stability of the difference type methods for linear volterra equations in hilbert spaces we study the stability of backward difference timestepping method for linear volterra equations of scalar type in a hilbert space framework the results and methods extend and simulate numerically those introduced by pruss for integrability with respect to continuous solutions

a method for partial fraction expansion of transfer matrices a general method for partial fraction expansion of transfer matrices is presented and a general formula is given the main ideas of the procedure include 1 no required derivatives for the case of repeated eigenvalues 2 all the matrix residues are found at the same time and 3 there is no need for knowledge of the minimal polynomial the procedure can be easily applied to digital computer application and fills a void in the computer method for finding the partial fraction expansion in the case of repeated eigenvalues the results are presented in structured form using kronecker products in a way that facilitates easy implementation using a variety of standard software packages

a comparative study of three boundary element approaches to calculate the transient response of viscoelastic solids with unbounded domains abstract as an alternative to domain discretization methods the boundary element method bem provides a powerful tool for the calculation of dynamic structural response in frequency and time domain field equations of motion and boundary conditions are cast into boundary integral equations bie which are discretized only on the boundary fundamental solutions are used as weighting functions in the bie which fulfil the sommerfeld radiation condition i e the energy radiation into a surrounding medium is modelled correctly therefore infinite and semi infinite domains can be effectively treated by the method the soil represents such a semi infinite domain in soil structure interaction problems the response to vibratory loads superimposed to static pre loads can often be calculated by linear viscoelastic constitutive equations conventional viscoelastic constitutive equations can be generalized by taking fractional order time derivatives into account in the present paper two time domain bem approaches including generalized viscoelastic behaviour are compared with the laplace domain bem approach and subsequent numerical inverse transformation one of the presented time domain approaches uses an analytical integration of the elastodynamic bie in a time step viscoelastic constitutive properties are introduced after laplace transformation by means of an elastic viscoelastic correspondence principle the transient response is obtained by inverse transformation in each time step the other time domain approach is based on the so called convolution quadrature method in this formulation the convolution integral in the bie is numerically approximated by a quadrature formula whose weights are determined by the same laplace transformed fundamental solutions used in the first method and a linear multistep method a numerical study of wave propagation problems in 3 d viscoelastic continuum is performed for comparing the three bem formulations

on the convergence of multistep methods for nonlinear stiff differential equations convergence estimates are given fora stable multistep methods applied to singularly perturbed differential equations and nonlinear parabolic problems the approach taken here combines perturbation arguments with frequency domain techniques

polynomially bounded error estimates for trapezoidal rule convolution quadrature abstract we clarify the dependence with respect to the time variable of some estimates about the convergence of the trapezoidal rule based convolution quadrature method applied to hyperbolic problems this requires a careful investigation of the article of lehel banjai where the first convergence estimates were introduced and of some technical results from a classical paper of christian lubich

the numerical solution of linear multi term fractional differential equations systems of equations in this paper we show how the numerical approximation of the solution of a linear multi term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity we begin by showing how our method applies to a simple class of problems and we give a convergence result we solve the bagley torvik equation as an example we show how the method can be applied to a general linear multi term equation and give two further examples

fractional evolution dirac like equations some properties and a discrete von neumann type analysis a system of fractional evolution equations results from employing the tool of the fractional calculus and following the method used by dirac to obtain his well known equation from klein gordon s one it represents a possible interpolation between dirac and diffusion and wave equations in one space dimension in this paper some analytical properties typical of the general solution of this system of equations are obtained and necessary stability bounds for a numerical scheme approximating such equations are found through the classical discrete von neumann type analysis the non local property of the time fractional differential operator leads to discretizations in terms of series here the analytical methods usually employed in the study of the stability of discrete schemes when dealing with integer order differential equations have been adapted to the complexity of the real order case

adaptive discretization of fractional order viscoelasticity using sparse time history an efficient numerical method to integrate the constitutive response of fractional order viscoelasticity is developed the method can handle variable time steps to overcome the problem of the growing amount of data that has to be stored and used in each time step we introduce sparse quadrature we use an internal variable formulation of the viscoelastic equations where the internal variable is of stress type the rate equation that governs the evolution of the internal variable involves a fractional integral and can be identified as a volterra integral equation of the second kind with a weakly singular kernel for the numerical integration of the rate equation we adopt the finite element method in time in particular the discontinuous galerkin method with piecewise constant basis functions is used a priori and a posteriori error estimates are proved an adaptive strategy based on the a posteriori error estimate is developed finally the precision and effectiveness of the method are demonstrated by comparing the numerical solutions with analytical solutions

theoretical aspects of the application of convolution quadrature to scattering of acoustic waves in this paper we address several theoretical questions related to the numerical approximation of the scattering of acoustic waves in two or three dimensions by penetrable non homogeneous obstacles using convolution quadrature cq techniques for the time variable and coupled boundary element method finite element method for the space variable the applicability of cq to waves requires polynomial type bounds for operators related to the operator s 2 in the right half complex plane we propose a new systematic way of dealing with this problem both at the continuous and semidiscrete in space cases we apply the technique to three different situations scattering by a group of sound soft and hard obstacles by homogeneous and non homogeneous obstacles

transient elastodynamic antiplane crack analysis of anisotropic solids abstract transient elastodynamic analysis of an antiplane crack in anisotropic solids is presented a time domain traction boundary integral equation bie method is applied for this purpose the traction bie is hypersingular and has the crack opening displacement as its fundamental unknown quantity unlike the usual time domain bie method the present formulation applies a convolution quadrature developed by lubich lubich c 1988a b convolution quadrature and discretized operational calculus numer math 52 129 145 part i 413 425 part ii which requires only the laplace domain instead of the time domain green s functions the spatial variation of the crack opening displacement is approximated by an infinite series of chebyshev polynomials which take the local behavior of the crack opening displacement at crack tips into account by using a galerkin method the time domain bie is converted into a system of linear algebraic equations which can be solved step by step special attention is devoted to the computation of dynamic stress intensity factors of an antiplane crack in generally anisotropic solids numerical results for isotropic solids are presented and compared with the well known analytical results of thau and lu thau s a lu t h 1970 diffraction of transient horizontal shear waves by a finite crack and a finite rigid ribbon int j enggn sci 8 857 874 to check the accuracy and efficiency of the present time domain bie method the effect of the material anisotropy on the dynamic stress intensity factors is analyzed via several numerical examples

recursive algorithm for computing the matrix partial fraction expansion of transfer matrices having multiple quadratic poles a new method for determining matrix coefficients of partial fraction expansion pfe of transfer matrices tm with distinct and or multiple poles is presented the pfe coefficient matrices of tm are obtained recursively and the matrix formulation for the matrix pfe is amenable to computing

a second order accurate numerical method for a fractional wave equation we study a generalized crank nicolson scheme for the time discretization of a fractional wave equation in combination with a space discretization by linear finite elements the scheme uses a non uniform grid in time to compensate for the singular behaviour of the exact solution at t 0 with appropriate assumptions on the data and assuming that the spatial domain is convex or smooth we show that the error is of order k2 h2 where k and h are the parameters for the time and space meshes respectively

fast numerical solution of weakly singular volterra integral equations abstract we give a fortran program for the numerical solution of abel volterra convolution equations of the second and first kind we use a fractional linear multistep method bdf4 1 2 and use fast fourier transform techniques to exploit the convolution structure a detailed description of the computation of the quadrature weights using fft is given numerical examples are included

rapid solution of the wave equation in unbounded domains in this paper we propose and analyze a new fast method for the numerical solution of time domain boundary integral formulations of the wave equation we employ lubich s convolution quadrature method for the time discretization and a galerkin boundary element method for the spatial discretization the coefficient matrix of the arising system of linear equations is a triangular block toeplitz matrix possible choices for solving the linear system arising from the above discretization include the use of fast fourier transform fft techniques and the use of data sparse approximations by using fft techniques the computational complexity can be reduced substantially while the storage cost remains unchanged and is typically high using data sparse approximations the gain is reversed i e the computational cost is approximately unchanged while the storage cost is substantially reduced the method proposed in this paper combines the advantages of these two approaches first the discrete convolution related to the block toeplitz system is transformed into the discrete fourier image thereby arriving at a decoupled system of discretized helmholtz equations with complex wave numbers a fast data sparse e g fast multipole or panel clustering method can then be applied to the transformed system additionally significant savings can be achieved if the boundary data are smooth and time limited in this case the right hand sides of many of the helmholtz problems are almost zero and hence can be disregarded finally the proposed method is inherently parallel we analyze the stability and convergence of these methods thereby deriving the choice of parameters that preserves the convergence rates of the unperturbed convolution quadrature we also present numerical results which illustrate the predicted convergence behavior

a time optimal solution for the path cover problem on cographs we show that the notoriously difficult problem of finding and reporting the smallest number of vertex disjoint paths that cover the vertices of a graph can be solved time and work optimally for cographs our result implies that for this class of graphs the task of finding a hamiltonian path can be solved time and work optimally in parallel it was open for more than 10 years to find a time and work optimal parallel solution for this important problem our contribution is to offer an optimal solution to this important problem we begin by showing that any algorithm that solves an instance of size n of the problem must take logn time on the crew even if an infinite number of processors are available we then go on to show that this time lower bound is tight by devising an erew algorithm that given an n vertex cograph g represented by its cotree finds and reports all the paths in a minimum path cover in o logn time using n logn processors

interface dynamic stiffness matrix approach for three dimensional transient multi region boundary element analysis a novel substructuring method is developed for the coupling of boundary element and finite element subdomains in order to model three dimensional multi region elastodynamic problems in the time domain the proposed procedure is based on the interface stiffness matrix approach for static multi region problems using variational principles together with the concept of duhamel integrals unit impulses are applied at the boundary of each region in order to evaluate the impulse response matrices of the duhamel convolution integrals although the method is not restricted to a special discretization technique the regions are discretized using the boundary element method combined with the convolution quadrature method this results in a time domain methodology with the advantages of performing computations in the laplace domain which produces very accurate and stable results as verified on test examples in addition the assembly of the boundary element regions and the coupling to finite elements are greatly simplified and more efficient finally practical applications in the area of soil structure interaction and tunneling problems are shown copyright 2009 john wiley sons ltd

statistics of nonlinear stochastic dynamical systems under levy noises by a convolution quadrature approach this paper describes a novel numerical approach to find the statistics of the non stationary response of scalar nonlinear systems excited by levy white noises the proposed numerical procedure relies on the introduction of an integral transform of the wiener hopf type into the equation governing the characteristic function once this equation is rewritten as partial integro differential equation it is then solved by applying the method of convolution quadrature originally proposed by lubich here extended to deal with this particular integral transform the proposed approach is relevant for two reasons 1 statistics of systems with several different drift terms can be handled in an efficient way independently from the kind of white noise 2 the particular form of wiener hopf integral transform and its numerical evaluation both introduced in this study are generalizations of fractional integro differential operators of potential type and grunwald letnikov fractional derivatives respectively

finite element approximation of diffusion equations with convolution terms approximation of solutions to diffusion equations with memory represented by convolution integral terms is considered such problems arise from modeling of flows in fissured media convergence of the method is proved and results of numerical experiments confirming the theoretical results are presented the advantages of implementation of the algorithm in a multiprocessing environment are discussed

two fully discrete schemes for fractional diffusion and diffusion wave equations with nonsmooth data we consider initial boundary value problems for the subdiffusion and diffusion wave equations involving a caputo fractional derivative in time we develop two fully discrete schemes based on the piecewise linear galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward euler method second order backward difference method and establish error estimates optimal with respect to the regularity of problem data these two schemes are first and second order accurate in time for both smooth and nonsmooth data extensive numerical experiments for two dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity

a new physiological boundary condition for hemodynamics we propose a new physiologically based outflow boundary condition for hemodynamics under general transient regimes this is in contrast to previous studies that impose restrictions of temporal periodicity the new condition is analyzed and its numerical implementation is discussed in detail we show that existing impedance boundary conditions can be viewed as numerical approximations of the new condition our study provides a partial justification for using some of these existing conditions beyond the periodic problems for which they were designed moreover the new condition has better stability properties the theoretical results are illustrated by numerical experiments pertaining to cerebral blood flow

uniform l1 error bounds for the semidiscrete solution of a volterra equation with completely monotonic convolution kernel abstract we study the second order continuous time galerkin approximation of the solution ut 0t t s au s ds 0 u 0 v t 0 where a is an elliptic partial differential operator and t is completely monotonic and locally integrable but not constant error estimates are derived in the norm of lt1 0 lx2

an analysis of the rayleigh stokes problem for a generalized second grade fluid we study the rayleigh stokes problem for a generalized second grade fluid which involves a riemann liouville fractional derivative in time and present an analysis of the problem in the continuous space semidiscrete and fully discrete formulations we establish the sobolev regularity of the homogeneous problem for both smooth and nonsmooth initial data v v including v in l 2 omega v l2 a space semidiscrete galerkin scheme using continuous piecewise linear finite elements is developed and optimal with respect to initial data regularity error estimates for the finite element approximations are derived further two fully discrete schemes based on the backward euler method and second order backward difference method and the related convolution quadrature are developed and optimal error estimates are derived for the fully discrete approximations for both smooth and nonsmooth initial data numerical results for one and two dimensional examples with smooth and nonsmooth initial data are presented to illustrate the efficiency of the method and to verify the convergence theory

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

a linear recognition algorithm for cographs cographs are the graphs formed from a single vertex under the closure of the operations of union and complement another characterization of cographs is that they are the undirected graphs with no induced paths on four vertices cographs arise naturally in such application areas as examination scheduling and automatic clustering of index terms furthermore it is known that cographs have a unique tree representation called a cotree using the cotree it is possible to design very fast polynomial time algorithms for problems which are intractable for graphs in general such problems include chromatic number clique determination clustering minimum weight domination isomorphism minimum fill in and hamiltonicity in this paper we present a linear time algorithm for recognizing cographs and constructing their cotree representation

a numerical method for a partial integro differential equation a method is considered for the integration in time of a partial integro differential equation the discretization technique employed is patterned after an idea of ch lubich error bounds are derived for both smooth and nonsmooth initial data

initial conditions contribution in a bem formulation based on the convolution quadrature method this work presents a two dimensional boundary element method bem formulation for the analysis of scalar wave propagation problems the formulation is based on the so called convolution quadrature method cqm by means of which the convolution integral presented in time domain bem formulations is numerically substituted by a quadrature formula whose weights are computed using the laplace transform of the fundamental solution and a linear multistep method this bem formulation was initially developed for scalar wave propagation problems with null initial conditions in order to overcome this limitation this work presents a general procedure that enables one to take into account non homogeneous initial conditions after replacing the initial conditions by equivalent pseudo forces the numerical results included in this work show the accuracy of the proposed bem formulation and its applicability to such kind of analysis copyright 2006 john wiley sons ltd

runge kutta convolution quadrature methods for well posed equations with memory runge kutta based convolution quadrature methods for abstract well posed linear and homogeneous volterra equations non necessarily of sectorial type are developed a general representation of the numerical solution in terms of the continuous one is given the error and stability analysis is based on this representation which for the particular case of the backward euler method also shows that the numerical solution inherits some interesting qualitative properties such as positivity of the exact solution numerical illustrations are provided

on the quadrature error in operational quadrature methods for convolutions we analyze the quadrature error associated with operational quadrature methods for convolution equations the assumptions are that the convolution kernel is inl 1 and that its laplace transform is analytic and bounded in an obtuse sector of the complex plane under these circumstances the laplace transform has a slow variation property which admits a fourier analysis of the quadrature error we provide generall p error estimates assuming suitable smoothness conditions on the function under convolution

on the discretization in time for a parabolic integrodifferential equation with a weakly singular kernel ii nonsmooth initial data abstract backward euler methods are considered for a parabolic integrodifferential equation with a memory term containing a weakly singular kernel the integral term is treated by means of a convolution quadrature suggested by lubich some results from lubich s discretized fractional calculus play a crucial role in error analyses the stability and convergence properties of the time discretizations are derived and are applied to the semidiscrete equations built by the galerkin finite element methods in the space variables

laplace transforms for approximation of highly oscillatory volterra integral equations of the first kind abstract this paper focuses on laplace and inverse laplace transforms for approximation of volterra integral equations of the first kind with highly oscillatory bessel kernels where the explicit formulae for the solution of the first kind integral equations are derived from which the integral equations can also be efficiently calculated by the clenshaw curtis filon type methods furthermore by applying the asymptotics of the solution some simpler formulas for approximating the solution for large values of the parameters are deduced preliminary numerical results are presented based on the approximate formulae and the explicit formulae which are compared with the convolution quadrature and numerical inverse laplace transform methods all these methods share that the costs the same independent of large values of frequencies

on the k path partition of graphs the k path partition problem is to partition a graph into the minimum number of paths so that none of them has length more than k for a given positive integer k the problem is a generalization of the hamiltonian path problem and the problem of partitioning a graph into the minimum number of paths the k path partition problem remains np complete on the class of chordal bipartite graphs if k is part of the input and we show that it is np complete on the class of comparability graphs even for k 3 on the positive side we present a polynomial time solution for the problem with any k on bipartite permutation graphs which form a subclass of chordal bipartite graphs

wave propagation in a simplified modelled poroelastic continuum fundamental solutions and a time domain boundary element formulation in finite element formulations for poroelastic continua a representation of biot s theory using the unknowns solid displacement and pore pressure is preferred such a formulation is possible either for quasi static problems or for dynamic problems if the inertia effects of the fluid are neglected contrary to these formulations a boundary element method bem for the general case of biot s theory in time domain has been published wave propagation in viscoelastic and poroelastic continua a boundary element approach lecture notes in applied mechanics springer berlin heidelberg new york 2001 if the advantages of both methods are required it is common practice to couple both methods however for such a coupled fe be procedure a bem for the simplified dynamic biot theory as used in fem must be developed therefore here the fundamental solutions as well as a be time stepping procedure is presented for the simplified dynamic theory where the inertia effects of the fluid are neglected further a semi analytical one dimensional solution is presented to check the proposed be formulation finally wave propagation problems are studied using either the complete biot theory as well as the simplified theory these examples show that no significant differences occur for the selected material copyright 2005 john wiley sons ltd

a new space time energetic formulation for wave propagation analysis in layered media by bems in this paper considering a dirichlet neumann problem for 1d wave propagation analysis in layered media reformulated in terms of boundary integral equations we compare two classical weak formulations for the boundary element method with a new space time energetic formulation continuity and coerciveness of the bilinear form corresponding to this last formulation are proved several numerical results will be presented pointing out the numerical properties of the derived linear systems instability phenomena arising using one of the classical techniques can be prevented choosing suitable time steps in the discretization phase of course with a higher computational cost with respect to the new procedure which appears to be unconditionally stable copyright 2008 john wiley sons ltd

a fully dynamic algorithm for recognizing and representing proper interval graphs in this paper we study the problem of recognizing and representing dynamically changing proper interval graphs the input to the problem consists of a series of modifications to be performed on a graph where a modification can be a deletion or an addition of a vertex or an edge the objective is to maintain a representation of the graph as long as it remains a proper interval graph and to detect when it ceases to be so the representation should enable one to efficiently construct a realization of the graph by an inclusion free family of intervals this problem has important applications in physical mapping of dna we give a near optimal fully dynamic algorithm for this problem it operates in o log n worst case time per edge insertion or deletion we prove a close lower bound of omega log n log log n log b amortized time per operation in the cell probe model with word size b we also construct optimal incremental and decremental algorithms for the problem which handle each edge operation in o 1 time as a byproduct of our algorithm we solve in o log n worst case time the problem of maintaining connectivity in a dynamically changing proper interval graph

implicit runge kutta methods for the discretization of time domain integral equations implicit runge kutta based schemes are proposed for solving the time domain integral equations of electromagnetic theory the proposed technique maps a laplace domain equation to a domain equation using the butcher tableau of the implicit runge kutta scheme a discrete time domain system is recovered by computing the inverse transform numerically the resulting technique is capable of third or fifth order accuracy in time and is stable independent of time step numerical results illustrate the accuracy and stability of the technique

an algorithm for approximate solutions to weakly filtered synchronous control systems and nonlinear renewal processes a multi time perturbation algorithm is derived to study certain synchronous control systems and nonlinear renewal processes the new methods presented here study singular perturbation problems for nonlinear volterra integro differential equations whose kernels are close to the dirac delta function the algorithm provides useful approximate solutions to weakly filtered phase locked loop circuits and to nonlinear renewal equations describing certain population dynamics

effects of material gradients on transient dynamic mode iii stress intensity factors in a fgm abstract this paper presents a transient dynamic crack analysis for a functionally graded material fgm by using a hypersingular time domain boundary integral equation method the spatial variations of the material parameters of the fgm are described by an exponential law a numerical solution procedure is developed for solving the hypersingular time domain traction bie to avoid the use of time dependent green s functions which are not available for general fgm a convolution quadrature formula is adopted for approximating the temporal convolution while a galerkin method is applied for the spatial discretization of the hypersingular time domain traction bie numerical results for the transient dynamic stress intensity factors for a finite crack in an infinite and linear elastic fgm subjected to an impact anti plane crack face loading are presented and discussed the effects of the material gradients of the fgm on the transient dynamic stress intensity factors and their dynamic overshoot over the corresponding static stress intensity factors are analyzed

linear time algorithms for the hamiltonian problems on distance hereditary graphs a hamiltonian path of a graph g is a simple path that contains each vertex of g exactly once a hamiltonian cycle of a graph is a simple cycle with the same property the hamiltonian path resp cycle problem involves testing whether a hamiltonian path resp cycle exists in a graph the 1hp resp 2hp problem is to determine whether a graph has a hamiltonian path starting from a specified vertex resp starting from a specified vertex and ending at the other specified vertex the hamiltonian problems include the hamiltonian path hamiltonian cycle 1hp and 2hp problems a graph is a distance hereditary graph if each pair of vertices is equidistant in every connected induced subgraph containing them in this paper we present a unified approach to solving the hamiltonian problems on distance hereditary graphs in linear time

diffraction of waves on square lattice by semi infinite crack in this paper a discrete analogue of sommerfeld half plane diffraction is investigated the two dimensional problem of diffraction on a square lattice of a plane transverse wave by a semi infinite crack is solved the discrete wiener hopf method has been used to obtain the exact solution of the discrete helmholtz equation with input data prescribed on the crack boundary sites due to a time harmonic incident wave it is established that there exists a unique saddle point for the diffraction integral and its properties are characterized an asymptotic approximation of the solution in the far field is provided and for some values of the frequency it is compared with the numerical solution of the diffraction problem using a finite grid a low frequency approximation of the solution in integral form recovers the classical continuum solution at sufficiently large frequency in the pass band the effects due to discreteness and anisotropy appear the analysis is relevant to a 5 point discretization bas

a polynomial solution to the k fixed endpoint path cover problem on proper interval graphs we study a variant of the path cover problem namely the k fixed endpoint path cover problem or kpc for short given a graph g and a subset t of k vertices of v g a k fixed endpoint path cover of g with respect to t is a set of vertex disjoint paths p that covers the vertices of g such that the k vertices of t are all endpoints of the paths in p the kpc problem is to find a k fixed endpoint path cover of g of minimum cardinality note that if t is empty or equivalently k 0 the stated problem coincides with the classical path cover problem the kpc problem generalizes some path cover related problems such as the 1hp and 2hp problems which have been proved to be np complete note that the complexity status for both 1hp and 2hp problems on interval graphs remains an open question damaschke 1993 9 in this paper we show that the kpc problem can be solved in linear time on the class of proper interval graphs that is in o n m time on a proper interval graph on n vertices and m edges the proposed algorithm is simple requires linear space and also enables us to solve the 1hp and 2hp problems on proper interval graphs within the same time and space complexity

on the k path cover problem for cacti in this paper we investigate the k path cover problem for graphs which is to find the minimum number of vertex disjoint k paths that cover all the vertices of a graph the k path cover problem for general graphs is np complete though notable applications of this problem to database design network vlsi design ring protocols and code optimization efficient algorithms are known for only few special classes of graphs in order to solve this problem for cacti i e graphs where no edge lies on more than one cycle we introduce the so called steiner version of the k path cover problem and develop an efficient algorithm for the steiner k path cover problem for cacti which finds an optimal k path cover for a given cactus in polynomial time

time domain boundary integral equation modeling of heat transmission problems this paper investigates the numerical modeling of a time dependent heat transmission problem by the convolution quadrature boundary element method it introduces the latest theoretical development into the error analysis of the numerical scheme semigroup theory is applied to obtain stability in the spatially semidiscrete scheme functional calculus is employed to yield convergence in the fully discrete scheme we compare the results to a more traditional laplace domain approach

weighted l1 paley wiener theorem with applications to stability of the linear multi step methods for volterra equations in hilbert spaces abstract we study the weighted l 1 paley wiener theorem with applications to stability of the linear multi step time discretization for a linear volterra equation of scalar type in a hilbert space the results extend to weighted l 1 stability some recent l 1 stability theorem due to the author numer math 109 2008 571 595 our theoretical results are applied to investigate the asymptotic behavior as t n of the numerical solutions of certain integro differential equations in hilbert space

the approximability of the weighted hamiltonian path completion problem on a tree given a graph the hamiltonian path completion problem is to find an augmenting edge set such that the augmented graph has a hamiltonian path in this paper we show that the hamiltonian path completion problem will unlikely have any constant ratio approximation algorithm unless np p this problem remains hard to approximate even when the given subgraph is a tree moreover if the edge weights are restricted to be either 1 or 2 the hamiltonian path completion problem on a tree is still np hard then it is observed that this problem is strongly np hard so it does not have any fully polynomial time approximation scheme fptas unless np p when the given tree is a k tree we give an approximation algorithm with performance ratio 1 5

an optimal path cover algorithm for cographs abstract the class of cographs or complement reducible graphs arises naturally in many different areas of applied mathematics and computer science in this paper we present an optimal algorithm for determining a minimum path cover for a cograph g in case g has a hamiltonian path cycle our algorithm exhibits the path cycle as well

convolution quadrature and discretized operational calculus ii operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations singular integrals power and logarithmic singularities finite part integrals multiple timescale convolution volterra integral equations wiener hopf integral equations frequency domain conditions which determine the stability of such equations can be carried over to the discretization

on accurate product integration rules for linear fractional differential equations this paper addresses the numerical solution of linear fractional differential equations with a forcing term competitive and highly accurate product integration rules are derived by starting from an equivalent formulation in terms of a volterra integral equation with a generalized mittag leffler function in the kernel the error analysis is reported and aspects related to the computational complexity are treated numerical tests confirming the theoretical findings are presented

potential evaluation in space time bie formulations of 2d wave equation problems in this paper we examine the computation of the potential generated by space time bie representations associated with dirichlet and neumann problems for the 2d wave equation in particular we consider the efficient evaluation of the convolution time integral that appears in the potential representation for this we propose two simple quadrature rules which appear more efficient than the currently used ones both are of gaussian type the classical gauss jacobi quadrature rule in the case of a dirichlet problem and the classical gauss radau quadrature rule in the case of a neumann problem both of them give very accurate results by using a few quadrature nodes as long as the potential is evaluated at a point not very close to the boundary of the pde problem domain to deal with this latter case we propose an alternative rule which is defined by a proper combination of a gauss legendre formula with one of product type the latter having only 5 nodes the proposed quadrature formulas are compared with the second order bdf lubich convolution quadrature formula and with two higher order lubich formulas of runge kutta type an extensive numerical testing is presented this shows that the new proposed approach is very competitive from both the accuracy and efficiency points of view

uniform l 1 behavior for time discretization of a volterra equation with completely monotonic kernel ii convergence a previous work on the time discretization for the solution of a volterra equation with completely monotonic convolution kernel is extended the considered time discretization method comes from part i x da ima j numer anal 22 2002 pp 133 151 where the backward euler is combined with order one convolution quadrature approximating the integral in the present paper the convergence properties of the discretization in time are given in the l t 1 0 infty h bigcap l t infty 0 infty h norm

dynamic analyses of plane frames by integral equations for bars and timoshenko beams abstract since especially for higher frequencies timoshenko s theory gives more reliable results than euler bernoulli s theory systems of beams like frames under arbitrary dynamic excitations should be analyzed on the basis of this refined theory in this paper after deriving the basic fundamental solutions of a lateral unit point force and of a unit single moment the deflection and the rotation of the beam cross section are given as integral equation forms of the governing second order differential equation system in the laplace and the frequency domain since in frameworks axial displacements also occur the well known integral equations for bars under tension or compression are added to complete the modelling of plane frame structures from these the unknown boundary values may be exactly determined via point collocation at the beam ends and solution of the resulting algebraic system with these boundary values the exact deflection and rotation can also be found at arbitrary interior points finally the convolution quadrature method is applied to the laplace domain integral equation formulation in order to analyze the time dependent wave propagation process two examples are presented to demonstrate the method s exactness compared with conventional finite element results a single beam where the shift of the resonance frequencies when using the timoshenko theory instead of euler bernoulli s theory is shown and a two storey framework where in addition to the frequence response the causality of the propagating wave is studied

an energy approach to space time galerkin bem for wave propagation problems in this paper we consider dirichlet or neumann wave propagation problems reformulated in terms of boundary integral equations with retarded potential starting from a natural energy identity a space time weak formulation for 1d integral problems is briefly introduced and continuity and coerciveness properties of the related bilinear form are proved then a theoretical analysis of an extension of the introduced formulation for 2d problems is proposed pointing out the novelty with respect to existing literature results at last various numerical simulations will be presented and discussed showing unconditional stability of the space time galerkin boundary element method applied to the energetic weak problem copyright 2009 john wiley sons ltd

solutions to the discrete airy equation application to parabolic equation calculations in the case of the equidistant discretization of the airy differential equation discrete airy equation the exact solution can be found explicitly this fact is used to derive a discrete transparent boundary condition tbc for a schrodinger type equation with linear varying potential which can be used in parabolic equation simulations in underwater acoustics and for radar propagation in the troposphere we propose different strategies for the discrete tbc and show an efficient implementation finally a stability proof for the resulting scheme is given a numerical example in the application to underwater acoustics shows the superiority of the new discrete tbc

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

uniform error estimates of galerkin methods for monotone abel volterra integral equations on the half line we consider galerkin methods for monotone abel volterra integral equations of the second kind on the half line the l2 theory follows from kolodner s theory of monotone hammerstein equations we derive the loo theory from the l2 theory by relating the l2and loo spectra of operators of the form x b ax to one another here denotes convolution and b e l1 and a e loo as an extra condition we need b t o t a 1 with a 0 we also prove the discrete analogue in particular we verify that the galerkin matrix satisfies the discrete conditions

fast convolution quadrature for the wave equation in three dimensions this work addresses the numerical solution of time domain boundary integral equations arising from acoustic and electromagnetic scattering in three dimensions the semidiscretization of the time domain boundary integral equations by runge kutta convolution quadrature leads to a lower triangular toeplitz system of size n this system can be solved recursively in an almost linear time o n log 2 n but requires the construction of o n dense spatial discretizations of the single layer boundary operator for the helmholtz equation this work introduces an improvement of this algorithm that allows to solve the scattering problem in an almost linear time the new approach is based on two main ingredients the near field reuse and the application of data sparse techniques exponential decay of runge kutta convolution weights w n h d outside of a neighborhood of d n h where h is a time step allows to avoid constructing the near field i e singular and near singular integrals for most of the discretizations of the single layer boundary operators near field reuse the far field of these matrices is compressed with the help of data sparse techniques namely h matrices and the high frequency fast multipole method numerical experiments indicate the efficiency of the proposed approach compared to the conventional runge kutta convolution quadrature algorithm

runge kutta convolution quadrature for the boundary element method abstract time domain boundary element formulations can be established either directly in time domain or via laplace or fourier domain somewhere in between are the convolution quadrature based boundary element formulations which utilize the laplace domain fundamental solution but establish a time stepping procedure up to now in applications mostly backward differential formulas of second order are used as the underlying multistep method however in recent mathematical literature also runge kutta methods have been applied here the use of runge kutta methods is explained in detail and some numerical studies are given in these studies the backward difference based procedures are compared to runge kutta methods for a non smooth problem an l 2 norm of the error is used as the basis of comparison the convergence of which is investigated theoretically as well the results confirm that the usage of the new techniques is preferable with regard to less numerical oscillations in the solution and better representation of wave fronts

an analysis of the modified l1 scheme for time fractional partial differential equations with nonsmooth data we introduce a modified l1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous c

matrix partial fraction expansions of rational matrices a general and rigorous derivation is made of a new formula established recently in the literature for matrix partial fraction expansion of a rational matrix the procedure by use of a minimal jordan realization of the rational matrix provides as a byproduct general expressions for residue matrices in terms of products of columns of the output matrix and rows of the input matrix of the realization

systems based decomposition schemes for the approximate solution of multi term fractional differential equations we give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi term fractional differential equations using the caputo form of the fractional derivative the schemes we compare are based on conversion of the original problem into a system of equations we review alternative approaches and consider how the most appropriate numerical scheme may be chosen to solve a particular equation

long time behavior of the two grid finite element method for fully discrete semilinear evolution equations with positive memory abstract based on two grid discretizations two fully discrete finite element algorithms for semilinear parabolic integro differential equations with positive memory are proposed with the backward euler scheme for the temporal discretization the basic idea of the space two grid finite element algorithms is to approximate the semilinear equations on a coarse space grid and to solve the linearized equations on a finer space grid at each time step to further decreases the amount of computational work a space time two grid algorithm based on a coarse space grid with large time stepsize t and a finer space grid with small time stepsize t for the evolutional equations is proposed in this paper the sharp long time stability and error estimates for the standard finite element method the space two grid finite element method and the space time two grid finite element method are derived it is showed that the two grid algorithms long time stability and error estimates are similar to those of the direct resolution of the semilinear problem on a fine grid

strong order of convergence of a fully discrete approximation of a linear stochastic volterra type evolution equation in this paper we investigate a discrete approximation in time and in space of a hilbert space valued stochastic process u t t 0 t satisfying a stochastic linear evolution equation with a positive type memory term driven by an additive gaussian noise the equation can be written in an abstract form as du t 0 b t s au s ds dt dw t 0 t u 0 u0 h where w is a q wiener process on h l d and where the main example of b we consider is given by b t t 1 0 0 such that a has finite trace and that q is bounded from h into d a for some real with 1 1 the discretization is achieved via an implicit euler scheme and a laplace transform convolution quadrature in time parameter t t n and a standard continuous finite element method in space parameter h let un h be the discrete solution at t n t we show that e un h u t 1 2 o h t for any 1 1 2 and 1 1 m kovacs department of mathematics and statistics university of otago po box 56 dunedin 9054 new zealand e mail mkovacs maths otago ac nz j printems laboratoire d analyse et de mathematiques appliquees cnrs umr 8050 61 avenue du general de gaulle universite paris est 94010 creteil france e mail printems u pec fr 2 mihaly kovacs jacques printems

nonsmooth data error estimates for approximations of an evolution equation with a positive type memory term we study the numerical approximation of an integro differential equation which is intermediate between the heat and wave equations the proposed discretization uses convolution quadrature based on the first and second order backward difference methods in time and piecewise linear finite elements in space optimal order error bounds in terms of the initial data and the inhomogeneity are shown for positive times without assumptions of spatial regularity of the data

weak convergence of a fully discrete approximation of a linear stochastic evolution equation with a positive type memory term abstract in this paper we are interested in the numerical approximation of the marginal distributions of the hilbert space valued solution of a stochastic volterra equation driven by an additive gaussian noise this equation can be written in the abstract ito form as d x t 0 t b t s a x s d s d t d w q t t 0 t x 0 x 0 h where w q is a q wiener process on the hilbert space h and where the time kernel b is the locally integrable potential t 2 1 2 or slightly more general the operator a is unbounded linear self adjoint and positive on h our main assumption concerning the noise term is that a 1 2 q 1 2 is a hilbert schmidt operator on h for some 0 1 the numerical approximation is achieved via a standard continuous finite element method in space parameter h and an implicit euler scheme and a laplace convolution quadrature in time parameter t t n we show that for h r twice continuously differentiable test function with bounded second derivative e x h n e x t c ln t h 2 t t h 2 for any 0 1 this is essentially twice the rate of strong convergence under the same regularity assumption on the noise

optimal greedy algorithms for indifference graphs abstract a fundamental problem in social sciences and management is understanding and predicting decisions made by individuals various groups or the society as a whole in this context one important concept is the notion of indifference we characterize the class of indifference graphs that is graphs which arise in the process of quantifying indifference relations in particular we show that these graphs are characterized by the existence of a special ordering of their vertices as it turns out this ordering leads naturally to optimal greedy algorithms for a number of computational problems including coloring finding a shortest path between two vertices computing a maximum matching the center and a hamiltonian path

a fully discrete bem fem scheme for transient acoustic waves abstract we study a symmetric bem fem coupling scheme for the scattering of transient acoustic waves by bounded inhomogeneous anisotropic obstacles in a homogeneous field an incident wave in free space interacts with the obstacles and produces a combination of transmission and scattering the transmitted part of the wave is discretized in space by finite elements while the scattered wave is reduced to two fields defined on the boundary of the obstacles and is discretized in space with boundary elements we choose a coupling formulation that leads to a symmetric system of integro differential equations the retarded boundary integral equations are discretized in time by convolution quadrature and the interior field is discretized in time with the trapezoidal rule we show that the scattering problem generates a c 0 group of isometries in a hilbert space and use associated estimates to derive stability and convergence results we provide numerical experiments and simulations to validate our results and demonstrate the flexibility of the method

analysis of a temporal discretization for a semilinear fractional diffusion equation abstract this paper considers a temporal discretization of a semilinear fractional diffusion equation with nonsmooth initial data with appropriately graded temporal grids first order temporal accuracy in the norm of l 2 0 t l 2 is derived by a new discrete gronwall s inequality and then first order temporal accuracy in the norm of l 0 t l 2 is established for 1 2 1 finally numerical results are provided to verify the theoretical results

optimal path cover problem on block graphs and bipartite permutation graphs abstract the optimal path cover problem is to find a minimum number of vertex disjoint paths which together cover all the vertices of the graph in this paper we present linear time algorithms for the optimal path cover problem for the class of block graphs and bipartite permutation graphs

dynamics of structures coupled with elastic media a review of numerical models and methods abstract this paper reviews issues and developments in the field of structure environment interaction problems in which the environment is an elastic body possibly unbounded it covers in particular the fields of soil structure interaction ground borne noise and vibration emitted by transportation systems and wave diffraction by obstacles in an elastic medium the general setting for a linear bounded structure coupled to an unbounded linear environment through a bounded interface is first recalled and the domain decomposition technique classically used for its description is put up extensions for the cases of nonlinear structure and uncertain environment are then discussed finally the ongoing research in the fields of unbounded interface moving interface and multiple interfaces are reviewed and summarized

boundary integral solvers for an evolutionary exterior stokes problem this paper proposes and analyzes a full discretization of the exterior transient stokes problem with dirichlet boundary conditions the method is based on a single layer boundary integral representation using galerkin semidiscretization in the space variables and multistep convolution quadrature in time convergence estimates are based on a laplace domain analysis which translates into a detailed study of the exterior brinkman problem some numerical experiments are provided

the numerical analysis on a volterra equation with asymptotically periodic solution we consider order one operational quadrature methods on a certain integro differential equation of volterra type on 0 with piecewise linear convolution kernels the forms of discretization solution are patterned after a continuous one of hannsgen 1979 2 an l 1 remainder stability and an error bound are derived