successive overrelaxation iteration for the stability of large scale systems abstract the successive overrelaxation iteration is used to investigate the stability of large scale systems and some sufficient conditions such that the zero solution of an overall system is asymptotically stable are obtained which only depend upon the fundamental matrices of the variational isolated subsystems and the interconnected matrices

variable step size predictor corrector schemes for second kind volterra integral equations in this paper a family of implicit multistep methods for the solution of volterra integral equations is derived these methods together with an explicit euler predictor permit the use of a variable step size when solving integral equations means of controlling the error and stability by varying the step size and the order of the method are described extensive numerical results are presented

nonpolynomial spline collocation for volterra equations with weakly singular kernels volterra integral equations of the second kind with weakly singular kernels possess in general solutions which are not smooth near the left endpoint of the interval of integration since ordinary polynomial spline collocation cannot lead to high order convergence we introduce special nonpolynomial spline spaces which are modelled after the structure of these solutions near the point of nonsmooth behavior collocation in these spaces will once more lead to high order methods analogous results are derived for volterra integro differential equations with weakly singular kernels

lipschitz stability for nonlinear volterra integrodifferential systems unlike uniform stability the linearized system inherits the property of uniform lipschitz stability from its original nonlinear system in this paper we investigate sufficient conditions for uniform lipschitz stability of nonlinear integro differential systems through their associated variational systems

convergence analysis of discretization methods for nonlinear first kind volterra integral equations an existence and uniqueness result is given for nonlinear volterra integral equations of the first kind this permits by means of analogous discrete manipulations a general convergence analysis for a wide class of discretization methods for nonlinear first kind volterra integral equations to be presented a concept of optimal consistency allows twosided error bounds to be derived

implicit runge kutta nystrom methods for general second order volterra integro differential equations abstract in this paper we derive m stage runge kutta nystrom methods for the numerical solution of general second order volterra integro differential equations these implicit methods are based on collocation techniques in certain polynomial spline spaces the question concerning the attainable order of convergence of such one step methods is completely answered

observer design for nonlinear systems with discrete time measurements this paper focuses on the development of asymptotic observers for nonlinear discrete time systems it is argued that instead of trying to imitate the linear observer theory the problem of constructing a nonlinear observer can be more fruitfully studied in the context of solving simultaneous nonlinear equations in particular it is shown that the discrete newton method properly interpreted yields an asymptotic observer for a large class of discrete time systems while the continuous newton method may be employed to obtain a global observer furthermore it is analyzed how the use of broyden s method in the observer structure affects the observer s performance and its computational complexity an example illustrates some aspects of the proposed methods moreover it serves to show that these methods apply equally well to discrete time systems and to continuous time systems with sampled outputs

the numerical treatment of volterra integro differential equations with unbounded delay abstract initial value problems for ordinary and partial volterra integro differential equations with infinite and finite delay arise in the mathematical modelling of various physical and biological phenomena in this paper we give a survey of some recent developments in the numerical treatment of such functional equations with the focus being on equations with infinite delay in addition we will discuss a number of numerical aspects that are not yet understood well

analysis of explosion for nonlinear volterra equations this survey paper presents some analytical tools for determining the existence of a blow up solution for a wide class of nonlinear volterra integral equations the related phenomenon of quenching has also been investigated using similar techniques in both cases lower and upper bound estimates for the critical time as well as asymptotic solution behavior in the key limit are established this paper surveys recent results in the analysis of such problems and proposes problems for future study

on global superconvergence of iterated collocation solutions to linear second kind volterra integral equations it is well known that in contrast to fredholm integral equations iterated collocation solutions based on collocation at the gauss points to volterra integral equations of the second kind exhibit optimal discrete superconvergence only at the mesh points here we show that some degree of global superconvergence is possible on the entire interval

the piecewise polynomial collocation method for nonlinear weakly singular volterra equations second kind volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration using an adaptation of the analysis originally developed for nonlinear weakly singular fredholm integral equations we present a complete discussion of the optimal global and local order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear volterra integral equations with algebraic or logarithmic singularities in their kernels

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

on some partial integral inequalities in n independent variables in this paper comparison method will be used to establish a number of fundamental partial integral inequalities in n independent variables these inequalities can be used as ready and powerful tools in developing the qualitative theory of partial differential and integral equations in n independent variables

backward differentiation type formulas for volterra integral equations of the second kind numerical integration formulas are discussed which are obtained by differentiation of the volterra integral equation and by applying backward differentiation formulas to the resulting integro differential equation in particular the stability of the method is investigated for a class of convolution kernels the accuracy and stability behaviour of the method proposed in this paper is compared with that of i a block implicit runge kutta scheme and ii the scheme obtained by applying directly a quadrature rule which is reducible to the backward differentiation formulas the present method is particularly advantageous in the case of stiff volterra integral equations

runge kutta theory for volterra integrodifferential equations the present paper develops the theory of general runge kutta methods for volterra integrodifferential equations the local order is characterized in terms of the coefficients of the method we investigate the global convergence of mixed and extended runge kutta methods and give results on asymptotic error expansions in a further section we construct examples of methods up to order 4

on some integral inequalities in n independent variables and their applications abstract using young s technique which is based on a riemann function we unify some inequalities of young and pachpatte in n independent variables we can use these inequalities in the analysis of a class of nonlinear hyperbolic partial integrodifferential equations to study the uniqueness boundedness continuous dependence comparison stability and numerical computations

integral inequalities in measure spaces the inequality 2 is usually known as a gronwall bellman type integral inequality special cases of 1 and 2 both continuous and discrete have been studied by many authors for example see 1 wx 4 6 7 9 and they have important applications first we give an explicit upper bound for the solutions of 2 then by using this result we obtain estimates for the solutions of 1 at the same time we deduce a method to get various explicit upper bounds for thesolutions of 2 by applying this method we can obtain much better estimates than the first one finally we consider the integral equationcorresponding to 1 we prove the existence and the uniqueness of the solution and we show that this solution is the best possible estimate forthe solutions of 1 we give some examples to illustrate the results

boundedness of discrete volterra equations abstract investigation of the boundedness of the solutions of volterra equations with discrete time by means of the direct liapunov method and comparison theorems is performed boundedness conditions of solutions of some particular classes of equations are derived using such conditions stability of direct quadrature methods for second kind volterra equations is analyzed

asymptotic stability for volterra intergrodifferential systems in this paper the successive overrelaxation iteration and the variation of parameters formula are used to discuss the stability of linear and nonlinear volterra integrodifferential equations some sufficient conditions are obtained such that the trivial solution of the volterra integrodifferential equations is asymptotically stable these stability conditions are given directly from the coefficients of the equations

on monotone abel volterra integral equations on the half line in this paper we reanalyze the trapezoidal method for the solution of nonlinear abel volterra integral equations on the half line we prove the convergence of the method in the uniform norm provided the nonlinearity is lipschitz continuous and strictly monotone

improving the accuracy of collocation solutions of volterra integral equations of the first kind by local interpolation we study the numerical solution of volterra integral equations of the first kind by collocation methods using piecewise polynomials of degreep we show that when superconvergence occurs the error in the collocation solution at the node points alternates in sign independent from the kernel or the exact solution evaluation at certain points of interpolating polynomials of degreep 2 annihilates the leading term in the asymptotic expansion of the error thus yielding convergence of orderp 3 the derivation of the asymptotic expansion of the error for the general equation is reduced to the special case of numerical differentiation by the use of an asymptotic continuity result for the projectors associated with the collocation method under compact perturbations we present some numerical results on the improved order of convergence forp 2 and 3

runge kutta theory for volterra integral equations of the second kind the present paper develops the theory of general runge kutta methods for volterra integral equations of the second kind the order conditions are derived by using the theory of p series which for our problem reduces to the theory of v series these results are then applied to two special classes of runge kutta methods introduced by pouzet and by bel tyukov

high order methods for the numerical solution of volterra integro differential equations abstract if a first order volterra integro differential equation is solved by collocation in the space of continuous polynomial splines of degree m 1 with collocation occuring at the gauss legendre points then the resulting approximation u converges at its knots like o h 2m while its derivative u exhibits only o h m convergence this paper deals with the question of how to choose the collocation points so that both u and u converge like o h q with q maximal

on some variation of parameter methods for integrodifferential integral and quasilinear partial integrodifferential equations a nonlinear variation of parameter method of bernfeld and lord for systems of ordinary integrodifferential and integral equations is corrected and extended representation formulas for certain solutions of perturbations of such systems are obtained a correction and extension of a similar recent result of pachpatte for a quasilinear partial integrodifferential equation is also given some applications are given to stability and boundedness properties of solutions of systems of integrodifferential and integral equations

on selecting models for nonlinear time series abstract constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models or to choose between competing classes this paper discusses a method of building nonlinear models of possibly chaotic systems from data while maintaining good robustness against noise the models that are built are close to the simplest possible according to a description length criterion the method will deliver a linear model if that has shorter description length than a nonlinear model we show how our models can be used for prediction smoothing and interpolation in the usual way we also show how to apply the results to identification of chaos by detecting the presence of homoclinic orbits directly from time series

a unified approach to stability in integrodifferential equations via liapunov functions in this paper we investigate the problems of stability uniform stability uniform lipschitz stability asymptotic stability and uniform asymptotic stability in three classes of integrodifferential systems these are linear nonautonomous systems nonlinear systems and nonlinearly perturbed nonlinear systems a simple method for constructing liapunov functions will be used to unify the treatment of all of the above mentioned stability notions relative to the various classes of integrodifferential systems in our investigation of linear systems we benefited from the work in 4 8 123 for nonlinear systems and their perturbations we have used some of the results obtained recently in is 101 in particular we have used an extension of alekseev s l nonlinear variation of constants formula which was developed in lo the approach we use here was first formulated in 3 and utilized to study uniform lipschitz stability 2 of nonlinear differential systems the method has been extended in 12 to cover all other stability notions in nonlinear differential systems we now give a description of the contents of the paper section 1 includes all the basic definitions and preliminaries in section 2 we investigate the various stability notions of nonautonomous integrodifferential systems this extends the work in 6 81 which mainly dealt with autonomous systems we recall here from 4 that in linear systems the notions of uniform stability and uniform lipschitz stability are equivalent it was shown in 4 however that for nonlinear systems uniform

solutions of n point boundary value problems associated with nonlinear summary difference equations variation of parameter methods play a fundamental role in understanding solutions of perturbed nonlinear differential as well as difference equations this paper is devoted to the study of n point boundary value problems associated with systems of nonlinear first order summary difference equations by using the nonlinear variation of parameter methods new variational formulae which provide connections between the solutions of initial value problems and n point boundary value problems are obtained an iterative scheme for computing approximated solutions of the boundary value problems is also provided

superconvergence of collocation methods for volterra and abel integral equations of the second kind this paper deals with the question of the attainable order of convergence in the numerical solution of volterra and abel integral equations by collocation methods in certain piecewise polynomial spaces and which are based on suitable interpolatory quadrature for the resulting moment integrals the use of a nonlinear variation of constants formula for the representation of the error function in terms of the defect allows for a unified treatment of equations with continuous and weakly singular kernels

collocation methods for nonlinear volterra integro differential equations with infinite delay in this paper we study the numerical solution of nonlinear volterra integrodifferential equations with infinite delay by spline collocation and related runge kutta type methods the kernel function in these equations is of the form k t 8 y t y s with a representative example given by volterra s population equation where we have k t a y t y 8 a t s g y t y 8

order of convergence of one step methods for volterra integral equations of the second kind nice proofs of convergence and asymptotic expansions are known for one step methods for ordinary differential equations it is shown that these proofs can be generalized in a natural way to extended one step methods for volterra integral equations of the second kind furthermore the convergence of mixed one step methods is investigated for both types general volterra runge kutta methods are considered as examples

stability results for one step discretized collocation methods in the numerical treatment of volterra integral equations this paper is concerned with the stability analysis of the discretized collocation method for the second kind volterra integral equation with degenerate kernel a fixed order recurrence relation with variable coefficients is derived and local stability conditions are given independent of the discretization local stability and stability with respect to an isolated perturbation of some methods are proved the reliability of the derived stability conditions is shown by numerical experiments

natural continuous extensions of runge kutta methods for volterra integral equations of the second kind and their applications we consider a very general class of runge kutta methods for the numerical solution of volterra integral equations of the second kind which includes as special cases all the more important methods which have been considered in the literature the main purpose of this paper is to define and prove the existence of the natural continuous extensions nce s of runge kutta methods i e piecewise polynomial functions which extend the approximation at the grid points to the whole interval of integration the particular properties required of the nce s allow us to construct the tail approximations which are quite efficient in terms of kernel evaluations

a survey of recent advances in the numerical treatment of volterra integral and integro differential equations abstract the present survey paper samples recent advances in the numerical analysis of volterra integral equations of the first and second kind and of integro differential equations including equations with weakly singular kernels except for some important earlier references the discussion focuses on the development which has taken place during the last dozen years a fairly extensive bibliography selected to be representative rather than comprehensive complements the paper

stability results for collocation methods for volterra integral equations abstract stability behavior is investigated for polynomial spline collocation applied to volterra integral equations with special emphasis on a weakly singular test equation the characterization of the corresponding stability domain is related to stability results for solutions of finite recursion relations as they occur for methods applied to ordinary differential equations estimates of the stability domains are established and as a consequence conditions for a 2 stability are obtained exploiting these estimates collocation approximations with constant linear and continuous splines are considered in more depth and several numerical illustrations are presented

lipschitz stability of nonlinear systems of differential equations in s the authors introduced the notion of lipschitz stability in differential equations this notion lies somewhere between uniform stability on one side and the notions of asymptotic stability in variation 3 and uniform stability in variation 4 on the other side however lipschitz stability is new only as a nonliner phenomenon since it coincides with uniform stability in linear systems s an important feature of lipschitz stability is that unlike uniform stability the linearized system inherits the property of lipschitz stability from the original nonlinear system 5 in this paper we pursue the study of lipschitz stability that started in s using essentially the techniques of liapunov functions then we give sufficient conditions for the lipschitz stability of certain nonlinearly pertur bed nonlinear systems such systems include among other equations certain integrodifferential and functional differential equations then we give an example which can be investigated successfully using our results but cannot be handled by any previous techniques or results 9

qualitative behaviour and stability of solutions of discretised nonlinear volterra integral equations of convolution type abstract in this work we consider equations of the form y t g t t o k t s y s ds te r and their discretised versions obtained through application of quadrature to of the form y n g n h j 0 n w n j j j y j n j e n subject to certain conditions on k g and the weights wj one purpose of the paper is to show how the discussion of qualitative behaviour and stability for can be mimicked in a discussion of we first describe corduneanu s 1973 discussion of stability for re presenting his material in a modified form which lends itself to adaptation for our discussion of we give a stability result for we then demonstrate the stability behaviour of some simple quadrature rules applied to an illustrative example equation and we observe that for a particular family of quadrature rules the qualitative behaviour of solutions to the example equation is preserved under the discretisation

fft techniques in the numerical solution of convolution equations abstract we discuss the efficient use of fft routines to determine discrete convolutions required in the numerical solution of volterra and abel integral and integro differential equations the paper is based on a series of technical reports produced by the authors which include fortran 77 listings of computer codes these codes employ the fft routiens constructed for the nag library

a stability analysis of the trapezoidal method for volterra integral equations with completely positive kernels abstract the solution of the volterra integral equation with completely positive kernel y t 0t b t s y s ds u0 0t b t s g s ds t 0 is nonnegative and nonincreasing provided that g is nonincreasing and 0 g t u0 for any t 0 we prove that under some additional hypotheses this property is inherited by the solution of the recurrence relation resulting from applying the trapezoidal method to this equation

numerical solution of non linear volterra integral equations this paper deals with non linear volterra integral equations of the type y x f x 0x h t x y t y x dt convergence criteria are given in the same sense of the maximum and ca norms for the numerical solution of this type of volterra integral equation several numerical methods are compared

the numerical solution of weakly singular volterra integral equations by collocation on graded meshes since the solution of a second kind volterra integral equation with weakly singular kernel has in general unbounded derivatives at the left endpoint of the interval of integration its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates in this paper we investigate the convergence rates with respect to graded meshes and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation

lipschitz stability of nonlinear systems of differential equations ii liapunov functions in s the authors introduced the notion of lipschitz stability in differential equations this notion lies somewhere between uniform stability on one side and the notions of asymptotic stability in variation 3 and uniform stability in variation 4 on the other side however lipschitz stability is new only as a nonliner phenomenon since it coincides with uniform stability in linear systems s an important feature of lipschitz stability is that unlike uniform stability the linearized system inherits the property of lipschitz stability from the original nonlinear system 5 in this paper we pursue the study of lipschitz stability that started in s using essentially the techniques of liapunov functions then we give sufficient conditions for the lipschitz stability of certain nonlinearly perturbed nonlinear systems such systems include among other equations certain integrodifferential and functional differential equations then we give an example which can be investigated successfully using our results but cannot be handled by any previous techniques or results 9

stability of difference volterra equations direct liapunov method and numerical procedure abstract a procedure to construct liapunov functionals for discrete volterra equations is proposed using this procedure stability conditions are derived for general volterra difference equations some applications of the proposed procedure for obtaining stability conditions for linear multistep methods for volterra integro differential equations are presented

superconvergence of numerical solutions to volterra integral equations with singularities in this paper we discuss the beta polynomial discrete collocation method based on practical meshes for volterra integral equations with weakly singular kernels it will be shown that superconvergence properties may be obtained by using appropriate collocation parameters and by a simple cheap postprocessing of the original discrete collocation solution

experimental control of a chaotic pendulum with unknown dynamics using delay coordinates unstable periodic orbits upos of an experimental chaotic pendulum were stabilized using a semi continuous control method scc applying control actions several times per cycle one advantage of this method compared to a one map based control method such as the ott grebogi yorke method phys rev lett 64 1196 1990 is the applicability to systems with relatively large unstable eigenvalues and or high noise levels compared to a continuous type of feedback control as was proposed by pyragas phys lett a 170 421 1992 the advantage is that the controller settings can be measured from experimental data because the control method uses delay coordinates only one variable has to be measured this paper describes an scc method using delay coordinates the extraction of upos from time series how the effect of the control parameter can be measured the effect on the control in case of an error in the estimate of the upo and how this error can be reduced to obtain more stable control

entropic uncertainty bound for open pointer based simultaneous measurements of conjugate observables we discuss the information entropy for a general open pointer based simultaneous measurement and show how it is bound from below this entropic uncertainty bound is a direct consequence of the structure of the entropy and can be obtained from the formal solution of the measurement dynamics furthermore the structural properties of the entropy allow us to give an intuitive interpretation of the noisy influence of the pointers and the environmental heat bath on the measurement results

moduli of semistable logarithmic connections under the riemann hilbert correspondence which is an equivalence of categories local systems on a nonsingular complex projective variety x correspond to pairs e s v where is a locally free sheaf on x and v 8 q1 is an integrable connection on f generalizing this correspondence to noncomplete quasi projective varieties deligne see d showed that the correct algebro geometric objects which correspond to local systems on such a variety y are the so called regular connections on y the condition of regularity on algebraic connections on y can be expressed as follows fix a hironaka completion x of y that is nonsingular projective variety x which contains y as an open subvariety such that the complement s x y is a smooth divisor with normal crossings a logarithmic connection f y v on x with singularities over s is by definition a torsion free coherent sheaf y on x together with a map v 9 21 log s 9 which satisfies c linearity and the leibniz rule where i14 logs is the sheaf of rational 1 forms on x with logarithmic poles on s the curvature of the connection v is assumed to be zero finally an algebraic connection e f v on y is regular if and only if there exists a logarithmic connection f on x with singularities over s which extends e if such an extension f exists the underlying sheaf f can be chosen to be locally free and will be called a logarithmic lattice for e a logarithmic lattice for a given regular connection is of course not unique but there is a canonical way of choosing it called deligne s construction which only depends on the choice of a set theoretic section of the map exp c c as such a section is not even continuous the deligne lattice does not behave well in families of regular connections simpson has given in s a construction of a moduli scheme for nonsingular connections on a projective variety x however a simple example shows that it would be unreasonable to expect in general the existence of a moduli scheme for regular connections on a quasi projective variety y for this let y be the affine line minus the origin with coordinate x the differential equation dy dx tylx 0 with parameter t defines a family of regular connections on the sheaf y parametrized by the affine line t with coordinate t we can see that t1 and t2 parametrize isomorphic connections if and only if t1 t2

controlling chaotic dynamical systems abstract we describe a method that converts the motion on a chaotic attractor to a desired attracting time periodic motion by making only small time dependent perturbations of a control parameter the time periodic motion results from the stabilization of one of the infinite number of previously unstable periodic orbits embedded in the attractor the present paper extends that of ott grebogi and yorke phys rev lett 64 1990 1196 allowing for a more general choice of the feedback matrix and implementation to higher dimensional systems the method is illustrated by an application to the control of a periodically impulsively kicked dissipative mechanical system with two degrees of freedom resulting in a four dimensional map the double rotor map a key issue addressed is that of the dependence of the average time to achieve control on the size of the perturbations and on the choice of the feedback matrix

optimal control of nonlinear systems to given orbits abstract using optimal control techniques we derive and demonstrate the use of an explicit single step control method for directing a nonlinear system to a target orbit and keeping it there we require that control values remain near the uncontrolled settings the full nonlinearity of the problem in state space variables is retained the one step of the control is typically a composition of known or learned maps over a the time required to learn the state b the time to compute the control and c the time to apply the control no special targeting is required yet the time to control is quite rapid working with the dynamics of a well studied nonlinear electrical circuit we show how this method works efficiently and accurately in two situations when the known circuit equations are used and when control is performed only on a poincare section of the reconstructed phase space in each case because the control rule is known analytically the control strategy is computationally efficient while retaining high accuracy the target locations on the selected target trajectory at each control stage are determined dynamically by the initial conditions and the system dynamics and the target trajectory is an approximation to an unstable periodic orbit of the uncontrolled system a linear stability analysis shows that dissipation in the dynamical system is essential for reaching a controllable state

higher dimensional targeting this paper describes a procedure to steer rapidly successive iterates of an initial condition on a chaotic attractor to a small target region about any prespecified point on the attractor using only small controlling perturbations such a procedure is called targeting previous work on targeting for chaotic attractors has been in the context of one and two dimensional maps here it is shown that targeting can also be done in higher dimensional cases the method is demonstrated with a mechanical system described by a four dimensional mapping whose attractor has two positive lyapunov exponents and a lyapunov dimension of 2 8 the target is reached by making very small successive changes in a single control parameter in one typical case 35 iterates on average are required to reach a target region of diameter 10 sup minus 4 as compared to roughly 10 sup 11 iterates without the use of the targeting procedure

continuous collocation approximations to solutions of first kind volterra equations in this paper we give necessary and sufficient conditions for convergence of continuous collocation approximations of solutions of first kind volterra integral equations the results close some longstanding gaps in the theory of polynomial spline collocation methods for such equations the convergence analysis is based on a runge kutta or ode approach

applications and numerical analysis of partial differential volterra equations a brief survey this article contains a concise survey of the numerical analysis of volterra equations and leads up to some recent results on a posteriori error estimation for finite element approximations

nonlinear variation of parameters formula for integro differential equations of volterra type on developpe une variation non lineaire de la formule des parametres pour des equations integrodifferentielles on etudie le probleme de la continuite et de la differentiabilite des solutions des equations integrodifferentielles non lineaires et on obtient la relation entre les derivees

systems of multidimensional volterra integral equations and inequalities on obtient des bornes superieures explicites pour les solutions u x d equations integrales de volterra lineaires ou non lineaires on demontre des theoremes d existence globale

stability of volterra integrodifferential systems abstract the purpose of this paper is to use a decomposition aggregation method and liapunov functionals to discuss the stability of large scale systems described by liner volterra integrodifferential equations

nonlinear variation of parameter methods for summary difference equations in several independent variables in this paper we shall present two new nonlinear variation of parameters formulae for the summary difference equations in several independent variables a corrected variation of parameters formula for the corresponding integrodifferential equation is also provided some applications that dwell upon the importance of these new results are also included

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

flat parabolic vector bundles on elliptic curves we describe the moduli space of logarithmic rank 2 connections on elliptic curves with 2 poles

symplectic manifolds and isomonodromic deformations abstract we study moduli spaces of meromorphic connections with arbitrary order poles over riemann surfaces together with the corresponding spaces of monodromy data involving stokes matrices natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint generalising the atiyah bott approach this enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of jimbo miwa and ueno thereby putting the existing results for the six painleve equations and schlesinger s equations into a uniform framework

on the stability of volterra runge kutta methods we investigate the stability of extended runge kutta methods for volterra integral equations of the second kind as test equations we choose linear convolution equations with positive definite l 1 kernels our results are strongly connected with the contractivity of runge kutta methods for nonlinear dissipative differential equations

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

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations

comparison results for multidimensional difference equations on compare les solutions u x de l equation aux differences non lineaire x u u x f x u x avec les solutions x et x des inegalites aux differences non lineaires correspondantes x n x f x x et x n x f x x

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