time discretization of an integro differential equation of parabolic type the stability and convergence properties of two time discretizations of an integro differential equation of parabolic type are studied the methods reduce to the backward euler and crank nicolson methods if the integral term is absent the integral term is approximated in each case by a quadrature rule with relatively high order truncation error so that a relatively large time step can be used for the quadrature in order to reduce the memory and computational requirements of the method

polynomial poisson algebras for classical superintegrable systems with a third order integral of motion we present polynomial poisson algebras for the eight classical potentials in two dimensional euclidian space that separate in cartesian coordinates and allow a third order integral of motion some of the classical superintegrable potentials do not coincide with quantum ones but are their singular limits we show that all bounded trajectories in these potentials are periodic

classical and quantum superintegrability with applications a superintegrable system is roughly speaking a system that allows more integrals of motion than degrees of freedom this review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta we present a classification of second order superintegrable systems in two dimensional riemannian and pseudo riemannian spaces it is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis the determining equations for the existence of integrals of motion of arbitrary order in real euclidean space e2 are presented and partially solved for the case of third order integrals a systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order the algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally the relation between superintegrability and the classification of orthogonal polynomials is analyzed

general nth order integrals of motion in the euclidean plane the general form of an integral of motion that is a polynomial of order n in the momenta is presented for a hamiltonian system in two dimensional euclidean space the classical and the quantum cases are treated separately emphasizing both the similarities and the differences between the two the main application will be to study nth order superintegrable systems that allow separation of variables in the hamilton jacobi and schrodinger equations respectively

superconvergence and asymptotic expansion for semidiscrete bilinear finite volume element approximation of the parabolic problem abstract we first derive the asymptotic expansion of the bilinear finite volume element for the linear parabolic problem by employing the energy embedded method on uniform grids and then obtain a high accuracy combination pointwise formula of the derivatives for the finite volume element approximation based on the above asymptotic expansion furthermore we prove that the approximate derivatives have the convergence rate of order two numerical experiments confirm the theoretical results

on fully discrete galerkin approximations for partial integro differential equations of parabolic type the subject of this work is the application of fully discrete galerkin finite element methods to initial boundary value problems for linear partial integro differential equations of parabolic type we investigate numerical schemes based on the pade discretization with respect to time and associated with certain quadrature formulas to approximate the integral term a preliminary error estimate is established which contains a term related to the quadrature rule to be specified in particular we consider quadrature rules with sparse quadrature points so as to limit the storage requirements without sacrificing the order of overall convergence for the backward euler scheme the crank nicolson scheme and a third order pade type scheme the specific quadrature rules analyzed are based on the rectangular the trapezoidal and simpson s rule for all the schemes studied optimal order error estimates are obtained in the case that the solution of the problem is smooth enough since this is important for our error analysis we also discuss the regularity of the exact solutions of our equations high order regularity results with respect to both space and time are given for the solution of problems with smooth enough data 17 refs

finite element methods for partial volterra integro differential equations on two dimensional unbounded spatial domains abstract the precise and effective way for solving a partial volterra integro differential equation on unbounded spatial domain is to derive artificial boundary conditions for some feasible and prescribed computational domain thus the unsolvable problem on unbounded domain is converted to the computational problem on bounded domain which is often called reduced problem a feasible numerical method will then be applied to solve the reduced problem with the artificial boundary conditions which often exhibit nonlocal properties in this paper the finite element method for the spatial space and the recently developed discontinuous galerkin time stepping method for the temporal space are considered to solve the reduced problem the convergence rate explicitly with those interesting parameters spatial grid size h temporal mesh size k the location of the artificial boundary d and the truncation number m is obtained this paper is the first theoretical analysis of the numerical methods for volterra integro differential equations on unbounded spatial domains

artificial boundary conditions for parabolic volterra integro differential equations on unbounded two dimensional domains in this paper we study the numerical solution of parabolic volterra integro differential equations on certain unbounded two dimensional spatial domains the method is based on the introduction of a feasible artificial boundary and the derivation of corresponding artificial fully transparent boundary conditions two examples illustrate the application and numerical performance of the method

numerical solution of parabolic integro differential equations by the discontinuous galerkin method the numerical solution of a parabolic equation with memory is considered the equation is first discretized in time by means of the discontinuous galerkin method with piecewise constant or piecewise linear approximating functions the analysis presented allows variable time steps which as will be shown call then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data the combination with finite element discretizat in space is also studied

time discretization via laplace transformation of an integro differential equation of parabolic type we consider the discretization in time of an inhomogeneous parabolic integro differential equation with a memory term of convolution type in a banach space setting the method is based on representing the solution as an integral along a smooth curve in the complex plane which is evaluated to high accuracy by quadrature using the approach in recent work of lopez fernandez and palencia this reduces the problem to a finite set of elliptic equations with complex coefficients which may be solved in parallel the method is combined with finite element discretization in the spatial variables to yield a fully discrete method the paper is a further development of earlier work by the authors which on the one hand treated purely parabolic equations and on the other an evolution equation with a positive type memory term

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

error estimates for discontinuous galerkin approximations of implicit parabolic equations we analyze the classical discontinuous galerkin method for implicit parabolic equations symmetric error estimates for schemes of arbitrary order are presented the ideas developed allow certain assumptions frequently required in previous work to be relaxed for example different discrete spaces are allowed at each time step and the spatial operator is not required to be self adjoint or independent of time error estimates are posed in terms of projections of the exact solution onto the discrete spaces and are valid under the minimal regularity guaranteed by the natural energy estimate these projections are local and enjoy optimal approximation properties when the solution is sufficiently regular

long time numerical solution of a parabolic equation with memory long time stability and convergence properties of two time discreti zation methods for an integro differential equation of parabolic type are studied the methods are based on the standard backward euler and second order back ward differencing methods the memory term is approximated by a quadrature rule with emphasis on such rules with reduced computational memory require ments discretization of the spatial partial differential operators by the finite element method is also considered

darboux points and integrability analysis of hamiltonian systems with homogeneous rational potentials abstract we study integrability in the liouville sense for natural hamiltonian systems with a homogeneous rational potential v q strong necessary conditions for the integrability of such systems were obtained by analysis of differential galois groups of variational equations along certain particular solutions these conditions have the form of arithmetic restrictions on eigenvalues of the hessians v d calculated at non zero solutions d of the equation grad v d d such solutions are called proper darboux points it was recently proved that for generic polynomial homogeneous potentials there exist universal relations between eigenvalues of hessians of the potential taken at all proper darboux points the existence of such relations for rational potentials seems to be a difficult question one of the reasons is the presence of points of indeterminacy of the potential and its gradient nevertheless for two degrees of freedom we prove that such a relation exists this result is important because it allows us to show that the set of admissible values for hessian eigenvalues at a proper darboux point for potentials satisfying the necessary conditions for integrability is finite in turn this provides a tool for classification of integrable rational potentials it was also recently shown that for polynomial homogeneous potentials additional necessary conditions for integrability can be deduced from the existence of improper darboux points that is points d that are non zero solutions of grad v d 0 these new conditions also take the form of arithmetic restrictions imposed on eigenvalues of v d in this paper we prove that for rational potentials improper darboux points give the same necessary conditions for integrability

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold in this paper we prove that the two dimensional superintegrable systems with quadratic integrals of motion on a manifold can be classified by using the poisson algebra of the integrals of motion there are six general fundamental classes of superintegrable systems analytic formulas for the involved integrals are calculated in all the cases all the known superintegrable systems are classified as special cases of these six general classes

addition theorems and the drach superintegrable systems we propose a new construction of the polynomial integrals of motion related to the addition theorems as an example we reconstruct drach systems and get some new two dimensional superintegrable stackel systems with third fifth and seventh order integrals of motion

darboux points and integrability of hamiltonian systems with homogeneous polynomial potential in this paper we study the integrability of natural hamiltonian systems with a homogeneous polynomial potential the strongest necessary conditions for their integrability in the liouville sense have been obtained by a study of the differential galois group of variational equations along straight line solutions these particular solutions can be viewed as points of a projective space of dimension smaller by one than the number of degrees of freedom we call them darboux points we analyze in detail the case of two degrees of freedom we show that except for a radial potential the number of darboux points is finite and it is not greater than the degree of the potential moreover we analyze cases when the number of darboux points is smaller than maximal for two degrees of freedom the above mentioned necessary condition for integrability can be expressed in terms of one nontrivial eigenvalue of the hessian of potential calculated at a darboux point we prove that for a given potential these nontrivial eigenvalues calculated for all darboux points cannot be arbitrary because they satisfy a certain relation which we give in an explicit form we use this fact to strengthen maximally the necessary conditions for integrability and we show that in a generic case for a given degree of the potential there is only a finite number of potentials which satisfy these conditions we also describe the nongeneric cases as an example we give a full list of potentials of degree four satisfying these conditions then investigating the differential galois group of higher order variational equations we prove that except for one discrete family among these potentials only those which are already known to be integrable are integrable we check that a finite number of potentials from the exceptional discrete family are not integrable and we conjecture that all of them are not integrable

space time spectral method for a weakly singular parabolic partial integro differential equation on irregular domains abstract the spectral method is proposed for the partial integro differential equations with a weakly singular kernel on irregular domains the space discretization is based on the nodal spectral element method using the lagrange polynomials basis associated with the gauss lobatto legendre quadrature nodes also the model is discretized in time with the legendre spectral galerkin method the discretization leads to conversion of the problem to a sylvester matrix equation which can be solved efficiently by the qz algorithm gardiner et al 1992 the convergence of the method is proven by providing a priori l 2 error estimate numerical results illustrate the efficiency and spectral accuracy of the proposed method

finite element methods for parabolic and hyperbolic partial integro differential equations on formule plusieurs methodes de galerkin a temps discret pour des equations integrodifferentielles paraboliques

time discretization of parabolic problems by the hp version of the discontinuous galerkin finite element method the discontinuous galerkin finite element method dgfem for the time discretization of parabolic problems is analyzed in the context of the hp version of the galerkin method error bounds which are explicit in the time steps as well as in the approximation orders are derived and it is shown that the hp dgfem gives spectral convergence in problems with smooth time dependence in conjunction with geometric time partitions it is proved that the hp dgfem results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms for the h version dgfem algebraically graded time partitions are determined that give the optimal algebraic convergence rates a fully discrete hp scheme is discussed exemplarily for the heat equation the use of certain mesh design principles for the spatial discretizations yields exponential rates of convergence in time and space numerical examples confirm the theoretical results

discontinuous galerkin finite element method for parabolic problems in this paper we develop a time and its corresponding spatial discretization scheme based upon the assumption of a certain weak singularity of u t t l 2 w email protected u email protected 2 for the discontinuous galerkin finite element method for one dimensional parabolic problems optimal convergence rates in both time and spatial variables are obtained a discussion of automatic time step control method is also included

error estimates for semidiscrete finite element methods for parabolic integro differential equations application des methodes d elements finis aux equations integro differentielles de type parabolique avec un noyau integral forme par un operateur aux derivees partielles d ordre 2

the weak galerkin method for solving the incompressible brinkman flow the brinkman equations are used to describe the dynamics of fluid flows in complex porous media with the high variability in the permeability coefficients which may take extremely large or small values this paper is devoted to the numerical analysis of a family of weak galerkin wg finite element methods for solving the time dependent brinkman problems this wg method is equipped with stable finite elements consisting of usual polynomials of degree k 1 for the velocity and polynomials of degree k 1 for the pressure the velocity element is enhanced by polynomials of degree k on the interface of the finite element partition all the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly defined spaces we further establish a priori error estimates in l 2 norm and h 1 norm and we provide a few numerical experiments to illustrate the behavior of the proposed scheme and confirm our theoretical findings regarding optimal convergence of the approximate solutions

on algebraic construction of certain integrable and super integrable systems abstract we propose a new construction of two dimensional natural bi hamiltonian systems associated with a very simple lie algebra the presented construction allows us to distinguish three families of super integrable monomial potentials for which one additional first integral is quadratic and the second one can be of arbitrarily high degree with respect to the momenta many integrable systems with additional integrals of degree greater than two in momenta are given moreover an example of a super integrable system with first integrals of degree two four and six in the momenta is found

time discretization of fractional subdiffusion equations via fractional resolvent operators abstract in this work we study time discretization of subdiffusion equations that is fractional differential equations of order 0 1 assuming that a is the generator of a fractional resolvent family s t t 0 which allows to write the solution to the subdiffusion equation t u t a u t f t as a variation of constants formula we find an interesting connection between s t t 0 and a discrete resolvent family s n n n and then by using the properties of s t t 0 we study the existence of solutions to the discrete subdiffusion equation c u n a u n f n n n where based on the backward euler method for a 0 given c u n is an approximation of t u t at time t n n we study simultaneously the fractional derivative in the caputo and riemann liouville sense we also provide error estimates and some experiments to illustrate the results

an hp version discontinuous galerkin method for integro differential equations of parabolic type we study the numerical solution of a class of parabolic integro differential equations with weakly singular kernels we use an hp version discontinuous galerkin dg method for the discretization in time we derive optimal hp version error estimates and show that exponential rates of convergence can be achieved for solutions with singular temporal behavior near t 0 caused by the weakly singular kernel moreover we prove that by using nonuniformly refined time steps optimal algebraic convergence rates can be achieved for the h version dg method we then combine the dg time stepping method with a standard finite element discretization in space and present an optimal error analysis of the resulting fully discrete scheme our theoretical results are numerically validated in a series of test problems

cα regularity for non autonomous linear integrodifferential equations of parabolic type abstract linear integrodifferential equations in general banach space are studied and applications are given to linear integrodifferential partial differential equations

analytic resolvent operators for integral equations in banach space on etudie l equation integrodifferentielle x t a 0 x t 0 t b t s x s ds f t t 0 x 0 x 0 ed a x et l equation integrale x t 0 t a t s x s ds f t dans un espace de banach x les operateurs a a 0 b t et a t sont des operateurs clos sur x a domaine dense

transformation of the stackel matrices preserving superintegrability if we take a superintegrable stackel system and make variables faster or slower that is equivalent to a trivial transformation of the stackel matrix and potentials then we obtain an infinite family of superintegrable systems with explicitly defined additional integrals of motion we present some examples of such transformations associated with angle variables expressed via logarithmic functions if we take a superintegrable stackel system and make variables faster or slower that is equivalent to a trivial transformation of the stackel matrix and potentials then we obtain an infinite family of superintegrable systems with explicitly defined additional integrals of motion we present some examples of such transformations associated with angle variables expressed via logarithmic functions

superintegrable systems and riemann roch theorem in algebraic geometry there is a reduction algorithm that transforms the unreduced divisor into a unique reduced divisor which existence is guaranteed by the riemann roch theorem we discuss application of this algorithm to construction of finite dimensional superintegrable systems with n degrees of freedom identifying coordinates of the reduced divisor with integrals of motion

ritz volterra projections to finite element spaces and applications to integrodifferential and related equations the object of this paper is to investigate the convergence of finite element approximations to solutions of parabolic and hyperbolic integrodifferential equations and also of equations of sobolev and viscoelasticity type the concept of ritz volterra projection will be seen to unify much of the analysis for the different types of problems optimal order error estimates are obtained in l p for 2 leqq p infty and almost optimal order pointwise results given

a galerkin method for two dimensional hyperbolic integro differential equation with purely integral conditions the present paper is devoted to the solution for two dimensional hyperbolic integro differential equations subject to purely integral conditions first it is demonstrated the existence and uniqueness of the solution under certain conditions for the numerical approach a galerkin method based on least squares is proposed the numerical examples illustrate the technique and show the efficiency of the proposed method

finite element approximation of a parabolic integro differential equation with a weakly singular kernel we give error estimates for the numerical solution by means of the galerkin finite element method of an integro differential equation of parabolic type with a memory term containing a weakly singular kernel optimal order estimates are shown for spatially semidiscrete and completely discrete methods special attention is paid to the regularity of the exact solution

non commutative banach function spaces in this paper we survey some aspects of the theory of non commutative banach function spaces that is spaces of measurable operators associated with a semi finite von neumann algebra these spaces are also known as non commutative symmetric spaces the theory of such spaces emerged as a common generalization of the theory of classical commutative rearrangement invariant banach function spaces in the sense of w a j luxemburg and a c zaanen and of the theory of symmetrically normed ideals of bounded linear operators in hilbert space in the sense of i c gohberg and m g krein these two cases may be considered as the two extremes of the theory in the first case the underlying von neumann algebra is the commutative algebra l on some measure space with integration as trace in the second case the underlying von neumann algebra is b the algebra of all bounded linear operators on a hilbert space with standard trace important special cases of these non commutative spaces are the non commutative l p spaces which correspond in the commutative case with the usual l p spaces on a measure space and in the setting of symmetrically normed operator ideals they correspond to the schatten p classes mathfrak s p