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

spectral methods for weakly singular volterra integral equations with smooth solutions we propose and analyze a spectral jacobi collocation approximation for the linear volterra integral equations vies of the second kind with weakly singular kernels in this work we consider the case when the underlying solutions of the vies are sufficiently smooth in this case we provide a rigorous error analysis for the proposed method which shows that the numerical errors decay exponentially in the infinity norm and weighted sobolev space norms numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence

using a dpg method to validate dma experimental calibration of viscoelastic materials abstract a discontinuous petrov galerkin dpg method is used to solve the time harmonic equations of linear viscoelasticity it is based on a broken primal variational formulation which is very similar to the classical primal variational formulation used in galerkin methods but has additional interface variables at the boundaries of the mesh elements both the classical and broken formulations are proved to be well posed in the infinite dimensional setting and the resulting discretization is proved to be stable a full h p convergence analysis is also included and the analysis is verified using computational simulations the method is particularly useful as it carries its own natural arbitrary p a posteriori error estimator which is fundamental for solving problems with localized solution features this proves to be useful when validating calibration models of dynamic mechanical analysis dma experiments indeed different dma experiments of epoxy and silicone resins were successfully validated to within 5 of the quantity of interest using the numerical method

convergence analysis of the multistep legendre pseudo spectral method for volterra integral equations with vanishing delays in this paper we extend the single step pseudo spectral method for second kind volterra integral equations with vanishing variable delays to the multistep pseudo spectral method we also analyze the convergence of the hp version of the multistep pseudo spectral method under the l2 norm and show that the scheme enjoys high order accuracy and can be implemented in a stable and efficient manner in addition it is very suitable for long time calculations and large step size situations numerical results show good agreement with the theoretical analysis

symmetrically coupled higher order nonlinear schrodinger equations singularity analysis and integrability the integrability of a system of two symmetrically coupled higher order nonlinear schrodinger equations with parameter coefficients is tested by means of singularity analysis it is proven that the system passes the painleve test for integrability only in ten distinct cases of which two are new for one of the new cases a lax pair and a multi field generalization are obtained for the other one the equations of the system are uncoupled by a nonlinear transformation

sinh gordon equation painleve property and backlund transformation abstract the painleve property for partial differential equation proposed by weiss et al is studied for the sinh gordon equation we performed a singular point analysis to show that the painleve property is a valuable analytical test for integrability and we found in a remarkably straightforward manner the backlund transformation of the sinh gordon equation

graded mesh approximation in weighted sobolev spaces and elliptic equations in 2d we study the approximation properties of some general finite element spaces constructed using improved graded meshes in our results either the approximating function or the function to be approximated or both are in a weighted sobolev space we consider also the l p version of these spaces the finite element spaces that we define are obtained from em conformally invariant families of finite elements no affine invariance is used stressing the use of elements that lead to higher regularity finite element spaces we prove that for a suitable grading of the meshes one obtains the usual optimal approximation results we provide a construction of these spaces that does not lead to long skinny triangles our results are then used to obtain l 2 error estimates and h m quasi optimal rates of convergence for the fem approximation of solutions of strongly elliptic interface boundary value problems

a posteriori error estimates for mixed fem in elasticity a residue based reliable and efficient error estimator is established for finite element solutions of mixed boundary value problems in linear planar elasticity the proof of the reliability of the estimator is based on helmholtz type decompositions of the error in the stress variable and a duality argument for the error in the displacements the efficiency follows from inverse estimates the constants in both estimates are independent of the lame constant lambda and so locking phenomena for lambda to infty are properly indicated the analysis justifies a new adaptive algorithm for automatic mesh refinement

weighted sobolev spaces and regularity for polyhedral domains we prove a regularity result for the poisson problem u f u p g on a polyhedral domain p r3 using the babuska kondratiev spaces kam p these are weighted sobolev spaces in which the weight is given by the distance to the set of edges 4 33 in particular we show that there is no loss of kam regularity for solutions of strongly elliptic systems with smooth coefficients we also establish a trace theorem for the restriction to the boundary of the functions in kam p

a new strategy for the numerical solution of nonlinear volterra integral equations with vanishing delays abstract the aim of this paper is to develop a high order algorithm for nonlinear volterra integral equations with vanishing variable delays the algorithm is a non trivial extension of the single step methods this method is based on the variational form so the algorithm is relatively simple this scheme enjoys high order accuracy and can be implemented in a stable and efficient manner we prove the existence and uniqueness of the numerical solution and derive error estimates of the algorithm numerical results show a good agreement with the theoretical analysis

constrained anisotropic elastic materials in unilateral contact with or without friction we consider time independent problems of linear homogeneous anisotropic elasticiy for materials with internal constraints that are in frictional or frictionless contact with a rigid support

generalized laguerre pseudospectral method based laguerre interpolation abstract in this paper we develop the generalized laguerre pseudospectral method for the second order ordinary differential equation based on the modified generalized laguerre approximation which is very efficient for the long time numerical simulations of dynamical systems the numerical solution possesses the spectral accuracy the global convergence of the proposed algorithm is proved numerical results demonstrate the effectiveness of the proposed method and coincide well with theoretical analysis

a multistep legendre pseudo spectral method for volterra integral equations in this paper we extend the single step pseudo spectral method for volterra integral equations of the second kind to the multistep pseudo spectral method we also analyze the convergence of the hp version of the multistep pseudo spectral method under the l2 norm and show that the scheme enjoys high order accuracy and can be implemented in a stable and efficient manner in addition it is very suitable for long time calculations and large step size situations numerical experiments confirm theoretical expectations

locking effects in the finite element approximation of elasticity problems we consider the finite element approximation of the 2d elasticity problem when the poisson ratiov is close to 0 5 it is well known that the performance of certain commonly used finite elements deteriorates asv 0 a phenomenon calledlocking we analyze this phenomenon and characterize the strength of the locking androbustness of varioush version schemes using triangular and rectangular elements we prove that thep andh p versions are free of locking with respect to the error in the energy norm a generalization of our theory to the 3d problem is also discussed

finite element analysis for the axisymmetric laplace operator on polygonal domains abstract let l r 2 r r 2 z 2 we consider the equation l u f on a bounded polygonal domain with suitable boundary conditions derived from the three dimensional axisymmetric poisson s equation we establish the well posedness regularity and fredholm results in weighted sobolev spaces for possible singular solutions caused by the singular coefficient of the operator l as r 0 and by non smooth points on the boundary of the domain in particular our estimates show that there is no loss of regularity of the solution in these weighted sobolev spaces besides by analyzing the convergence property of the finite element solution we provide a construction of improved graded meshes such that the quasi optimal convergence rate can be recovered on piecewise linear functions for singular solutions the introduction of a new projection operator from the weighted space to the finite element subspace certain scaling arguments and a calculation of the index of the fredholm operator together with our regularity results are the ingredients of the finite element estimates

error analysis of some galerkin least squares methods for the elasticity equations the recent technique of stabilizing mixed finite element methods by augmenting the galerkin formulation with least squares terms calculated separately on each element is considered the error analysis is performed in a unified manner yielding improved results for some methods introduced earlier in addition a new formulation is introduced and analyzed method is used within two different contexts the first is in connection with the plain strain or three dimensional problems near the incompressible limit for which the poisson ratio equals one half it is well known that for these problems a standard lower order displacement finite element method will break down by now it is also well known how a working method can be obtained a new unknown the pressure is introduced and discretized with a finite element space different from that used for the displacements the finite element method obtained falls into the class of saddlepoint problems for which there is a general theory developed by brezzi and babuska 5 7 the application of this theory shows that the method converges optimally if the discrete spaces for the displacement and pressure satisfy a stability condition for a particular choice of displacement and pressure approximations it can be a difficult task to verify this babuska brezzi or inf sup condition during the last decade this problem has been intensively studied and the performances of many combinations are now known cf 18 and 23 26 for some recent extensions in addition there exists a general technique utilizing bubble functions by which a stable method can

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

on stabilized mixed methods for generalized stokes problem based on the velocity pseudostress formulation a priori error estimates abstract in this paper we present an augmented mixed formulation applied to generalized stokes problem and uses it as state equation in an optimal control problem the augmented scheme is obtained adding suitable least squares terms to the corresponding velocity pseudostress formulation of the generalized stokes problem to ensure the existence and uniqueness of solution at continuous and discrete levels we prove coerciveness of the corresponding augmented bilinear form and using approximation properties of the respective discrete subspaces we deduce the optimal rate of convergence as by product and considering the associated optimal control problem we derive error estimates for the approximated control unknown finally we present several numerical examples confirming the theoretical properties of this approach

on the stability of bdms and peers elements using fortin operators we give a new proof of stability for stenberg s family of bdms elements in linear elasticity our approach yields the inf sup condition with respect to the standard norms which is indispensable for a posteriori error analysis furthermore our technique allows the construction of another family of finite elements strongly related to the classical peers element the given estimates are robust for nearly incompressible materials

a generalized block by block method for solving linear volterra integral equations abstract one of the numerical methods for solving linear volterra integral equations is block by block method which is explained in l m delves j l mohamed computational methods for integral equations cambridge university press 1985 l m delves j walsh numerical solution of integral equations oxford university press 1974 and p k kyte p puri computational methods for linear integral equations birkhauser boston 2002 in this article we explain a general method for constructing block by block systems for solving volterra integral equations then we deduce some of the special cases especially the linz s block by block method which explained in these references

an augmented mixed finite element method for 3d linear elasticity problems in this paper we introduce and analyze a new augmented mixed finite element method for linear elasticity problems in 3d our approach is an extension of a technique developed recently for plane elasticity which is based on the introduction of consistent terms of galerkin least squares type we consider non homogeneous and homogeneous dirichlet boundary conditions and prove that the resulting augmented variational formulations lead to strongly coercive bilinear forms in this way the associated galerkin schemes become well posed for arbitrary choices of the corresponding finite element subspaces in particular raviart thomas spaces of order 0 for the stress tensor continuous piecewise linear elements for the displacement and piecewise constants for the rotation can be utilized moreover we show that in this case the number of unknowns behaves approximately as 9 5 times the number of elements tetrahedrons of the triangulation which is cheaper by a factor of 3 than the classical peers in 3d several numerical results illustrating the good performance of the augmented schemes are provided

a posteriori error estimators for mixed finite element methods in linear elasticity three a posteriori error estimators for peers and bdms elements in linear elasticity are presented one residual error estimator and two estimators based on the solution of auxiliary local problems with different boundary conditions all of them are reliable and efficient with respect to the standard norm and furthermore robust for nearly incompressible materials

a family of mixed finite elements for the elasticity problem a new mixed finite element formulation for the equations of linear elasticity is considered in the formulation the variables approximated are the displacement the unsymmetric stress tensor and the rotation the rotation act as a lagrange multiplier introduced in order to enforce the symmetry of the stress tensor based on this formulation a new family of both two and three dimensional mixed methods is defined optimal error estimates which are valid uniformly with respect to the poisson ratio are derived finally a new postprocessing scheme for improving the displacement is introduced and analyzed

well posedness of the fundamental boundary value problems for constrained anisotropic elastic materials we consider the equations of linear homogeneous anisotropic elasticity admitting the possibility that the material is internally constrained and formulate a simple necessary and sufficient condition for the fundamental boundary value problems to be well posed for materials fulfilling the condition we establish continuous dependence of the displacement and stress on the elastic moduli and ellipticity of the elasticity system as an application we determine the orthotropic materials for which the fundamental problems are well posed in terms of their young s moduli shear moduli and poisson ratios finally we derive a reformulation of the elasticity system that is valid for both constrained and unconstrained materials and involves only one scalar unknown in addition to the displacements for a two dimensional constrained material a further reduction to a single scalar equation is outlined

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

connection aspects of nonlinear lattice equations in 1 1 dimensions considers two nonlinear lattice equations and studies some geometrical features the author demonstrates that two nonlinear partial differential equations are deduced from the fact that the sl 2 r connection has zero curvature

simplified reproducing kernel method and convergence order for linear volterra integral equations with variable coefficients abstract this paper proposes a simplified reproducing kernel method to solve the linear volterra integral equations with variable coefficients the main idea of the method is to establish a reproducing kernel direct space that can be used in volterra integral equations and in the first time this paper analyzes the convergence order and stability of the approximate solution then the uniform convergence of the numerical solution is proved and the time consuming schmidt orthogonalization process is avoided the proposed method is proved to be stable and is not less than the second order convergence the algorithm is proved to be feasible and stable through some numerical examples

well posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks we prove a regularity result for the anisotropic linear elasticity equation p u rm div left boldmath mathsf c cdot nabla u right f with mixed displacement and traction boundary conditions on a curved polyhedral domain omega subset mathbb r 3 in weighted sobolev spaces mathcal k m 1 a 1 omega for which the weight is given by the distance to the set of edges in particular we show that there is no loss of mathcal k m a regularity our curved polyhedral domains are allowed to have cracks we establish a well posedness result when there are no neighboring traction boundary conditions and a 0 that depends on p on the boundary conditions and on the domain our results extend to other strongly elliptic systems and higher dimensions

convergence analysis of the jacobi spectral collocation methods for volterra integral equations with a weakly singular kernel in this paper a jacobi collocation spectral method is developed for volterra integral equations of second kind with a weakly singular kernel we use some function transformation and variable transformations to change the equation into a new volterra integral equation deflned on the standard interval i1 1 so that the solution of the new equation possesses better regularity and the jacobi orthogonal polynomial theory can be applied conveniently in order to obtain high order accuracy for the approx imation the integral term in the resulting equation is approximated by using jacobi spectral quadrature rules the convergence analysis of this novel method is based on the lebesgue constants corresponding to the lagrange interpolation polynomials poly nomials approximation theory for orthogonal polynomials and the operator theory the spectral rate of convergence for the proposed method is established in the l 1 norm and weighted l 2 norm numerical results are presented to demonstrate the efiectiveness of the proposed method

discontinuous galerkin approximations for near incompressible and near inextensible transversely isotropic bodies abstract this work studies discontinuous galerkin dg approximations of the boundary value problem for homogeneous transversely isotropic linear elastic bodies low order approximations on triangles are adopted with the use of three interior penalty dg methods viz nonsymmetric symmetric and incomplete it is known that these methods are uniformly convergent in the incompressible limit for the isotropic case this work focuses on behaviour in the inextensible limit for transverse isotropy an error estimate suggests the possibility of extensional locking a feature that is confirmed by numerical experiments under integration of the extensional edge terms is proposed as a remedy this modification is shown to lead to an error estimate that is consistent with locking free behaviour numerical tests confirm the uniformly convergent behaviour at an optimal rate of the under integrated scheme

fractional linear multistep methods for abel volterra integral equations of the second kind fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular volterra integral equations the proposed methods are convergent of the order of the underlying multistep method also in the generic case of solutions which are not smooth at the origin the stability properties stability region a stability a a stability are closely related to those of the underlying linear multistep method