perfect and subperfect regulation in linear multivariable control systems this paper is concerned with the synthesis of a high gain feedback control called perfect regulation p r for linear multivariable systems with external signals the existence condition for p r is derived in which the minimum phase property plays an essential role the use of an observer is discussed for achieving p r under restricted state observation some asymptotically ideal feedback properties are demonstrated such as the complete desensitization the complete servo performance with decoupling and the complete disturbance rejection which formulate the loop tightness for multivariable systems a design method for applying p r to non minimum phase systems called subperfect regulation s p r is proposed based on the factorization of the plant transfer function matrix into the minimum phase part and the totally non minimum phase part it is a multivariable extension of a well known design technique for scalar systems to overcome the difficulty of phase non minimality computational procedures for p r and s p r are discussed an illustrative example is shown

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

analysis of the truncation errors in the fast multipole method for scattering problems discretisation of the integral equations of acoustic scattering yields large dense systems of linear equations using the fast multipole method an approximate solution to these systems can be computed with a low operation count when implementing the method various infinite sums must be truncated in this paper sharp computable bounds on the errors of these truncations are derived which could form the basis for an automatic selection of truncation length this choice will guarantee a given solution accuracy whilst minimising the operation count of the fast multipole algorithm

fast and stable contour integration for high order divided differences via elliptic functions in this paper we will present a new method for evaluating high order divided differences for certain classes of analytic possibly operator values functions this is a classical problem in numerical mathematics but also arises in new applications such as e g the use of generalized convolution quadrature to solve retarded potential integral equations the functions which we will consider are allowed to grow exponentially to the left complex half plane and the interpolation points are scattered in a large real interval our approach is based on the representation of divided differences as contour integral and we will employ a subtle parameterization of the contour in combination with a quadrature approximation by the trapezoidal rule

desynchronization transitions in nonlinearly coupled phase oscillators abstract we consider the nonlinear extension of the kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter a bifurcation analysis of the transition from fully synchronous state to partial synchrony is performed we demonstrate that for small ensembles it is typically mediated by stable cluster states that disappear with creation of heteroclinic cycles while for a larger number of oscillators a direct transition from full synchrony to a periodic or a quasiperiodic regime occurs

effects of time delay on symmetric two species competition subject to noise noise and time delay act simultaneously on real ecological systems the lotka volterra model of symmetric two species competition with noise and time delay was investigated in this paper by means of stochastic simulation we find that i the time delay induces the densities of the two species to periodically oscillate synchronously ii the stationary probability distribution function of the two species densities exhibits a transition from multiple to single stability as the delay time increases iii the characteristic correlation time for the sum of the two species densities squared exhibits a nonmonotonic behavior as a function of delay time our results have the implication that the combination of noise and time delay could provide an efficient tool for understanding real ecological systems

cubic spline interpolation based time domain equivalent source method for modeling transient acoustic radiation abstract transient acoustic radiation from an arbitrarily shaped source is modeled using a cubic spline interpolation based time domain equivalent source method in the method the cubic spline interpolation function is introduced to approximate the equivalent source strengths in the time domain for establishing the relationships between the acoustic quantities in the field the normal acceleration on the source surface and the equivalent sources and the least squares solution of the equivalent source strengths at each time step is determined by an iterative solving process with the singular value decomposition technique the proposed method has the abilities of calculating the acoustic quantities directly in the time domain and applying to arbitrarily shaped sources numerical examples of calculating the transient acoustic fields from a baffled planar piston and an impulsively accelerating sphere demonstrate that the proposed method can model the transient acoustic radiation very well the experimental results also further verify the validity of the proposed method

on the multistep time discretization of linear initial boundary value problems and their boundary integral equations convergence estimates in terms of the data are shown for multistep methods applied to non homogeneous linear initial boundary value problems similar error bounds are derived for a new class of time discrete and fully discrete approximation schemes for boundary integral equations of such problems e g for the single layer potential equation of the wave equation in both cases the results are obtained from convergence and stability estimates for operational quadrature approximations of convolutions these estimates which are also proved here depend on bounds of the laplace transform of the distributional convolution kernel outside the stability region scaled by the time stepsize and on the smoothness of the data

the effects of time delay on stochastic resonance in a bistable system with correlated noises the effects of time delay on stochastic resonance sr in a bistable system with time delay correlated noises and periodic signal are studied by using the theory of signal to noise ratio snr the expression of the snr is derived under the adiabatic limit and the small delay time approximation it is found that i for the case of no correlations between multiplicative and additive noise the delay time can enhance the snr as a function of the multiplicative noise intensity and it can restrain the snr as a function of the additive noise intensity d ii for the case of correlations between multiplicative and additive noise can induce a minimum and maximum in curve of the snr as a function of and can intensively restrain the snr as a function of the d and there is a critical value of delay tim c 0 1 in the height of the snr peak with change of i e when takes value blow c t he boosts up the snr as a function of the strength of correlations between multiplicative and additive noise however when takes value above c t he restrains that

direct singular perturbation analysis of high gain and cheap control problems a new and direct approach to the analysis and design of high gain feedback and cheap control problems is developed transformation of such singular problems to singular perturbation models of conventional type early at the onset of analysis or design allows a direct application of the readily available singular perturbation literature the approach permits a simple characterization of a multivariable root loci under a high gain feedback and b asymptotic behavior of optimal closed loop poles state and control trajectories performance index and optimal transfer function as the control cost coefficient in the performance index goes to zero

information flow through a chaotic channel prediction and postdiction at finite resolution we reconsider the persistence of information under the dynamics of the logistic map in order to discuss communication through a nonlinear channel where the sender can set the initial state of the system with finite resolution and the recipient measures it with the same accuracy we separate out the contributions of global phase space shrinkage and local phase space contraction and expansion to the uncertainty in predicting and postdicting the state of the system we determine how the amplification parameter the time lag and the resolution influence the possibility for communication a novel clockwork representation for real numbers is introduced that allows for a visualization of the flow of information between scales

stability and convergence of collocation schemes for retarded potential integral equations time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations solving such equations numerically is both complicated and computationally intensive and numerical methods often prove to be unstable collocation schemes are easier to implement than full finite element formulations but little appears to be known about their stability and convergence here we derive and analyze some new stable collocation schemes for the single layer equation for transient acoustic scattering and use spatial fourier and temporal laplace transform techniques to demonstrate that such stable schemes are second order convergent

a galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions we consider retarded boundary integral formulations of the three dimensional wave equation in unbounded domains our goal is to apply a galerkin method in space and time in order to solve these problems numerically in this approach the computation of the system matrix entries is the major bottleneck we will propose new types of finite dimensional spaces for the time discretization they allow variable time stepping variable order of approximation and simplify the quadrature problem arising in the generation of the system matrix substantially the reason is that the basis functions of these spaces are globally smooth and compactly supported in order to perform numerical tests concerning our new basis functions we consider the special case that the boundary of the scattering problem is the unit sphere in this case explicit solutions of the problem are available which will serve as reference solutions for the numerical experiments

transparent boundary conditions for wave propagation in fractal trees convolution quadrature approach in this work we propose high order transparent boundary conditions for the weighted wave equation on a fractal tree with an application to the modeling of sound propagation in a human lung this article follows the recent work 29 dedicated to the mathematical analysis of the corresponding problem and the construction of low order absorbing boundary conditions the method proposed in this article consists in constructing the exact trans parent boundary conditions for the semi discretized problem in the spirit of the convolution quadrature method developed by ch lubich we analyze the stability and convergence of the method and propose an efficient algorithm for its implementation the exposition is concluded with numerical experiments

energy estimates for galerkin semidiscretizations of time domain boundary integral equations in this paper we present a battery of results related to how galerkin semidiscretization in space affects some formulations of wave scattering and propagation problems when retarded boundary integral equations are used

bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators abstract we study the bifurcation and dynamical behaviour of the system of n globally coupled identical phase oscillators introduced by hansel mato and meunier in the cases n 3 and n 4 this model has been found to exhibit robust slow switching oscillations that are caused by the presence of robust heteroclinic attractors this paper presents a bifurcation analysis of the system in an attempt to better understand the creation of such attractors we consider bifurcations that occur in a system of identical oscillators on varying the parameters in the coupling function these bifurcations preserve the permutation symmetry of the system we then investigate the implications of these bifurcations for the sensitivity to detuning i e the size of the smallest perturbations that give rise to loss of frequency locking for n 3 we find three types of heteroclinic bifurcation that are codimension one with symmetry on varying two parameters in the coupling function we find three curves giving a an s 3 transcritical homoclinic bifurcation b a saddle node heteroclinic bifurcation and c a z 3 heteroclinic bifurcation we also identify several global bifurcations with symmetry that organize the bifurcation diagram these are codimension two with symmetry for n 4 oscillators we determine many but not all codimension one bifurcations with symmetry including those that lead to a robust heteroclinic cycle a robust heteroclinic cycle is stable in an open region of parameter space and unstable in another open region furthermore we verify that there is a subregion where the heteroclinic cycle is the only attractor of the system while for other parts of the phase plane it can coexist with stable limit cycles we finish with a discussion of bifurcations that appear for this coupling function and general n as well as for more general coupling functions

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

truncation error analysis of multipole expansion the multilevel fast multipole algorithm is based on the multipole expansion which has numerical error sources such as truncation of the addition theorem numerical integration and interpolation anterpolation of these we focus on the truncation error and discuss its control precisely the conventional selection rule fails when the buffer size is small compared to the desired numerical accuracy we propose a new approach and show that the truncation error can be controlled and predicted regardless of the number of buffer sizes

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

adaptive time discretization for retarded potentials in this paper we will present advanced discretization methods for solving retarded potential integral equations we employ a c infty c partition of unity method in time and a conventional boundary element method for the spatial discretization one essential point for the algorithmic realization is the development of an efficient method for approximation the elements of the arising system matrix we present here an approach which is based on quadrature for non analytic c infty c functions in combination with certain chebyshev expansions furthermore we introduce an a posteriori error estimator for the time discretization which is employed also as an error indicator for adaptive refinement numerical experiments show the fast convergence of the proposed quadrature method and the efficiency of the adaptive solution process

a galerkin scheme for the time domain integral equation of acoustic scattering from a hard surface the transient acoustic scattered wave from a hard surface is computed using a space time integral equation a variational formulation of this integral equation is obtained from an energy identity leading to a galerkin type numerical procedure by choosing an appropriate basis for the discretized functional space this scheme is of the marching in time type with matrix coefficients this technique is applied to a sphere a box and dihedron targets

almost invariant subspaces an approach to high gain feedback design part ii almost conditionally invariant subspaces in part i of this paper almost controlled invariant subspaces were studied in this part we will consider their duals the almost conditionally invariant subspaces these concepts give immediately by dualization of the almost disturbance decoupling control by state feedback the solution of the almost disturbance decoupled estimation problem finally we consider the problem of approximate disturbance decoupling by measurement feedback and it is shown that this problem is solvable to any arbitrary degree of accuracy if and only if 1 almost disturbance decoupling by state feedback and 2 almost disturbance decoupled estimation of the to be controlled output are both possible

fast analysis of transient acoustic wave scattering from rigid bodies using the multilevel plane wave time domain algorithm the analysis of transient wave scattering from rigid bodies using integral equation based techniques is computationally intensive if carried out using classical schemes the evaluation of the velocity potential on the surface of a three dimensional scatterer represented in terms of ns spatial basis functions for nt time steps requires o ntns2 operations the recently developed plane wave time domain pwtd algorithm permits the rapid evaluation of transient fields that are generated by bandlimited source distributions it has been shown that incorporation of the pwtd algorithm into integral equation based solvers in a two level setting reduces the computational complexity of a transient analysis to o ntns1 5 logns in this paper it is shown that casting the pwtd scheme into a multilevel framework permits the analysis of transient acoustic surface scattering phenomena in o ntnslog2ns operations using o ntns memory numerical examples that demonstrate the efficacy of the multilevel implementation are also presented

derivation and implementation of the boundary integral formula for the convective acoustic wave equation in time domain kirchhoff s formula for the convective wave equation is derived using the generalized function theory the generalized convective wave equation for a stationary surface is obtained and the integral formulation the convective kirchhoff s formula is derived the formula has a similar form to the classical kirchhoff s formula but an additional term appears due to a moving medium effect for convenience the additional term is manipulated to a final form as the classical kirchhoff s formula the frequency domain boundary integral can be obtained from the current time domain boundary integral form the derived formula is verified by comparison with the analytic solution of source in the uniform flow the formula is also utilized as a boundary integral equation time domain boundary element method bem analysis using the boundary integral equation is conducted and the results show good agreement with the analytical solution the formula derived here can be useful for sound radiation and scattering by arbitrary bodies in a moving medium in the time domain

electromagnetic scattering from 3 d curved dielectric bodies using time domain integral equations a boundary integral equation bie approach is developed to calculate transient scattering from dielectric bodies the treatment is directly in terms of the e and h fields rather than magnetic and electric currents it employs curvilinear quadratic modeling which allows accurate representation of arbitrarily shaped curved bodies the treatment is isoparametric with the same quadratic representation of the spatial field variation and with the temporal variation modeled by similar quadratic elements integration employs high order gaussian quadrature with special treatment of the singular and hypersingular integrals that arise the treatment is implicit requiring the solution of a sparse matrix equation at each timestep this adds only trivially to the cost at each timestep and by freeing the timestep from the constraint that it be smaller than the smallest nodal spacing can greatly reduce the number of timesteps that must be employed additionally it produces stable results without resort to the averaging processes proposed elsewhere example calculations of scattering from a sphere a cube and an almond are presented and compared with earlier published transient results and with results from a frequency domain treatment good agreement and improved accuracy is found

time discretization of parabolic boundary integral equations in time dependent boundary integral equations a boundary element method in space can be coupled with a different type of discretization in time for the latter a procedure based on linear multistep methods is proposed which is applicable whenever the laplace transform of the fundamental solution is known the stability properties of the method are obtained from those of the underlying multistep method in the absence of a space discretization the numerical solution given by the proposed method is identical to that of a semi discretization in time of the partial differential equation by the underlying multistep method the theory is presented for the single layer potential equation of the heat equation convergence estimates which are pointwise in time and expressed in terms of the boundary data are obtained for full discretizations using galerkin or collocation boundary element methods in space numerical examples are included

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

a singular perturbation canonical form of invertible systems determination of multivariate root loci a new canonical form of multivariable invertible systems is given this form is suitable for the singular perturbation analysis of such singular problems as multi variable asymptotic root loci under high gain feedback cheap control state variable estimation with weak measurement noise etc appropriate selection and decomposition of state variables into slow and fast changing groups is a key feature that loads to well defined singular perturbation models the selection of state variables is accomplished without explicitly calculating any eigenvalues of the given system but by knowing the direct relationship of the output and its differential coefficients with the input the use of the canonical form to characterize the multivariable root loci under high gain feedback is demonstrated

special coordinate basis for multivariable linear systems finite and infinite zero structure squaring down and decoupling abstract a special coordinate basis for multivariable linear systems is introduced here several fundamental properties of linear systems regarding controllability stabiliza bility observability detectability invariant zeros decoupling zeros infinite zero structure effect of feedback on zero structure squaring down diagonal and triangular decoupling etc can be directly displayed in terms of the special coordinate basis moreover connections between the special coordinate basis and the important invariant subspaces of the geometric theory of linear systems are established

complex dynamics of an oscillator ensemble with uniformly distributed natural frequencies and global nonlinear coupling we consider large populations of phase oscillators with global nonlinear coupling for identical oscillators such populations are known to demonstrate a transition from completely synchronized state to the state of self organized quasiperiodicity in this state phases of all units differ yet the population is not completely incoherent but produces a nonzero mean field the frequency of the latter differs from the frequency of individual units here we analyze the dynamics of such populations in case of uniformly distributed natural frequencies we demonstrate numerically and describe theoretically i states of complete synchrony ii regimes with coexistence of a synchronous cluster and a drifting subpopulation and iii self organized quasiperiodic states with nonzero mean field and all oscillators drifting with respect to it we analyze transitions between different states with the increase of the coupling strength in particular we show that the mean field arises via a discontinuous transition for a further illustration we compare the results for the nonlinear model with those for the kuramoto sakaguchi model

encoding via conjugate symmetries of slow oscillations for globally coupled oscillators we study properties of the dynamics underlying slow cluster oscillations in two systems of five globally coupled oscillators these slow oscillations are due to the appearance of structurally stable heteroclinic connections between cluster states in the noise free dynamics in the presence of low levels of noise they give rise to long periods of residence near cluster states interspersed with sudden transitions between them moreover these transitions may occur between cluster states of the same symmetry or between cluster states with conjugate symmetries given by some rearrangement of the oscillators we consider the system of coupled phase oscillators studied by hansel et al phys rev e 48 3470 1993 in which one can observe slow noise driven oscillations that occur between two families of two cluster periodic states in the noise free case there is a robust attracting heteroclinic cycle connecting these families the two families consist of symmetric images of two inequivalent periodic orbits that have the same symmetry for n 5 oscillators one of the periodic orbits has one unstable direction and the other has two unstable directions examining the behavior on the unstable manifold for the two unstable directions we observe that the dimensionality of the manifold can give rise to switching between conjugate symmetry orbits by applying small perturbations to the system we can easily steer it between a number of different marginally stable attractors finally we show that similar behavior occurs in a system of phase energy oscillators that are a natural extension of the phase model to two dimensional oscillators we suggest that switching between conjugate symmetries is a very efficient method of encoding information into a globally coupled system of oscillators and may therefore be a good and simple model for the neural encoding of information

multiple time scale decomposition in cheap control problems singular control a linear quadratic regulator problem where a small parameter 2multiplies the control cost is considered as mu rightarrow 0 it exhibits a dynamic behavior with multiple time scales whereas for mu 0 it is a singular control problem at first a multiparameter singularly perturbed model of the given problem is constructed by transferring the singularity that exists in the performance index to the dynamic equations then the problem is decomposed into several subproblems of minimal order each pertaining to only one time scale the method developed permits under a single framework the characterization of asymptotic behavior of optimal closed loop poles state and control trajectories performance index and optimal transfer function as mu rightarrow 0 moreover the asymptotic solution brings into focus 1 the nature of singular control and its connection to transmission zeros and 2 the presence of various types of impulses and other higher order distributions in both state and control the approach unifies and extends significantly several ideas previously somewhat disjoint

diagonal form fast multipole singular boundary method applied to the solution of high frequency acoustic radiation and scattering summary the singular boundary method sbm is a recent strong form boundary collocation method that uses a linear combination of the fundamental solution of the governing equation to approximate the field variables because of its full interpolation matrix the sbm solution encounters the high computational complexity and storage requirement that limit its applications to large scale engineering problems this paper presents a way to overcome this drawback by introducing the diagonal form fast multipole method a diagonal form fast multipole singular boundary method is then developed to reduce the computational operations of the sbm with direct solvers from o n3 to o nlogn where n is the number of unknowns the proposed method works well for acoustic radiation and scattering with nondimensional wave number kd 110 k is the wave number and d the maximum diameter of the computational domain several numerical experiments are provided to show the efficiency and accuracy of the present strategy including the radiation from a nuclear submarine and the scattering from an airplane the numerical results clearly demonstrate that the present diagonal form fast multipole singular boundary method breaks the limitations of the sbm and successfully simulates the large scale acoustic problems with more than one million unknowns on a desktop computer copyright 2016 john wiley sons ltd

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

effects of nonresonant interaction in ensembles of phase oscillators we consider general properties of groups of interacting oscillators for which the natural frequencies are not in resonance such groups interact via nonoscillating collective variables like the amplitudes of the order parameters defined for each group we treat the phase dynamics of the groups using the ott antonsen ansatz and reduce it to a system of coupled equations for the order parameters we describe different regimes of cosynchrony in the groups for a large number of groups heteroclinic cycles corresponding to a sequential synchronous activity of groups and chaotic states where the order parameters oscillate irregularly are possible

cheap and singular controls for linear quadratic regulators a general not necessarily square and not necessarily invertible linear system with quadratic cost function where a small parameter 2multiplies the control cost is considered due to the cheapness of control a strong control action in the form of high gain feedback forces the given system to have slow and fast transients of a hierarchy of time scales coupled with high and low amplitude interactions by appropriate amplitude and time scaling of variables these interactions are normalized and the problem is decomposed into several nonsingular subproblems of minimal order each pertaining to only one time scale complete results characterizing the limiting behavior of the optimal performance index eigenvalues trajectory and control variables as mu rightarrow 0 are given all the different ways by which nonuniqueness can occur into singular control are discussed more importantly the method developed here allows the design of a high gain feedback system in terms of the design of several lower order subproblems thus it gives a strong impetus for practical implementation of the theory developed

wave formation by time delays in randomly coupled oscillators we study the dynamics of randomly coupled oscillators when interactions between oscillators are time delayed due to the finite and constant speed of coupling signals numerical simulations show that the time delays proportional to the euclidean distances between interacting oscillators can induce near regular waves in addition to near in phase synchronous oscillations even though oscillators are randomly coupled we discuss the stability conditions for the wave states and the in phase synchronous states

runge kutta convolution quadrature for operators arising in wave propagation an error analysis of runge kutta convolution quadrature is presented for a class of non sectorial operators whose laplace transform satisfies besides the standard assumptions of analyticity in a half plane re s 0 and a polynomial bound operatorname o s mu 1 there the stronger polynomial bound operatorname o s mu 2 in convex sectors of the form operatorname arg s leq pi 2 theta for 0 the order of convergence of the runge kutta convolution quadrature is determined by 2 and the underlying runge kutta method but is independent of 1 time domain boundary integral operators for wave propagation problems have laplace transforms that satisfy bounds of the above type numerical examples from acoustic scattering show that the theory describes accurately the convergence behaviour of runge kutta convolution quadrature for this class of applications our results show in particular that the full classical order of the runge kutta method is attained away from the scattering boundary

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

slow switching in globally coupled oscillators robustness and occurrence through delayed coupling the phenomenon of slow switching in populations of globally coupled oscillators is discussed this characteristic collective dynamics which was first discovered in a particular class of the phase oscillator model is a result of the formation of a heteroclinic loop connecting a pair of clustered states of the population we argue that the same behavior can arise in a wider class of oscillator models with the amplitude degree of freedom we also argue how such heteroclinic loops arise inevitably and persist robustly in a homogeneous population of globally coupled oscillators although a heteroclinic loop might seem to arise only exceptionally we find that it appears rather easily by introducing time delay into a population which would otherwise exhibit perfect phase synchrony we argue that the appearance of the heteroclinic loop induced by the delayed coupling is then characterized by transcritical and saddle node bifurcations slow switching arises when a system with a heteroclinic loop is weakly perturbed this will be demonstrated with a vector model by applying weak noises other types of weak symmetry breaking perturbations can also cause slow switching

experiments on oscillator ensembles with global nonlinear coupling we experimentally analyze collective dynamics of a population of 20 electronic wien bridge limit cycle oscillators with a nonlinear phase shifting unit in the global feedback loop with an increase in the coupling strength we first observe formation and then destruction of a synchronous cluster so that the dependence of the order parameter on the coupling strength is not monotonic after destruction of the cluster the ensemble remains nevertheless coherent i e it exhibits an oscillatory collective mode mean field we show that the system is now in a self organized quasiperiodic state predicted in rosenblum and pikovsky phys rev lett 98 064101 2007 in this state frequencies of all oscillators are smaller than the frequency of the mean field so that the oscillators are not locked to the mean field they create and their dynamics is quasiperiodic without a nonlinear phase shifting unit the system exhibits a standard kuramoto like transition to a fully synchronous state we demonstrate a good correspondence between the experiment and previously developed theory we also propose a simple measure which characterizes the macroscopic incoherence coherence transition in a finite size ensemble

efficient fast multipole method for low frequency scattering the solution of the helmholtz and maxwell equations using integral formulations requires to solve large complex linear systems a direct solution of those problems using a gauss elimination is practical only for very small systems with few unknowns the use of an iterative method such as gmres can reduce the computational expense most of the expense is then computing large complex matrix vector products the cost can be further reduced by using the fast multipole method which accelerates the matrix vector product for a linear system of size n the use of an iterative method combined with the fast multipole method reduces the total expense of the computation to n log n there exist two versions of the fast multipole method one which is based on a multipole expansion of the interaction kernel exp ikr r and which was first proposed by v rokhlin and another based on a plane wave expansion of the kernel first proposed by w c chew in this paper we propose a third approach the stable plane wave expansion spw fmm which has a lower computational expense than the multipole expansion and does not have the accuracy and stability problems of the plane wave expansion the computational complexity is n log n as with the other methods

multistep and multistage convolution quadrature for the wave equation algorithms and experiments we describe how a time discretized wave equation in a homogeneous medium can be solved by boundary integral methods the time discretization can be a multistep runge kutta or a more general multistep multistage method the resulting convolutional system of boundary integral equations belongs to the family of convolution quadratures of lubich the aim of this work is twofold it describes an efficient robust and easily parallelizable method for solving the semidiscretized system the resulting algorithm has the main advantages of time stepping methods and of fourier synthesis at each time step a system of linear equations with the same system matrix needs to be solved yet computations can easily be done in parallel the computational cost is almost linear in the number of time steps and only the laplace transform of the time domain fundamental solution is needed the new aspect of the algorithm is that all this is possible without ever explicitly constructing the weights of the convolution quadrature this approach also readily allows the use of modern data sparse techniques to efficiently perform computation in space we investigate theoretically and numerically to which extent hierarchical matrix mathcal h matrix techniques can be used to speed up the space computation the second aim of this article is to perform a series of large scale 3d experiments with a range of multistep and multistage time discretization methods the backward difference formula of order 2 bdf2 the trapezoid rule and the 3 stage radau iia methods are investigated in detail one of the conclusions of the experiments is that the radau iia method often performs overwhelmingly better than the linear multistep methods especially for problems with many reflections yet in connection with hyperbolic problems bdfs have so far been predominant in the literature on convolution quadrature

generalized convolution quadrature based on runge kutta methods in this paper we develop the runge kutta generalized convolution quadrature with variable time stepping for the numerical solution of convolution equations for time and space time problems and present the corresponding stability and convergence analysis for this purpose some new theoretical tools such as tensorial divided differences summation by parts with runge kutta differences and a calculus for runge kutta discretizations of generalized convolution operators such as an associativity property will be developed in this paper numerical examples will illustrate the stable and efficient behavior of the resulting discretization

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

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

dynamics of heterogeneous oscillator ensembles in terms of collective variables we consider general heterogeneous ensembles of phase oscillators sine coupled to arbitrary external fields starting with the infinitely large ensembles we extend the watanabe strogatz theory valid for identical oscillators to cover the case of an arbitrary parameter distribution the obtained equations yield the description of the ensemble dynamics in terms of collective variables and constants of motion as a particular case of the general setup we consider hierarchically organized ensembles consisting of a finite number of subpopulations whereas the number of elements in a subpopulation can be both finite or infinite next we link the watanabe strogatz and ott antonsen theories and demonstrate that the latter one corresponds to a particular choice of constants of motion the approach is applied to the standard kuramoto sakaguchi model to its extension for the case of nonlinear coupling and to the description of two interacting subpopulations exhibiting a chimera state with these examples we illustrate that although the asymptotic dynamics can be found within the framework of the ott antonsen theory the transients depend on the constants of motion the most dramatic effect is the dependence of the basins of attraction of different synchronous regimes on the initial configuration of phases

fast transient analysis of acoustic wave scattering from rigid bodies using a two level plane wave time domain algorithm it is well known that the computational cost associated with the application of classical time domain integral equation methods to the analysis of scattering from acoustical targets scales unfavorably with problem size indeed performing a three dimensional scattering analysis using these methods requires o ntns2 operations where ns denotes the number of basis functions that model the spatial field distribution over the surface of the scatterer and nt is the number of time steps in the analysis recently novel plane wave time domain algorithms that augment these classical methods and thereby reduce their high computational cost have been introduced this paper describes such a plane wave time domain algorithm within the context of the analysis of acoustic scattering from rigid bodies and outlines its incorporation into a time domain integral equation solver in a two level setting it is shown that the resulting scheme has a computational complexity of o ntns1 5 log ns examples comparing the accuracy

a new approach to the perfect regulation and the bounded peaking in linear multivariable control systems this paper presents a new approach to the perfect regulation p r and the bounded peaking b p in linear multivariable control systems based on the pole and the eigenvector assignment technique roughly speaking the p r represents an ideal control action which reduces the settling time to 0 while the b p simply means the bounded overshoot in this ideal situation simple frequency domain characterizations of the p r and the b p are derived they reveal some invariance properties of the p r and the b p that provide a powerful tool for achieving the p r and the b p the existence condition for the p r is derived which turns out to be identical to the result obtained in the optimal regulator theory this result is extended to a more general control objective of attaining the p r for one output while keeping the b p for another output a condition on which the p r is realized by an output feedback is also derived a simple design algorithm is proposed for achieving the p r which is essentially the eigenvector assignment procedure finally an application to a real system is discussed

elastodynamic direct boundary element methods with enhanced numerical stability properties evidence of numerical instabilities in two dimensional time domain direct boundary element methods is presented the e ects of numerical versus analytical integration of spatial integrals on stability are shown and two new time stepping algorithms are introduced and compared to existing formulations the so called new direct half step scheme and the epsilon scheme are shown to improve the numerical stability of direct boundary element methods copyright 1999 john wiley sons ltd

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

time domain integral equation solution for acoustic scattering from fluid targets the responses of fluid targets are computed using space time integral equations formulated in the time domain the incident pressure wave is a smoothed impulse with a gaussian shaped time dependence whose width is of the order of a target dimension a space time integral equation for the pressure field on the outside of the target surface and a space time integral equation for the pressure field on the inside of the target surface are solved simultaneously for the pressure and the pressure gradient by stepping on in time and making use of the boundary conditions continuity of pressure and displacement the farfield is then computed from these source fields the technique is applicable to targets of arbitrary contour and is demonstrated for a sphere and right circular cylinder at various angles of incidence fluid targets support interior compression waves sound hard and sound soft targets are treated as limiting cases of this formulation the technique has been verified for the case of a sphere by comparison with the response computed by classical expansions

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

numerical stability and convergence of approximations of retarded potential integral equations stability and convergence results for approximations of retarded potential integral equations are given such integral approximations arise in time marching schemes for direct scattering problems and are both extremely complicated and computationally intensive they are often observed to have exponentially increasing unstable solutions a stability analysis is performed which is shown to provide a way of predicting whether a given scheme is stable or not the proposed test is supported by numerical experiments and requires much less computation than the scheme itself it is also shown that if an approximation for this type of integral equation is both stable and consistent then it is convergent

suppressing the onset of instabilities in the time domain equivalent source method using a multistep approach numerical instability is an important issue that should be addressed in the time domain equivalent source method tesm this study proposes a multistep method to stabilize tesm when using the measured acoustic pressure data to optimize equivalent source strengths unlike the conventional single step method that solves each time step in the time marching process of tesm the proposed method performs a one time solution for several time steps the multistep solution can potentially reduce the accumulation rate of error and improves filtering effects by changing the structure of the matrix that needs to be inverted when the truncated singular value decomposition or tikhonov regularization is used in the time marching process numerical simulations with three examples demonstrate the effectiveness of the multistep method in improving the stability of solutions compared with the single step method effects of the number of merged time steps on the solutions are also discussed to guide the selection process finally the sensitivity of the multistep method to numerical parameters is investigated to demonstrate its consistency under different configurations of numerical parameters

a galerkin bem for transient acoustic scattering by an absorbing obstacle this paper deals with the numerical resolution of the so called time domain boundary integral equations the scattering problem by an absorbing obstacle serves as the model for this discussion a new system of retarded potential boundary integral equations rpbie is set up to solve the problem using an energy identity we are able to prove the unconditional stability for standard conforming galerkin approximation of the rpbie the actual space time boundary elements are described in some detail and their implementation has confirmed that stability in a wide range of situations and for large run times never reached before

phase reduction of weakly perturbed limit cycle oscillations in time delay systems abstract the phase reduction method is applied to a general class of weakly perturbed time delay systems exhibiting periodic oscillations the adjoint equation with an appropriate initial condition for the infinitesimal phase response curve of a time delay system is derived the method is demonstrated numerically for the mackey glass equation as well as for a chaotic rossler system subject to a delayed feedback control dfc we show that the profile of the phase response curve of a periodic orbit stabilized by the dfc algorithm does not depend on the control matrix this property is universal and holds for any dynamical system subject to the dfc

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

feedback mediated control of spiral waves abstract a theoretical approach is developed which allows one to describe the drift of a spiral wave mediated by a feedback signal taken as an integral of a system variable over a certain domain of an excitable medium the underlying reaction diffusion equations are reduced to an autonomous system of ordinary differential equations for the coordinates of the core center of the spiral wave a new method to determine the drift velocity field of the spiral core is proposed based on a superposition principle this approach is applied to analyze the drift velocity field for a feedback signal generated from a one point detector and from integration domains of different size and shape including global feedback it is shown that a variation of the shape of the integration domain can induce bifurcations that change the attractor structure of the drift velocity field these theoretical predictions are in good quantitative agreement with numerical simulations performed for the oregonator model and the complex ginzburg landau equation the results are also confirmed by experiments with the light sensitive belousov zhabotinsky reaction

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

design of a high gain regulator by the multiple time scale approach abstract in this paper a design procedure of the high gain feedback control for a linear time invariant system is presented to reveal the structure of the high gain system a new decoupling concept is introduced it divides a system into two parts a zero free part which is influential in the outputs and an unobservable part which contains all transmission zeros in view of the singular perturbation theory with multiple time scales the high gain feedback control with multiple gain parameters is designed for the zero free system the stabilizability of the unobservable part is discussed in relation to a minimal phase property of the plant the proposed design procedure consists only of elementary operations of matrices and an illustrative example shows its effectiveness

extending integral equation time domain acoustic scattering analysis to larger problems a method is presented to accelerate the execution of integral equation time domain analyses of exterior acoustic scattering problems conventionally these have costs which scale with the fifth power of the frequency of the excitation and practical limits to such computations are reached when bodies approach perhaps 5 10 wavelengths long the fast approach presented is based on exploiting the pulsed nature of the illumination to omit much nugatory calculation there is an associated slight accuracy loss this is investigated the method has costs which can scale with frequency to the power as low as 3 such that for example costs on a 18 5 wavelength body are reduced by a factor of about 28 with this factor itself increasing with roughly the square of the body size associated with the reduction in operations is a reduction in the scaling of storage required from the third to the second power of frequency examples of analysis of large scatterers are presented extending to a 22000 node almond copyright 1999 john wiley sons ltd

numerical solution of a heat diffusion problem by boundary element methods using the laplace transform this paper is concerned with a heat diffusion problem in a half space which is motivated by the detection of material defects using thermal measurements this problem is solved by inverting the laplace transform with respect to time on a contour in the complex plane using an exponentially convergent quadrature rule this leads to a finite number of time independent problems which can be solved in parallel using boundary integral equation methods we provide a full numerical analysis of this scheme on compact time intervals our results are formulated in a way that they can easily be used for other diffusion problems in exterior or interior domains

space time discretized retarded potential boundary integral operators quadrature for collocation methods we propose a novel discretization method for time domain boundary integral equations of the three dimensional wave equation the basic idea of the discussed approach is to treat time as if it were

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

transcritical bifurcations in nonintegrable hamiltonian systems we report on transcritical bifurcations of periodic orbits in nonintegrable two dimensional hamiltonian systems we discuss their existence criteria and some of their properties using a recent mathematical description of transcritical bifurcations in families of symplectic maps we then present numerical examples of transcritical bifurcations in a class of generalized henon heiles hamiltonians and illustrate their stabilities and unfoldings under various perturbations of the hamiltonians we demonstrate that for hamiltonians containing straight line librating orbits the transcritical bifurcation of these orbits is the typical case which occurs also in the absence of any discrete symmetries while their isochronous pitchfork bifurcation is an exception we determine the normal forms of both types of bifurcations and derive the uniform approximation required to include transcritically bifurcating orbits in the semiclassical trace formula for the density of states of the quantum hamiltonian we compute the coarse grained density of states in a specific example both semiclassically and quantum mechanically and find excellent agreement of the results

multistability in the kuramoto model with synaptic plasticity we present a simplified phase model for neuronal dynamics with spike timing dependent plasticity stdp for asymmetric experimentally observed stdp we find multistability a coexistence of a fully synchronized a fully desynchronized and a variety of cluster states in a wide enough range of the parameter space we show that multistability can occur only for asymmetric stdp and we study how the coexistence of synchronization and desynchronization and clustering depends on the distribution of the eigenfrequencies we test the efficacy of the proposed method on the kuramoto model which is de facto one of the sample models for a description of the phase dynamics in neuronal ensembles

analysis of three dimensional transient acoustic wave propagation using the boundary integral equation method this paper describes a boundary integral equation boundary element method for the solution of a variety of transient acoustic problems the spatial and temporal discretization employs quadratic isoparametric elements with high order gauss quadrature and the ensuing equations are implicit the implicit formulation both eliminates the instabilities reported in explicit treatments and permits a freedom of choice of timestep which can reduce costs dramatically the accuracy of the approach is demonstrated by comparison with the analytical solution for a sphere results for more demanding sphere cone sphere geometries extending to seven wavelengths long are presented and compared to those obtained from a related frequency domain approach

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

a least square iterative technique for solving time domain scattering problems an alternative method for the numerical solution of time domain integral equations pertaining to three dimensional transient scattering problems is developed and applied to two representative acoustic scattering configurations it is based on the iterative minimization of an integrated square error for the cases considered the method gives more accurate results than the frequently used marching on in time method within comparable computation times

a stable boundary element method for modeling transient acoustic radiation transient acoustic radiation from a closed axisymmetric three dimensional object is modeled using the time domain boundary element method the widely reported instability problems are overcome by reformulating the integral equation to obtain a burton and miller type equation in the time domain the stability of such an approach is mathematically justified and supported by subsequent numerical results the hypersingular integrals which arise are evaluated using a method valid for any surface discretization numerical results for the radiation of a spherical wave are presented and compared with an exact solution the accuracy and stability of the results are verified for several geometrically different radiating objects

correlation functions of an autonomous stochastic system with time delays abstract the auto correlation function and the cross correlation function of an autonomous stochastic system with time delays are investigated we obtain the distribution curves of the auto correlation function c x s and c y s and the cross correlation function c x y s and c y x s of the stochastic dynamic variables by the stochastic simulation method the delay time changes prominently the behaviors of the dynamical variables of an autonomous stochastic system which makes the auto correlation and the cross correlation of the autonomous stochastic system alternate oscillate periodically from positive to negative or from negative to positive decrease gradually and finally tends to zero with the decay time the delay time and the noise strength have important impacts for the auto correlation and the cross correlation of the autonomous stochastic delay system the delay time enhances the auto correlation and the cross correlation on the contrary the noise strength lowers the auto correlation and the cross correlation under the time delay by comparison we further show differences of the auto correlation and the cross correlation between the dynamical variables x and y

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

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems

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