration analysis

numerical analysis of free vibrations of laminated composite conical and cylindrical shells discrete singular convolution dsc approach a numerical study on the free vibration analysis for laminated conical and cylindrical shell is presented the analysis is carried out using love s first approximation thin shell theory and solved using discrete singular convolution dsc method numerical results in free vibrations of laminated conical and cylindrical shells are presented graphically for different geometric and material parameters free vibrations of isotropic cylindrical shells and annular plates are treated as special cases the effects of circumferential wave number number of layers on frequencies characteristics are also discussed the numerical results show that the present method is quite easy to implement accurate and efficient for the problems considered

homotopy analysis method for the sine gordon equation with initial conditions in this work approximate analytical solution of the sine gordon equation with initial conditions is obtained by the homotopy analysis method ham the ham solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions it is shown that the solutions obtained by the variational iteration method vim are only special cases of the ham solutions

geometry of the modulational instability iii homoclinic orbits for the periodic sine gordon equation in this paper the homoclinic geometric structure of the integrable sine gordon equation under periodic boundary conditions is developed specifically focus is given to orbits homoclinic to n tori simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given a labeling is provided which identifies and catalogues all such orbits these orbits are related in a one to one manner to linearized instabilities explicit formulas for all homoclinic orbits are given in terms of backlund transformations

dsc time domain solution of maxwell s equations a new computational algorithm the discrete singular convolution dsc is introduced for solving scattering and guided wave problems described by time domain maxwell s equations the dsc algorithm is utilized for the spatial discretization and the fourth order runge kutta scheme is used for the time advancing staggered meshes are used for electromagnetic fields four standard test problems a hollow air filled waveguide a dielectric slab loaded rectangular waveguide a shield microstrip line and a dielectric square are employed to illustrate the usefulness to test the accuracy and to explore the limitation of the dsc algorithm results are compared with those of finite difference scaling function multi resolution time domain and finite element based high frequency structure simulator numerical experiments indicate that the present algorithm is a promising approach for achieving high accuracy in electromagnetic wave computations

discrete singular convolution and its application to the analysis of plates with internal supports part 2 applications part 2 of this series of two papers presents the applications of the discrete singular convolution dsc algorithm the main purpose of this paper is to explore the utility test the accuracy and examine the convergence of the proposed approach for the vibration analysis of rectangular plates with internal supports both partial internal line supports and complex internal supports are considered for 21 square plates of various combinations of edge support conditions the effects of different size shape and topology of the internal supports and different boundary conditions on the vibration response of plates are investigated the partial internal line supports may vary from a central point support to a full range of cross or diagonal line supports several closed loop supports such as ring square and rhombus and their combinations are studied for complex internal supports convergence and comparison studies are carried out to establish the correctness and accuracy of the dsc algorithm the dsc results are compared with those in the available literature obtained by using other methods numerical results indicate that the dsc algorithm exhibits controllable accuracy for plate analysis and shows excellent flexibility in handling complex geometries boundary conditions and support conditions copyright 2002 john wiley sons ltd

travelling wave solutions for combined and double combined sine cosine gordon equations by the variable separated ode method abstract a combined sine cosine gordon and a double combined sine cosine gordon equations that combine the effects of sine and cosine are investigated the analysis rests mainly on the newly developed variable separated ode method sirendaoreji sun jiong a direct method for solving sinh gordon type equation phys lett a 298 2002 133 139 exact travelling wave solutions for the combined equations are formally derived by using this method the work emphasizes the power of the method that can be used in problems of identical nonlinearities

plate vibration under irregular internal supports this paper studies the problem of plate vibration under complex and irregular internal support conditions such a problem has its widely spread industrial applications and has not been addressed in the literature yet partially due to the numerical difficulties a novel computational method discrete singular convolution dsc is introduced for solving this problem the dsc algorithm exhibits controllable accuracy for approximations and shows excellent flexibility in handling complex geometries boundary conditions and internal support conditions convergence and comparison studies are carried out to check the validity and accuracy of the dsc method case studies are considered to the combination of a few different boundary conditions and irregular internal supports the latter are generated by using an image processing algorithm completely independent verifications are conducted by using the established pb 2 ritz method which is available for two relatively simpler support patterns the morphology of the first few eigenmodes is found to be localized to largest support free spatial regions

wavelets generated by using discrete singular convolution kernels this paper explores the connection between wavelet methods and an efficient computational algorithm the discrete singular convolution dsc many new dsc kernels are constructed and they are identified as wavelet scaling functions two approaches are proposed to generate wavelets from dsc kernels two well known examples the canny filter and the mexican hat wavelet are found to be special cases of the present dsc kernel generated wavelets a family of wavelet generators proposed in this paper are found to form an infinite dimensional lie group which has an invariant subgroup of translation and dilation if dsc kernels form an orthogonal system they are found to span a wavelet subspace in a multiresolution analysis

an overview of wavelet based multiresolution analyses in this paper an overview of wavelet based multiresolution analyses is presented first the continuous wavelet transform in its simplest form is discussed then the definition of a multiresolution analysis is given and how wavelets fit into it is shown the authors take a closer look at orthogonal biorthogonal and semiorthogonal wavelets the fast wavelet transform wavelets on an interval multidimensional wavelets and wavelet packets are discussed several examples of wavelet families are introduced and compared finally the essentials of two major applications are outlined data compression and compression of linear operators

discrete singular convolution finite subdomain method for the solution of incompressible viscous flows this paper proposes a discrete singular convolution finite subdomain method dsc fsm for the analysis of incompressible viscous flows in multiply connected complex geometries the dsc algorithm has its foundation in the theory of distributions a block structured grid of fictitious overlapping interfaces is designed to decompose a complex computational geometry into a finite number of subdomains in each subdomain the governing navier stokes equations are discretized by using the dsc algorithm in space and a third order runge kutta scheme in time information exchange between fictitious overlapping zones is realized by using the dsc interpolating algorithm the taylor problem with decaying vortices could be solved to machine precision with an excellent comparison against the exact solution the reliability of the proposed method is tested by simulating the flow in a lid driven cavity the utility of the dsc fsm approach is further illustrated by two other benchmark problems viz the flow over a backward facing step and the laminar flow past a square prism the present results compare well with the numerical and experimental data available in the literature

a unified approach for the solution of the fokker planck equation this paper explores the use of a discrete singular convolution algorithm as a unified approach for numerical integration of the fokker planck equation the unified features of the discrete singular convolution algorithm are discussed it is demonstrated that different implementations of the present algorithm such as global local galerkin collocation and finite difference can be deduced from a single starting point three benchmark stochastic systems the repulsive wong process the black scholes equation and a genuine nonlinear model are employed to illustrate the robustness and to test the accuracy of the present approach for the solution of the fokker planck equation via a time dependent method an additional example the incompressible euler equation is used to further validate the present approach for more difficult problems numerical results indicate that the present unified approach is robust and accurate for solving the fokker planck equation

a note on the numerical solution of high order differential equations numerical solution of high order differential equations with multi boundary conditions is discussed in this paper motivated by the discrete singular convolution algorithm the use of fictitious points as additional unknowns is proposed in the implementation of locally supported lagrange polynomials the proposed method can be regarded as a local adaptive differential quadrature method two examples an eigenvalue problem and a boundary value problem which are governed by a sixth order differential equation and an eighth order differential equation respectively are employed to illustrate the proposed method

tightness of the recentered maximum of the two dimensional discrete gaussian free field we consider the maximum of the discrete two dimensional gaussian free field gff in a box and prove that its maximum centered at its mean is tight settling a longstanding conjecture the proof combines a recent observation by bolthausen deuschel and zeitouni with elements from bramson s results on branching brownian motion and comparison theorems for gaussian fields an essential part of the argument is the precise evaluation up to an error of order 1 of the expected value of the maximum of the gff in a box related gaussian fields such as the gff on a two dimensional torus are also discussed 2011 wiley periodicals inc

cover times for brownian motion and random walks in two dimensions lett x denote the rst hitting time of the disc of radius centered at x for brownian motion on the two dimensional torus t 2 we prove that sup x2t2t x j log j 2 2 as 0 the same applies to brownian motion on any smooth compact connected two dimensional riemannian manifold with unit area and no boundary as a consequence we prove a conjecture due to aldous 1989 that the number of steps it takes a simple random walk to cover all points of the lattice torus z 2 is asymptotic to 4n 2 logn 2 determining these asymptotics is an essential step toward analyzing the fractal structure of the set of uncovered sites before coverage is complete so far this structure was only studied non rigorously in the physics literature we also establish a conjecture due to kesten and r ev esz that describes the asymptotics for the number of steps needed by simple random walk in z 2 to cover the disc of radius n

chaotic and homoclinic behavior for numerical discretizations of the nonlinear schro dinger equation abstract certain conservative discretizations of the nls can produce irregular behavior we consider the diagonal discretization as a conservative perturbation of the integrable discretization and study the homoclinic crossings in its nonlinear spectrum we find that irregularity sets in when two homoclinic structures are present and in this case many and continual homoclinic crossings occur throughout the irregular time series we indicate a melnikov analysis to study the consequences of this homoclinic behavior

discrete singular convolution and its application to the analysis of plates with internal supports part 1 theory and algorithm this paper presents a novel computational approach the discrete singular convolution dsc algorithm for analysing plate structures the basic philosophy behind the dsc algorithm for the approximation of functions and their derivatives is studied approximations to the delta distribution are constructed as either bandlimited reproducing kernels or approximate reproducing kernels unified features of the dsc algorithm for solving differential equations are explored it is demonstrated that different methods of implementation for the present algorithm such as global local galerkin collocation and finite difference can be deduced from a single starting point the use of the algorithm for the vibration analysis of plates with internal supports is discussed detailed formulation is given to the treatment of different plate boundary conditions including simply supported elastically supported and clamped edges this work paves the way for applying the dsc approach in the following paper to plates with complex support conditions which have not been fully addressed in the literature yet

a posteriori error analysis of the local discontinuous galerkin method for the sine gordon equation in one space dimension abstract in this paper we present asymptotically exact a posteriori error estimators of the local discontinuous galerkin ldg method for the nonlinear sine gordon equation in one space dimension we apply our recent optimal l 2 error estimates and superconvergence results baccouch 2018 to show that the ldg errors on each element can be split into two parts the first part is proportional to the p 1 degree radau polynomial and the second part converges with order p 3 2 in the l 2 norm when piecewise polynomials of degree at most p are used this new result is used to construct asymptotically exact a posteriori error estimates by solving a local steady problem on each element the proposed error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element we further prove that for smooth solutions these a posteriori error estimates for the solution and its derivative at a fixed time converge to the exact spatial errors in the l 2 norm under mesh refinement the theoretical order of convergence is proved to be p 3 2 while our computational results show higher o h p 2 convergence rate finally we prove that the global effectivity index converges to unity at o h 1 2 rate several numerical results are presented to validate the theoretical results

covering problems for brownian motion on spheres bounds are given on the mean time taken by a strong markov process to visit all of a finite collection of subsets of its state space these bounds are specialized to brownian motion on the surface of the unit sphere sigma p in r p this leads to bounds on the mean time taken by this brownian motion to come within a distance varepsilon of every point on the sphere and bounds on the mean time taken to come within varepsilon of every point or its opposite the second case is related to the grand tour a technique of multivariate data analysis that involves a search of low dimensional projections in both cases the bounds are asymptotically tight as varepsilon rightarrow 0 on sigma p for p geq 4

a numerical study of the large period limit of a zakharov shabat eigenvalue problem with periodic potentials deconinck and kutz 2006 j comput phys 219 296 321 developed an efficient algorithm for solving the zakharov shabat eigenvalue problem with periodic potentials numerically it is natural to use the same algorithm for solving the problem for non periodic potential decaying potentials defined over the whole real line using large periods in this paper we propose the use of a specific value of the floquet exponent our numerical results indicate that it can produce accurate results long before the period becomes large enough for the analytical convergence results of gardner 1997 j reine angew math 491 149 81 to be valid we also illustrate the rather complicated path to convergence of some nonlinear schr dinger potentials

geometry and boundary control of pattern formation and competition this paper presents the effective control of the formation and competition of cellular patterns simulation and theoretical analyses are carried out for pattern formation in a confined circular domain the cahn hilliard equation is solved with the zero flux boundary condition to describe the phase separation of binary mixtures a wavelet based discrete singular convolution algorithm is employed to provide high precision numerical solutions by extensive numerical experiments a set of cellular ordered state patterns are generated theoretical analysis is carried out by using the fourier bessel series modal decomposition shows that the pattern morphology of an ordered state pattern is dominated by a principal fourier bessel mode which has the largest fourier bessel decomposition amplitude interesting modal competition is also observed it is found that the formation and competition of cellular patterns are effectively controlled by the confined geometry and boundary condition 2002 elsevier science b v all rights reserved

a ray knight theorem for symmetric markov processes let x be a strongly symmetric recurrent markov process with state space s and let l t x denote the local time of x at x in s for a fixed element 0 in the state space s let tau t inf s l 0 s t the 0 potential density u 0 x y of the process x killed at t 0 inf s x s 0 is symmetric and positive definite let eta eta x x in s be a mean zero gaussian process with covariance e eta eta x eta y u 0 x y the main result of this paper is the following generalization of the classical second ray knight theorem for any b in r and t 0 l x tau t frac 1 2 eta x b 2 x in s frac 1 2 eta x sqrt 2 b 2 2 x in s text in law a version of this theorem is also given when x is transient

quasi wavelet based numerical method for volterra integro differential equations on unbounded spatial domains in this paper we study the numerical solution of the parabolic volterra integro differential equation with a memory term on one dimensional unbounded domain a new numerical method is proposed by applying the forward euler scheme in time and the quasi wavelet based numerical scheme in space some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method

free vibration analysis of symmetric laminated skew plates by discrete singular convolution technique based on first order shear deformation theory in this study free vibration of laminated skew plates was investigated discrete singular convolution dsc method is used for numerical solution of vibration problems the straight sided quadrilateral domain is mapped into a square domain in the computational space using a four node element by using the geometric transformation typical results are presented for different geometric parameters and boundary conditions it is concluded from the results that the skew angle have considerable influence on the variations of the frequencies with fibre orientation angle and number of layers in the laminate the results obtained by dsc method are compared with those obtained by analytical and numerical approaches it is shown that reasonable accurate results are obtained present work also indicates that the method of dsc is a promising and potential approach for computational mechanics copyright 2009 john wiley sons ltd

branching brownian motion seen from its tip it has been conjectured since the work of lalley and sellke ann probab 15 1052 1061 1987 that branching brownian motion seen from its tip e g from its rightmost particle converges to an invariant point process very recently it emerged that this can be proved in several different ways see e g brunet and derrida a branching random walk seen from the tip 2010 poissonian statistics in the extremal process of branching brownian motion 2010 arguin et al the extremal process of branching brownian motion 2011 the structure of this extremal point process turns out to be a poisson point process with exponential intensity in which each atom has been decorated by an independent copy of an auxiliary point process the main goal of the present work is to give a complete description of the limit object via an explicit construction of this decoration point process another proof and description has been obtained independently by arguin et al the extremal process of branching brownian motion 2011

a quasi periodic route to chaos in a near integrable pde abstract pattern formation and transitions to chaos are described for the damped ac driven one dimensional periodic sine gordon equation in a nonlinear schrodinger regime a generic quasi periodic route to intermittent chaos is exhibited in detail using a range of dynamical systems diagnostics in addition a nonlinear spectral transform is exploited i to identify and quantify coordinates of space time attractors in terms of a small number of soliton modes of the underlying integrable system ii to use these analytic coordinates to identify homoclinic orbits as possible sources of chaos and iii to demonstrate the significance of energy transfer between coherent and extended states in this chaotic system

discrete singular convolution for the prediction of high frequency vibration of plates theoretical analysis of high frequency vibrations is indispensable in a variety of engineering designs much effort has been made on this subject in the past few decades however there is no single technique which can be applied with confidence to various engineering structures for high frequency predictions at present this paper introduces a novel computational approach the discrete singular convolution dsc algorithm for high frequency vibration analysis of plate structures square plates with six distinct boundary conditions are considered to validate the proposed method a completely independent approach the levy method is employed to provide exact solutions for a comparison the proposed method is also validated by convergence studies we demonstrate the ability of the dsc algorithm for high frequency vibration analysis by providing extremely accurate frequency parameters for plates vibrating in the first 5000 modes

on the adaptive numerical solution of nonlinear partial differential equations in wavelet bases this work develops fast and adaptive algorithms for numerically solving nonlinear partial differential equations of the formut lu nf u where l and n are linear differential operators andf u is a nonlinear function these equations are adaptively solved by projecting the solutionuand the operators l and n into a wavelet basis vanishing moments of the basis functions permit a sparse representation of the solution and operators using these sparse representations fast and adaptive algorithms that apply operators to functions and evaluate nonlinear functions are developed for solving evolution equations for a wavelet representation of the solutionuthat containsnssignificant coefficients the algorithms update the solution usingo ns operations the approach is applied to a number of examples and numerical results are given

genealogy of extremal particles of branching brownian motion branching brownian motion describes a system of particles that diffuse in space and split into offspring according to a certain random mechanism by virtue of the groundbreaking work by m bramson on the convergence of solutions of the fisher kpp equation to traveling waves the law of the rightmost particle in the limit of large times is rather well understood in this work we address the full statistics of the extremal particles first second third largest etc in particular we prove that in the large t limit such particles descend with overwhelming probability from ancestors having split either within a distance of order 1 from time 0 or within a distance of order 1 from time t the approach relies on characterizing up to a certain level of precision the paths of the extremal particles as a byproduct a heuristic picture of branching brownian motion at the edge emerges which sheds light on the still unknown limiting extremal process 2011 wiley periodicals inc

dsc analysis of rectangular plates with non uniform boundary conditions this paper introduces the discrete singular convolution dsc for the vibration analysis of rectangular plates with non uniform and combined boundary conditions a systematic scheme is proposed for the treatment of boundary conditions required in the proposed approach the validity of the dsc approach for plate vibration is tested by using a large number of numerical examples that have a combination of simply supported clamped and transversely supported with non uniform elastic rotational restraint edges the present results are in excellent agreement with those in the literature

a numerical solution of the sine gordon equation using the modified decomposition method the decomposition method for solving the sine gordon equation has been implemented by using a number of initial values the explicit and numerical solutions of the equation are calculated in the form of convergent power series with easily computable components the present method performs extremely well in terms of accuracy efficiency simplicity stability and reliability

dsc solution for flow in a staggered double lid driven cavity a benchmark quality solution is presented for flow in a staggered double lid driven cavity obtained by using the wavelet based discrete singular convolution dsc the proposed wavelet based algorithm combines local methods flexibility and global methods accuracy and hence is a promising approach for achieving the high accuracy solution of the navier stokes equations block structured grids with pseudo overlapping subdomains are employed in the present simulation a third order runge kutta scheme is used for the temporal discretization quantitative results are presented apart from the qualitative fluid flow patterns the prevalence of rich features of flow morphology such as two primary vortex patterns merged single primary vortex patterns and secondary eddies makes this problem very attractive and interesting the problem is quite challenging for the possible existence of numerically induced asymmetric flow patterns and elliptic instability important computational issues like consistence convergence and reliability of the numerical scheme are examined the dsc algorithm is tested on the single lid driven cavity flow and the taylor problem with a closed form solution the double lid driven cavity simulations are cross validated with the standard second order finite volume method copyright 2003 john wiley sons ltd

comparison of discrete singular convolution and generalized differential quadrature for the vibration analysis of rectangular plates this paper presents a comprehensive comparison study between the discrete singular convolution dsc and the well known global method of generalized differential quadrature gdq for vibration analysis so as to enhance the understanding of the dsc algorithm the dsc method is implemented through lagrange s delta sequence kernel dsc lk which utilizes local lagrange polynomials to calculate weighting coefficients whereas the gdq requires global ones moreover it is shown that the treatments of boundary conditions and the use of grid systems are different in the two methods comparison study is carried out on 21 rectangular plates of different combinations of simply supported s clamped c and transversely supported with nonuniform elastic rotational restraint e edges and five rectangular plates of mixed supporting edges some of which with a range of aspect ratios and rotational spring coefficients all the results of the dsc lk agree very well with both those in the literature and newly computed gdq results furthermore it is observed that the dsc lk performs much better for plates vibrating at higher order eigenfrequencies unlike the gdq the dsc lk is numerically stable for problems which require a large number of grid points

the extremal process of branching brownian motion we prove that the extremal process of branching brownian motion in the limit of large times converges weakly to a cluster point process the limiting process is a randomly shifted poisson cluster process where the positions of the clusters is a poisson process with intensity measure with exponential density the law of the individual clusters is characterized as branching brownian motions conditioned to perform unusually large displacements and its existence is proved the proof combines three main ingredients first the results of bramson on the convergence of solutions of the kolmogorov petrovsky piscounov equation with general initial conditions to standing waves second the integral representations of such waves as first obtained by lalley and sellke in the case of heaviside initial conditions third a proper identification of the tail of the extremal process with an auxiliary process based on the work of chauvin and rouault which fully captures the large time asymptotics of the extremal process the analysis through the auxiliary process is a rigorous formulation of the cavity method developed in the study of mean field spin glasses

exact asymptotics of the freezing transition of a logarithmically correlated random energy model we consider a logarithmically correlated random energy model namely a model for directed polymers on a cayley tree which was introduced by derrida and spohn we prove asymptotic properties of a generating function of the partition function of the model by studying a discrete time analogy of the kpp equation thus translating bramson s work on the kpp equation into a discrete time case we also discuss connections to extreme value statistics of a branching random walk and a rescaled multiplicative cascade measure beyond the critical point

fourier bessel analysis of patterns in a circular domain this paper explores the use of the fourier bessel analysis for characterizing patterns in a circular domain a set of stable patterns is found to be well characterized by the fourier bessel functions most patterns are dominated by a principal fourier bessel mode n m which has the largest fourier bessel decomposition amplitude when the control parameter r is close to a corresponding non trivial root n m of the bessel function moreover when the control parameter is chosen to be close to two or more roots of the bessel function the corresponding principal fourier bessel modes compete to dominate the morphology of the patterns 2001 elsevier science b v all rights reserved

the variable separated ode and the tanh methods for solving the combined and the double combined sinh cosh gordon equations abstract exact travelling wave solutions for the combined sinh cosh gordon and the double combined sinh cosh gordon equation are formally derived the painleve property v eu will be used to back up the tanh method to derive a first set of solutions the newly developed variable separated ode method is also used to derive a second set of solutions each method provides travelling wave solutions of distinct physical structures

the tanh method exact solutions of the sine gordon and the sinh gordon equations in this work we use the tanh method for traveling wave solutions of the sine gordon and the sinh gordon equations several exact solutions of distinct physical structures are obtained the method is powerful with minimal algebra work and is demonstrated for four models

superconvergence of the local discontinuous galerkin method for the sine gordon equation in one space dimension abstract in this paper we present superconvergence results for the local discontinuous galerkin ldg method for the sine gordon nonlinear hyperbolic equation in one space dimension we identify a special numerical flux and a suitable projection of the initial conditions for the ldg scheme for which the l 2 norm of the ldg solution and its spatial derivative are of order p 1 when piecewise polynomials of degree at most p are used our numerical experiments demonstrate optimal order of convergence we further prove superconvergence toward particular projections of the exact solutions more precisely we prove that the ldg solution and its spatial derivative are o h p 3 2 super close to particular projections of the exact solutions while computational results show higher o h p 2 convergence rate our analysis is valid for arbitrary regular meshes and for p p polynomials with arbitrary p 1 numerical experiments validating these theoretical results are presented

a wavelet method for the characterization of spatiotemporal patterns this paper introduces a wavelet based method for the characterization of spatiotemporal patterns based on the wavelet multiresolution analysis two wavelet indices multiscale accumulative density mad and multiscale accumulative change mac are proposed for the characterization of the dynamics of the spatiotemporal patterns both indices are constructed by using orthogonal wavelet projection operators the mad is a measure of the spatial complexity of a pattern at a given time whereas the mac characterizes the spatial complexity of instantaneous change of the spatiotemporal patterns at a given time the ratio of the mad indices between the lowest and the highest scales reflects the order of coherence in a pattern the time series of both mad and mac provide the dynamical information of morphological pattern evolutions numerical experiments based on the cahn hilliard equation indicate that the proposed method is efficient for quantitatively characterizing the dynamics of the spatiotemporal patterns 2002 published by elsevier science b v

discrete singular convolution for the sine gordon equation abstract this paper explores the utility of a discrete singular convolution dsc algorithm for the integration of the sine gordon equation the initial values are chosen close to a homoclinic manifold for which previous methods have encountered significant numerical difficulties such as numerically induced spatial and temporal chaos a number of new initial values are considered including a case where the initial value is exactly on the homoclinic orbit the present algorithm performs extremely well in terms of accuracy efficiency simplicity stability and reliability

on homoclinic structure and numerically induced chaos for the nonlinear schro dinger equation it has recently been demonstrated that standard discretizations of the cubic nonlinear schrodinger nls equation may lead to spurious numerical behavior in particular the origins of numerically induced chaos and the loss of spatial symmetry are related to the homoclinic structure associated with the nls equation in this paper an analytic description of the homoclinic structure via soliton type solutions is provided and some consequences for numerical computations are demonstrated differences between an integrable discretization and standard discretizations are highlighted

on the numerical solution of the sine gordon equation in this the first of two papers on the numerical solution of the sine gordon equation we investigate the numerical behavior of a double discrete completely integrable discretization of the sine gordon equation for certain initial values in the vicinity of homoclinic manifolds this discretization admits an instability in the form of grid scale oscillations we clarify the nature of the instability through an analytical investigation supported by numerical experiments in particular a perturbation analysis of the associated linear spectral problem shows that the initial values used for the numerical experiments lie exponentially close to a homoclinic manifold this paves the way for the second paper where we use the non linear spectrum as a basis for comparing different numerical schemes

lagrange distributed approximating functionals lagrange distributed approximating functionals ldafs are proposed as the basis for a new collocation type method for accurately approximating functions and their derivatives both on and off discrete grids example applications are presented to illustrate the use of ldafs for solving the schr umlt o dinger equation and fokker planck equation ldafs are constructed by combining the daf concept with the lagrange interpolation scheme copyright ital 1997 ital the american physical society

chaotic transport and integrable instabilities in a near integrable many particle hamiltonian lattice abstract this paper is a report of detailed numerical experiments on a large dimensional hamiltonian lattice which arises as a nearly integrable discretization of the periodic sine gordon equation through four different one parameter families of initial data we focus on the interplay between numerically observable near integrable behavior quantified by modern statistical diagnostics and the integrable pde solutions and phase space geometry associated with the initial data transitions are observed within each family from quasiperiodic behavior to chaotic motion which is sometimes accompanied by transport toward equipartition these transitions in the near integrable system are correlated with integrable features of the initial data in particular transitions from neutrally stable to exponentially unstable sine gordon solutions and associated homoclinic orbits geometrically these correspond to transitions from neutrally stable tori to whiskered tori our results indicate that 1 positive lyapunov exponents our messure of chaos in near integrable orbits coincide one to one with integrable pde instabilities and homoclinic orbits 2 the statistical transition to equipartition of energy on a long fixed timescale occurs only if but not necessarily if there are integrable instabilities and associated whiskered tori the rate of transport appears to depend on the amplitude of the homoclinic degrees of freedom 3 isolated cases of discreteness induced instabilities and action transport are observed these return to quasiperiodic flow at finer discretizations 4 a simple measure such as energy of the initial data is not an accurate predictor of chaotic or recurrent behavior one family of initial data exhibits as the energy increases alternating windows of quasiperiodic and chaotic orbits

extremal geometry of a brownian porous medium the path w 0 t of a brownian motion on a d dimensional torus mathbb t d run for time t is a random compact subset of mathbb t d we study the geometric properties of the complement mathbb t d setminus w 0 t as t rightarrow infty for d ge 3 in particular we show that the largest regions in mathbb t d setminus w 0 t have a linear scale varphi d t d log t d 2 kappa d t 1 d 2 where kappa d is the capacity of the unit ball more specifically we identify the sets e for which mathbb t d setminus w 0 t contains a translate of varphi d t e and we count the number of disjoint such translates furthermore we derive large deviation principles for the largest inradius of mathbb t d setminus w 0 t as t rightarrow infty and the varepsilon cover time of mathbb t d as varepsilon downarrow 0 our results which generalise laws of large numbers proved by dembo et al electron j probab 8 15 1 14 2003 are based on a large deviation estimate for the shape of the component with largest capacity in mathbb t d setminus w rho t 0 t where w rho t 0 t is the wiener sausage of radius rho t with rho t chosen much smaller than varphi d t but not too small the idea behind this choice is that mathbb t d setminus w 0 t consists of lakes whose linear size is of order varphi d t connected by narrow channels we also derive large deviation principles for the principal dirichlet eigenvalue and for the maximal volume of the components of mathbb t d setminus w rho t 0 t as t rightarrow infty our results give a complete picture of the extremal geometry of mathbb t d setminus w 0 t and of the optimal strategy for w 0 t to realise extreme events

a novel approach for the analysis of high frequency vibrations abstract despite much effort in the past few decades the numerical prediction of high frequency vibrations remains a challenging task to the engineering and scientific communities due to the numerical instability of existing computational methods however such prediction is of crucial importance to certain problems of pressing practical concern as pointed out by langley and bardell 1998 the aeronautical journal102 287 297 this paper introduces the discrete singular convolution dsc algorithm for the prediction and analysis of high frequency vibration of structures both a beam and two span plates are employed as test examples to demonstrate the capability of the dsc algorithm for high frequency vibration analysis a completely independent approach the levy method is employed to provide exact solutions for a cross validation of the proposed method the reliability of the dsc results is also validated by convergence studies remarkably extremely accurate and stable results are obtained in this work e g the relative dsc errors for the first 7100 modes of the beam and the first 4500 modes of the two span plates are all

convergence properties of policy iteration this paper analyzes asymptotic convergence properties of policy iteration in a class of stationary infinite horizon markovian decision problems that arise in optimal growth theory these problems have continuous state and control variables and must therefore be discretized in order to compute an approximate solution the discretization may render inapplicable known convergence results for policy iteration such as those of puterman and brumelle math oper res 4 1979 pp 60 69 under certain regularity conditions we prove that for piecewise linear interpolation policy iteration converges quadratically also under more general conditions we establish that convergence is superlinear we show how the constants involved in these convergence orders depend on the grid size of the discretization these theoretical results are illustrated with numerical experiments that compare the performance of policy iteration and the method of successive approximations

a new algorithm for solving some mechanical problems this paper explores the utility of a discrete singular convolution algorithm for solving certain mechanical problems benchmark mechanical systems including plate vibrations and incompressible flows are employed to illustrate the robustness and to test accuracy of the present algorithm numerical results indicate that the present approach is very accurate efficient and reliable for solving the aforementioned problems

gumbel fluctuations for cover times in the discrete torus this work proves that the fluctuations of the cover time of simple random walk in the discrete torus of dimension at least three with large side length are governed by the gumbel extreme value distribution this result was conjectured for example in aldous and fill reversible markov chains and random walks on graphs in preparation we also derive some corollaries which qualitatively describe how covering happens in addition we develop a new and stronger coupling of the model of random interlacements introduced by sznitman ann math 2 171 3 2039 2087 2010 and random walk in the torus this coupling is used to prove the cover time result and is also of independent interest

travelling wave solutions for the mkdv sine gordon and the mkdv sinh gordon equations by using a variable separated ode method in this work we formally derive exact travelling wave solutions for the modified kdv sine gordon mkdv sine gordon equation and the modified kdv sinh gordon equation mkdv sinh gordon the proposed analysis depends mainly on a variable separated ode method two distinct sets of exact solitary wave solutions that possess distinct physical structures are formally derived for each equation the derived solutions include others results and introduce entirely new solutions

analyzing two dimensional sine gordon equation with the mesh free reproducing kernel particle ritz method abstract we consider an analysis of the non linear two dimensional sine gordon sg equation the analysis is performed using the mesh free reproducing kernel particle ritz method kp ritz method the mesh free kernel particle estimate is employed to approximate the 2d displacement field a system of discrete equations is obtained through application of the ritz minimization procedure to the energy expressions to validate the accuracy of the results and stability of the present method convergence studies were carried out based on influences of support size and number of nodes the present results were compared with results reported in extant literature and were found to be in good agreement with the literature

the subleading order of two dimensional cover times the varepsilon cover time of the two dimensional torus by brownian motion is the time it takes for the process to come within distance varepsilon 0 from any point its leading order in the small varepsilon regime has been established by dembo et al ann math 160 433 464 2004 in this work the second order correction is identified the approach relies on a multi scale refinement of the second moment method and draws on ideas from the study of the extremes of branching brownian motion

an adaptive wavelet vaguelette algorithm for the solution of pdes the paper first describes a fast algorithm for the discrete orthonormal wavelet transform and its inverse without using the scaling function this approach permits to compute the decomposition of a function into a lacunary wavelet basis i e a basis constituted of a subset of all basis functions up to a certain scale without modification the construction is then extended to operator adapted biorthogonal wavelets this is relevant for the solution of certain nonlinear evolutionary pdes where a priori information about the significant coefficients is available we pursue the approach described in j frohlich and k schneider europ j mech b fluids13 439 1994 which is based on the explicit computation of the scalewise contributions of the approximated function to the values at points of hierarchical grids here we present an improved construction employing the cardinal function of the multiresolution the new method is applied to the helmholtz equation and illustrated by comparative numerical results it is then extended for the solution of a nonlinear parabolic pde with semi implicit discretization in time and self adaptive wavelet discretization in space results with full adaptivity of the spatial wavelet discretization are presented for a one dimensional flame front as well as for a two dimensional problem

exact solutions to the double sinh gordon equation by the tanh method and a variable separated ode method new exact travelling wave solutions for the double sinh gordon equation and its generalized form are formally derived by using the tanh method and the variable separated ode method the painleve property v e u is employed to support the tanh method in deriving exact solutions the work emphasizes the power of the methods in providing distinct solutions of different physical structures

an analytical approach to the sine gordon equation using the modified homotopy perturbation method this paper deals with initial value problems for the sine gordon equation by using the modified homotopy perturbation method the advantage of this method that is its ability to provide the analytical or approximate solutions to linear and nonlinear equations makes it reliable for solving the sine gordon equation the numerical results are presented to show the efficiency of this method

free vibration analysis of symmetrically laminated thin composite plates by using discrete singular convolution dsc approach algorithm and verification this study presents a detailed procedure for the implementation of a discrete singular convolution dsc approach to the free vibration analysis of composite plates based on classical laminated plate theory clpt the approach performs a numerical solution of differential equation of motion by using a grid discretization based on distribution theory and wavelets in the paper firstly computational algorithm of the dsc method is presented then the accuracy of the computer code developed is verified by comparing dsc solutions with the exact results of simply supported isotropic thin beams fully simply supported one layer isotropic and specially orthotropic plates and also some symmetrically laminated thin composite plates orientated to become specially orthotropic besides dsc predictions for laminated composite plates with different boundary conditions and ply numbers for which there is no analytical solution are compared with those of several distinguished works available in the literature it is noteworthy that dsc results completely match with the exact solutions and are in perfect agreement with those of compared studies

comparison of the discrete singular convolution and three other numerical schemes for solving fisher s equation in this paper a discrete singular convolution dsc algorithm is introduced to solve fisher s equation to which obtaining an accurate and reliable traveling wave solution is a challenging numerical problem two novel numerical treatments a moving frame scheme and a new asymptotic scheme are designed to overcome the subtle difficulties involved in the numerical solution of fisher s equation the moving frame scheme is proposed to provide accurate and efficient spatial resolution of the problem the new asymptotic scheme is introduced to facilitate the correct prediction of the wave speed after long time integrations the resulting dsc algorithm is able to correctly predict long time traveling wave behavior the performance of the present algorithm is demonstrated through extensive numerical experiments as well as through a comparison with three other standard approaches i e methods of the fourier pseudospectral the accurate spatial derivatives and the crank nicolson schemes the spatial and temporal accuracies of all four methods are analyzed and numerically verified

a novel scheme for the discrete prediction of high frequency vibration response discrete singular convolution mode superposition approach abstract this study introduces a novel scheme for the discrete high frequency forced vibration analysis based on discrete singular convolution dsc and mode superposition ms approaches the accuracy of the dsc ms is validated for thin beams and plates by comparing with available analytical solutions the performance of the dsc ms is evaluated by predicting spatial distribution and discrete frequency spectra of the vibration response of thin plates with two different boundary conditions the frequency spectra of the time harmonic excitation forces are in the form of ideal and band limited white noise so that the natural modes in the frequency band are provoked the solution exposes high frequency response behaviour definitely therefore it is hoped with this paper to contribute the studies on the treatment of uncertainties in the high frequency design applications