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

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

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

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

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

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

sorting of permutations by cost constrained transpositions the problem of finding a minimum decomposition of a permutation in terms of transpositions with predetermined non uniform and non negative costs is addressed alternatively computing the transposition distance between two permutations where transpositions are endowed with arbitrary non negative costs is studied for such cost functions polynomial time constant approximation decomposition algorithms are described for metric path costs exact polynomial time decomposition algorithms are presented the algorithms in this paper represent a combination of viterbi type algorithms and graph search techniques for minimizing the cost of individual transpositions and dynamic programing algorithms for finding minimum cost decompositions of cycles the presented algorithms have a myriad of applications in information theory bioinformatics and algebra

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

bounds for permutation rate distortion we study the rate distortion relationship in the set of permutations endowed with the kendall tau metric and the chebyshev metric the ell infty metric this paper is motivated by the application of permutation rate distortion to the average case and worst case distortion analysis of algorithms for ranking with incomplete information and approximate sorting algorithms for the kendall tau metric we provide bounds for various distortion regimes while for the chebyshev metric we present bounds that are valid for all distortions and are especially accurate for small distortions in addition for the chebyshev metric we provide a construction for covering codes

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

algebraic cycles on a product of two hilbert modular surfaces in this paper we prove the tate conjecture for a product of two hilbert modular surfaces for non cm submotives

constructions of rank modulation codes rank modulation is a way of encoding information to correct errors in flash memory devices as well as impulse noise in transmission lines modeling rank modulation involves construction of packings of the space of permutations equipped with the kendall tau distance as our main set of results we present several general constructions of codes in permutations that cover a broad range of code parameters in particular we show a number of ways in which conventional error correcting codes can be modified to correct errors in the kendall space our constructions are nonasymptotic and afford simple encoding and decoding algorithms of essentially the same complexity as required to correct errors in the hamming metric as an example from binary bose chaudhuri hocquenghem codes we obtain codes correcting t kendall errors in n memory cells that support the order of n log2n t messages for any constant t 1 2 we give many examples of rank modulation codes with specific parameters turning to asymptotic analysis we construct families of rank modulation codes that correct a number of errors that grows with n at varying rates from n to n2 one of our constructions gives rise to a family of rank modulation codes for which the tradeoff between the number of messages and the number of correctable kendall errors approaches the optimal scaling rate

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

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

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

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

period relations and the tate conjecture for hilbert modular surfaces introduction 319 a sketch of the proof 320 0 notation and preliminaries 322 1 statement of the main results 328 2 twisting correspondences 331 3 period relations 334 4 hodge cycles 339 5 tare cycles 340 6 poles of l functions 341 7 abelian varieties 342 8 concluding remarks and open questions 343 acknowledgements 344 bibliography 344

an axiomatic approach to constructing distances for rank comparison and aggregation we propose a new family of distance measures on rankings derived through an axiomatic approach that consider the nonuniform relevance of the top and bottom of ordered lists and similarities between candidates the proposed distance functions include specialized weighted versions of the kendall distance and the cayley distance and are suitable for comparing rankings in a number of applications including information retrieval and rank aggregation in addition to proposing the distance measures and providing the theoretical underpinnings for their applications we also analyze algorithmic and computational aspects of weighted distance based rank aggregation we present an aggregation method based on approximating weighted distance measures by a generalized version of spearman s footrule distance as well as a markov chain method inspired by pagerank where transition probabilities of the markov chain reflect the chosen weighted distances

correlation clustering with constrained cluster sizes and extended weights bounds we consider the problem of correlation clustering on graphs with constraints on both the cluster sizes and the positive and negative weights of edges our contributions are twofold first we introduce the problem of correlation clustering with bounded cluster sizes second we extend the regime of weight values for which the clustering may be performed with constant approximation guarantees in polynomial time and apply the results to the bounded cluster size problem

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

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

error correction in flash memories via codes in the ulam metric we consider rank modulation codes for flash memories that allow for handling arbitrary charge drop errors unlike classical rank modulation codes used for correcting errors that manifest themselves as swaps of two adjacently ranked elements the proposed translocation rank codes account for more general forms of errors that arise in storage systems translocations represent a natural extension of the notion of adjacent transpositions and as such may be analyzed using related concepts in combinatorics and rank modulation coding our results include derivation of the asymptotic capacity of translocation rank codes construction techniques for asymptotically good codes as well as simple decoding methods for one class of constructed codes as part of our exposition we also highlight the close connections between the new code family and permutations with short common subsequences deletion and insertion error correcting codes for permutations and permutation codes in the hamming distance

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

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

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

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

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

a new correlation coefficient for comparing and aggregating non strict and incomplete rankings abstract we introduce a correlation coefficient that is designed to deal with a variety of ranking formats including those containing non strict i e with ties and incomplete i e unknown preferences the correlation coefficient is designed to enforce a neutral treatment of incompleteness whereby no assumptions are made about individual preferences involving unranked objects the new measure which can be regarded as a generalization of the seminal kendall tau correlation coefficient is proven to satisfy a set of metric like axioms and to be equivalent to a recently developed ranking distance function associated with kemeny aggregation in an effort to further unify and enhance both robust ranking methodologies this work proves the equivalence of an additional distance and correlation coefficient pairing in the space of non strict incomplete rankings these connections induce new exact optimization methodologies a specialized branch and bound algorithm and an exact integer programming formulation moreover the bridging of these complementary theories reinforces the singular suitability of the featured correlation coefficient to solve the general consensus ranking problem the latter premise is bolstered by an accompanying set of experiments on random instances which are generated via a herein developed sampling technique connected with the classic mallows distribution of ranking data associated experiments with the branch and bound algorithm demonstrate that as data becomes noisier the featured correlation coefficient yields relatively fewer alternative optimal solutions and that the aggregate rankings tend to be closer to an underlying ground truth shared by a majority

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

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

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

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

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