the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

conformal invariance and integrable models usually an integrable nonlinear partial differential equation can be transformed to its conformal invariant form schwartz form using the conformal invariance of the integrable models we can obtain many interesting results in this paper we will focus mainly in obtaining new symmetries and new integrable models starting from the conformal invariance of an integrable model one can obtain infinitely many non local symmetries many types of 1 1 and 2 1 dimensional new sine gordon or sinh gordon extensions are obtained from the conformal flow equations of the koerteweg de vries type equations many other kinds of integrable models can be obtained from the conformal constraints of the known integrable models

an implementation of the streamline upwind petrovgalerkin method for linear triangular elements abstract the weighting function of the streamline upwind petrov galerkin method contains a coefficient which has the dimension of time and is a function of element parameters e g element dimension element peclet number etc in this paper a formula for the coefficient applicable to linear triangular elements is presented the formula does not consist of the individual element parameters but rather of the element matrices only and so it is not only simple but also cost effective the validity of the formula is shown by several test problems and numerical examples

a study of solvers for nonlinear afc discretizations of convection diffusion equations abstract nonlinear discretizations are necessary for convection diffusion equations for obtaining accurate solutions that satisfy the discrete maximum principle the numerical solution of the arising nonlinear problems is often difficult this paper presents several approaches for solving the nonlinear problems of algebraic flux correction afc schemes for the kuzmin limiter and the bjk limiter comprehensive numerical studies are performed at examples that model the transport of energy from a body in a flow field in two and three dimensions it turns out that the most efficient approach from the point of view of computing times is a simple fixed point iteration because the iteration matrix possesses properties that can be exploited by the solvers of the arising linear systems of equations

complexity and robustness of frame structures abstract in structural engineering the word complex is often employed to define something that is difficult to understand or to solve anyhow even if the concept is extensively employed in the practice any proper definition has not been formulated yet taking one s cue from other scientific disciplines a metric for complexity based on the performance of the load paths through the structural scheme is proposed referring to frames characterized by large interaction between the beams an analysis based on graph theory is proposed a beam importance factor is then deduced and preliminary robustness properties are discussed three examples are proposed in order to examine the capabilities of the complexity indices and the beam importance factor the structural robustness is then worked out load redistribution which represents an important aspect in structural behavior should act in the structure in such a way that a damage would not cause collapse the effects of a prescribed damage on a frame are analyzed and commented

the painleve property and backlund transformations for the sequence of boussinesq equations we investigate the sequence of boussinesq equations by the method of singular manifolds for the boussinesq equation which is known to possess the painleve property a backlund transformation is defined this backlund transformation which is formulated in terms of the schwarzian derivative obtains the system of modified boussinesq equations and the resulting miura type transformation the modified boussinesq equations are found to be invariant under a discrete group of symmetries acting on the dependent variables by linearizing the miura transformation and modified equations the lax pair is readily obtained furthermore by a result of fokas and anderson the recursion operators defining the sequence of higher order boussinesq equations may be constructed from the miura transformation this allows the recursive definition of backlund transformations for this sequence of equations the recursion operator is found to preserve the discrete symmetries of the modified boussinesq equations this leads to the conclusion that the sequences of boussinesq and modified boussinesq equations identically possess the painleve property are meromorphic we also find that by a simple reduction the sequences of caudrey dodd gibbon and kuperschmidt equations are contained within the boussinesq sequence rational solutions are iteratively constructed for the boussinesq equation and a criterion is proposed for the existence of rational solutions of general integrable systems

infinitely many lax pairs and symmetry constraints of the kp equation starting from a known lax pair one can get some infinitely many coupled lax pairs infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways in this paper taking the well known kadomtsev petviashvili kp equation as a special example we show that infinitely many nonhomogeneous linear lax pairs can be obtained by using infinitely many symmetries differentiating the spectral functions with respect to the inner parameters using a known lax pair and the darboux transformations dt infinitely many nonhomogeneous nonlinear lax pairs can also be obtained by means of the infinitely many lax pairs dt and the conformal invariance of the schwartz form of the kp equation infinitely many new nonlocal symmetries can be obtained naturally infinitely many integrable models in 1 1 dimensions 2 1 dimensions 3 1 dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the kp equation related to the infinitely many lax pairs

physical complexity of variable length symbolic sequences abstract a measure called physical complexity is established and calculated for a population of sequences based on statistical physics automata theory and information theory it is a measure of the quantity of information in an organism s genome it is based on shannon s entropy measuring the information in a population evolved in its environment by using entropy to estimate the randomness in the genome it is calculated from the difference between the maximal entropy of the population and the actual entropy of the population when in its environment estimated by counting the number of fixed loci in the sequences of a population up until now physical complexity has only been formulated for populations of sequences with the same length here we investigate an extension to support variable length populations we then build upon this to construct a measure for the efficiency of information storage which we later use in understanding clustering within populations finally we investigate our extended physical complexity through simulations showing it to be consistent with the original

covolume upwind finite volume approximations for linear elliptic partial differential equations in this paper we study covolume upwind finite volume methods on rectangular meshes for solving linear elliptic partial differential equations with mixed boundary conditions to avoid non physical numerical oscillations for convection dominated problems nonstandard control volumes covolumes are generated based on local peclet s numbers and the upwind principle for finite volume approximations two types of discretization schemes with mass lumping are developed with use of bilinear or biquadratic basis functions as the trial space respectively some stability analyses of the schemes are presented for the model problem with constant coefficients various examples are also carried out to numerically demonstrate stability and optimal convergence of the proposed methods

universal computation and physical dynamics abstract a dynamical system is said to be computationally universal if it can be programmed through its initial condition to perform and digital computation such systems include traditional abstract computers such as turing machines and cellular automata as well as more physical models such as hard sphere gases and even a single particle in an appropriately shaped box because of the ability of any two such systems to simulate one another they are all equivalent in the range of computations they can perform and the global dynamics of any one of them provides a microcosm of all cause effect relations that can be expressed by deductive logic or numerical simulation this allows universal computers to be used to define in an objective manner that sort of complexity which increases when a self organizing system organizes itself

finite element solution for advection dominated thermal flows abstract a comparative study between an upwinding method supg with a discontinuity capturing term and the method of characteristics is presented in the case of advection diffusion problems numerical examples are given that demonstrate the superiority of the second method in terms of accuracy

deformations of the riccati equation by using miura type transformations using some different miura type transformations a c integrable ordinary differential equation the riccati equation is deformed to some different s integrable models such as the 1 1 dimensional and 2 1 dimensional sinh gordon equations and mikhailov dodd bullough equations

bell polynomial manipulations on the backlund transformations and lax pairs for some soliton equations with one tau function in the framework of bell polynomial manipulations under investigation hereby are three single field bilinearizable equations the 1 1 dimensional shallow water wave model boiti leon manna pempinelli model and 2 1 dimensional sawada kotera model based on the concept of scale invariance a direct and unifying bell polynomial scheme is employed to achieve the backlund transformations and lax pairs associated with those three soliton equations note that the bell polynomial expressions and bell polynomial typed backlund transformations for those three soliton equations can be respectively cast into the bilinear equations and bilinear backlund transformations with symbolic computation consequently it is also shown that the bell polynomial typed backlund transformations can be linearized into the corresponding lax pairs

analysis of a new stabilized higher order finite element method for advection diffusion equations abstract we consider a singularly perturbed advection diffusion two point boundary value problem whose solution has a single boundary layer based on piecewise polynomial approximations of degree k 1 a new stabilized finite element method is derived in the framework of a variation multiscale approach the method coincides with the supg method for k 1 but differs from it for k 2 estimates for the error to an appropriate interpolant are given in several norms on different types of meshes for k 1 enhanced accuracy is achieved by superconvergence postprocessing guarantees the same estimates for the error to the solution itself

pseudopotentials lax equations and backlund transformations for nonlinear evolution equations it is shown how to generate both lax equations and backlund transformations for nonlinear evolution equations using the concept of pseudopotentials associated with the properties of the riccati equation several examples are given modified korteweg de vries harry dym kaup kupershmidt and sawada kotera equations

quadratic streamline upwinding for finite element method solutions to 2 d convective transport problems abstract during the past decade finite element methods fems have been recognized to be more powerful tools in the solution of various flow problems as opposed to their predecessors finite difference methods on a fundamental level the fem involves approximations to the solution over a given trial space on a discrete mesh rather than finite difference approximations to the differential operators on a discrete mesh in this way the mechanics operators of the fem remain closer to the physical principles associated with a given problem in addition to this the fem incorporates boundary conditions very efficiently into the numerical formulation much current research is aimed at making these fems viable options for convection dominated flows in 1985 mizukami et al published two papers advancing the approach of streamline upwinding within linear elements for fems the purpose of this work is to demonstrate that streamline upwinding of higher ordered elements is a more accurate option the work goes on to show exactly how streamline upwinding effects the stability of iteratively solving the global algebraic system resulting from the fem

lattice boltzmann model for free surface flow and its application to filling process in casting a generalized lattice boltzmann model to simulate free surface is constructed in both two and three dimensions the proposed model satisfies the interfacial boundary conditions accurately a distinctive feature of the model is that the collision processes is carried out only on the points occupied partially or fully by the fluid to maintain a sharp interfacial front the method includes an anti diffusion algorithm the unknown distribution functions at the interfacial region are constructed according to the first order chapman enskog analysis the interfacial boundary conditions are satisfied exactly by the coefficients in the chapman enskog expansion the distribution functions are naturally expressed in the local interfacial coordinates the macroscopic quantities al the interface are extracted from the least square solutions of a locally linearized system obtained from the known distribution functions the proposed method does not require any geometric front construction and is robust for any interfacial topology simulation results of realistic filling process are presented rectangular cavity in two dimensions and hammer box campbell box sheffield box and motorblock in three dimensions to enhance the stability at high reynolds numbers various upwind type schemes are developed free slip and no slip boundary conditions are also discussed

the two singular manifold method i modified korteweg de vries and sine gordon equations we address the problem of finding by painleve analysis only the backlund transformation of partial differential equations pdes having two families of movable singularities with opposite principal parts such as the modified korteweg de vries mkdv sine gordon or nonlinear schrodinger equations this first paper gives an almost algorithmic method which extends the singular manifold method of weiss 1986 that is unable to handle these equations first with only one singular manifold at a time we obtain the darboux transformation second we assume that the ratio of two functions defining the singular manifolds satisfies the most general projective riccati system with undetermined coefficients the darboux transformation then generates a very small number of determining equations admitting a unique solution equivalent to the lax pair of the zakharov shabat ablowitz kaup newell segur scheme by the canonical linearization of the riccati system the method is here applied to the mkdv and sine gordon equations

prolongation structures for harry dym type equations and backlund transformations of cc ideals the authors examine the prolongation structures for a certain class of differential equations the interrelation between the incomplete set of commutator relations and the defining relations for kac moody algebras is discussed constant coefficient ideals are introduced for which backlund transformations are found in a purely algebraic manner

physical complexity of symbolic sequences a practical measure for the complexity of sequences of symbols strings is introduced that is rooted in automata theory but avoids the problems of kolmogorov chaitin complexity this physical complexity can be estimated for ensembles of sequences for which it reverts to the difference between the maximal entropy of the ensemble and the actual entropy given the specific environment within which the sequence is to be interpreted thus the physical complexity measures the amount of information about the environment that is coded in the sequence and is conditional on such an environment in practice an estimate of the complexity of a string can be obtained by counting the number of loci per string that are fixed in the ensemble while the volatile positions represent again with respect to the environment randomness we apply this measure to trna sequence data 2000 elsevier science b v all rights reserved

a new upwind function in stabilized finite element formulations using linear and quadratic elements for scalar convection diffusion problems abstract this article presents a study on the function of the peclet number that appears in the upwind function of the gls cau and saupg methods using linear and quadratic triangular elements as well as bilinear and quadratic quadrilateral elements the numerical experiments indicate that in the case of elements with faces on the outflow boundary the upwind function is strongly dependent on the geometry and on the degree of the interpolate polynomial a new function of the peclet number is therefore proposed to take these effects into consideration creating a new method of stabilization for higher order elements this new stabilized and accurate finite element formulation for convection dominated problems based on the idea of the saupg method uses the gls formulation as starting point

the wahlquist estabrook method for evolution equations with a small parameter a technique of approximating pseudopotentials an approach is suggested to construct approximate solutions for nonlinear partial differential equations non integrable by the inverse scattering transform the method is applied to find an approximate lax pair and conservation laws for the kawahara equation

painleve property and pseudopotentials for nonlinear evolution equations the link between the singularity manifold equation and the pseudopotential equations is found for nonlinear evolution equations both in 1 1 and 2 1 dimensions many examples are given burgers korteweg de vries modified korteweg de vries harry dym a higher order korteweg de vries sawada kotera kaup kuperschmidt kadomtsev petviashvili boussinesq modified kadomtsev petviashvili and 2 1 dimensional sawada kotera equations

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

a petrov galerkin finite element method for convection dominated flows an accurate upwinding technique for satisfying the maximum principle abstract in this paper we present a new petrov galerkin method for convection dominated flow problems which is conservative and satisfies the discrete maximum principle this method also possesses no spurious crosswind diffusion and gives very accurate solutions the method is representative of a class of methods which may be described as fixed mesh adaptive

pseudopotentials lax pairs and backlund transformations for some variable coefficient nonlinear equations using the method developed by nucci 1989 the author has obtained the pseudopotentials lax pairs backlund transformations and the singularity manifold equations of some variable coefficient nonlinear equations these equations are generalized korteweg de vries generalized modified korteweg de vries and generalized boussinesq equations

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers since the mid of the 1980s a number of methods have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers the paper gives a review and state of the art of these methods discusses their derivation proposes some alternative choices of parameters in the methods and categorizes them some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

approximating pseudopotentials scattering problems and backlund transformations for the korteweg de vries and modified korteweg de vries equations with perturbations approximating pseudopotentials associated with scattering problems and with backlund transformations are derived for the perturbed korteweg de vries and modified korteweg de vries equations recursion formulas for approximate conservation laws are obtained for the perturbed korteweg de vries equation as well as an example the auto backlund transformations are used to construct approximate one and two soliton solutions

the residual symmetry and exact solutions of the davey stewartson iii equation abstract through the truncated painleve expansion the residual symmetry of the davey stewartson iii system is derived this symmetry is localized to a properly prolonged system and the corresponding backlund transformation is obtained based on the transformation some exact solutions are generated furthermore several types of interaction wave solutions are obtained by using the consistent tanh expansion method

higher dimensional painleve integrable models from the kadomtsev petviashvili equation after embedding the kadomtsev petviashvili equation in higher dimensions and extending the painleve analysis approach to a new form such that the coefficients of the expansion around the singular manifold possess conformal invariance and contain explicit new space variables we can get infinitely many painleve integrable models in 3 1 dimensions and higher dimensions some concrete higher dimensional modified korteweg de vries type of extensions are given whether the models are lax integrable or integrable under other meanings remain still open

improving the speed and accuracy of projection type incompressible flow solvers abstract superseding so called first generation incompressible flow solvers of the projection type based on taylor galerkin advection second order pressure damping and element based data structures the current second generation solvers based on high order upwind advection fourth order pressure damping and edge based data structures have now been in use for half a decade and have proven remarkably robust and efficient for many large scale problems in order to achieve higher accuracy and speed these solvers have recently been enhanced in a variety of ways a substepping for advection b implicit treatment of advective terms via sgs and gmres lu sgs iterative solvers c fully implicit time accurate advancement of pressure and velocities and d linelet preconditioning for the pressure poisson equation the combined effect of these third generation improvements leads to speedups of the order of o 1 5 1 10 with similar or even better temporal accuracy as demonstrated on a variety of academic and industrial problems

modified equations rational solutions and the painleve property for the kadomtsev petviashvili and hirota satsuma equations we propose a method for finding the lax pairs and rational solutions of integrable partial differential equations that is when an equation possesses the painleve property a backlund transformation is defined in terms of an expansion about the singular manifold this backlund transformation obtains 1 a type of modified equation that is formulated in terms of schwarzian derivatives and 2 a miura transformation from the modified to the original equation by linearizing the ricati type miura transformation the lax pair is found on the other hand consideration of the distinct backlund transformations of the modified equations provides a method for the iterative construction of rational solutions this also obtains the lax pairs for the modified equations in this paper we apply this method to the kadomtsev petviashvili equation and the hirota satsuma equations

the investigation into the schwarz korteweg de vries equation and the schwarz derivative in 2 1 dimensions in this note we shall introduce a new integrable equation and the schwarz derivative in 2 1 dimensions first we show the existence of the lax pair for an equation which has the relation to the schwarz korteweg de vries skdv equation next we derive a new equation in 2 1 dimensions by using a well known higher dimensional manner to the lax pair for the skdv equation the 2 1 dimensional schwarz derivative is defined here finally we briefly discuss various results which we have obtained about the new equation

kdv extensions with painleve property by means of the conformal invariance mobious transformation invariance the well known kdv equation is extended to i a 1 1 dimensional space time symmetric form ii two 2 1 dimensional space isotropic forms and iii general 3 1 dimensional and n dimensional forms the extensions are proven to be integrable under the meaning that they possess the painleve property

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100