a priori l 2 error estimates for finite element methods for nonlinear diffusion equations with memory this paper studies finite element approximations to the solutions of the nonlinear diffusion equations with memory an elliptic projection with memory associated with our equations is defined and then used in the derivations of optimal error estimates for semidiscrete and crank nicolson finite element approximations

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

sharp l 2 error estimates and superconvergence of mixed finite element methods for non fickian flows in porous media a sharper l2 error estimate is obtained for the non fickian flow of fluid in porous media by means of a mixed ritz volterra projection instead of the mixed ritz projection used in r e ewing y lin and j wang acta math univ comenian n s 70 2001 pp 75 84 moreover local l2 superconvergence for the velocity along the gauss lines and for the pressure at the gauss points is derived for the mixed finite element method via the ritz volterra projection and global l2 superconvergence for the velocity and the pressure is also investigated by virtue of an interpolation postprocessing technique on the basis of the superconvergence estimates some useful a posteriori error estimators are presented for this mixed finite element method

spectral behavior of contractive noise we study the behavior of the spectra corresponding to quantum systems subjected to a contractive noise i e the environment reduces the accessible phase space of the system but the total probability is conserved we find that the number of long lived resonances grows as a power law in h but surprisingly there is no relationship between the exponent of this power law and the fractal dimension of the corresponding classical attractor this is in disagreement with the predictions of the fractal weyl law which has been established for open systems where the probability is lost under the effect of a projective opening

large time asymptotic and numerical solution of a nonlinear diffusion model with memory large time behavior of solutions and finite difference approximation of a nonlinear system of integro differential equations associated with the penetration of a magnetic field into a substance is studied two initial boundary value problems are investigated the first with homogeneous conditions on whole boundary and the second with nonhomogeneous boundary data on one side of lateral boundary the rates of convergence are also given mathematical results presented show that there is a difference between stabilization rates of solutions with homogeneous and nonhomogeneous boundary conditions the convergence of the corresponding finite difference scheme is also proved the decay of the numerical solution is compared with the analytical results

unconditionally stable numerical method for a nonlinear partial integro differential equation the paper presents an unconditionally stable numerical scheme to solve a nonlinear integro differential equation which arises in mathematical modeling of the penetration of a magnetic field into a substance if the temperature is kept constant throughout the material numerical scheme comprises of the galerkin finite element method jangveladze and kiguradze 2011 for the spatial discretization followed by an implicit finite difference scheme for the time stepping we extended the results for stability estimates to a non homogeneous problem and derived optimal order error estimates for the semidiscretized and fully discretized equations using h 0 1 projection further to show the efficiency the proposed numerical method is demonstrated via numerical example

quantum fractal eigenstates abstract we study quantum chaos in open dynamical systems and show that it is characterized by quantum fractal eigenstates located on the underlying classical strange repeller the states with longest life times typically reveal a scars structure on the classical fractal set

finite difference approximation of a nonlinear integro differential system finite difference approximation of the nonlinear integro differential system associated with the penetration of a magnetic field into a substance is studied the convergence of the finite difference scheme is proved the rate of convergence of the discrete scheme is given the decay of the numerical solution is compared with the analytical results proven earlier

linearly localized difference schemes for the nonlinear maxwell model of a magnetic field into a substance abstract a linearly localized difference scheme with the first order time approximation is proposed for solving a nonlinear maxwell model associated with the penetration of a magnetic field into a substance the new scheme is computationally efficient since the resulting algebra equations are linear and can be computed by the fast thomas algorithm without any newton type inner iterations it is also local in time that is only numerical solutions in one previous time level are necessary to update the current solutions such that it requires much less storage compared with the fully implicit method furthermore the exponential decaying behavior of difference solution which is analogous to that of the continuous solution is obtained to improve the time accuracy we apply the crank nicolson type time discretization to construct a second order linearly localized method numerical examples are presented to support our theoretical results

weyl law for fat fractals it has been conjectured that for a class of piecewise linear maps the closure of the set of images of the discontinuity has the structure of a fat fractal that is a fractal with positive measure an example of such maps is the sawtooth map in the elliptic regime in this work we analyze this problem quantum mechanically in the semiclassical regime we find that the fraction of states localized on the unstable set satisfies a modified fractal weyl law where the exponent is given by the exterior dimension of the fat fractal

eigenvalue equalization filtered x algorithm for the multichannel active noise control of stationary and nonstationary signals the fxlms algorithm which is extensively used in active noise control exhibits frequency dependent convergence behavior this leads to degraded performance for time varying and multiple frequency signals a new algorithm called the eigenvalue equalization filtered x least mean squares ee fxlms has been developed to overcome this limitation without increasing the computational burden of the controller the algorithm is easily implemented for either single or multichannel control the magnitude coefficients of the secondary path transfer function estimate are altered while preserving the phase for a reference signal that has the same magnitude at all frequencies the secondary path estimate is given a flat response over frequency for a reference signal that contains tonal components of unequal magnitudes the magnitude coefficients of the secondary path are adjusted to be the inverse magnitude of the reference tones both modifications reduce the variation in the eigenvalues of the filtered x autocorr

global smooth solutions for a class of parabolic integrodifferential equations the existence and uniqueness of smooth global large data solutions of a class of quasilinear partial integrodifferential equations in one space and one time dimension are proved if the integral kernel behaves like t near t 0 o with a 2 3 an existence and regularity theorem for linear equations with variable coefficients that are related to this type is also proved in arbitrary space dimensions and with no restrictions for o

role of short periodic orbits in quantum maps with continuous openings we apply a recently developed semiclassical theory of short periodic orbits to the continuously open quantum tribaker map in this paradigmatic system the trajectories are partially bounced back according to continuous reflectivity functions this is relevant in many situations that include optical microresonators and more complicated boundary conditions in a perturbative regime the shortest periodic orbits belonging to the classical repeller of the open map a cantor set given by a region of exactly zero reflectivity prove to be extremely robust in supporting a set of long lived resonances of the continuously open quantum maps moreover for steplike functions a significant reduction in the number needed is obtained similarly to the completely open situation this happens despite a strong change in the spectral properties when compared to the discontinuous reflectivity case in order to give a more realistic interpretation of these results we compare with a fresnel type reflectivity function

convergence analysis of the multi variable filtered x lms algorithm with application to active noise control averaging analysis is used to study the asymptotic behavior of the multichannel filtered x lms algorithm for the rejection of narrowband and broadband stationary disturbances a multivariable strictly positive real spr condition on the multi input multi output mimo plant model error is shown to be sufficient for convergence

geometric bounds on the density of resonances for semiclassical problems in this paper we shall give upper bounds on the number of resonances in certain regions in the complex plane close to the real axis for semiclassical operators like h 2 v x when h is small

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

finite element methods with numerical quadrature for parabolic integrodifferential equations we study the effect of numerical quadrature in space on semidiscrete and fully discrete piecewise linear finite element methods for parabolic integrodifferential equations we give error estimates for both smooth and nonsmooth initial data our results generalize those obtained for the lumped mass method

google matrix dynamical attractors and ulam networks we study the properties of the google matrix generated by a coarse grained perron frobenius operator of the chirikov typical map with dissipation the finite size matrix approximant of this operator is constructed by the ulam method this method applied to the simple dynamical model generates directed ulam networks with approximate scale free scaling and characteristics being in certain features similar to those of the world wide web with approximate scale free degree distributions as well as two characteristics similar to the web a power law decay in pagerank that mirrors the decay of pagerank on the world wide web and a sensitivity to the value ensuremath alpha in pagerank the simple dynamical attractors play here the role of popular websites with a strong concentration of pagerank a variation in the google parameter ensuremath alpha or other parameters of the dynamical map can drive the pagerank of the google matrix to a delocalized phase with a strange attractor where the google search becomes inefficient

fractal weyl law for quantum fractal eigenstates the properties of the resonant gamow states are studied numerically in the semiclassical limit for the quantum chirikov standard map with absorption it is shown that the number of such states is described by the fractal weyl law and their husimi distributions closely follow the strange repeller set formed by classical orbits nonescaping in future times for large matrices the distribution of escape rates converges to a fixed shape profile characterized by a spectral gap related to the classical escape rate

numerical blow up analysis of linearly implicit euler method for nonlinear parabolic integro differential equations abstract the linearly implicit euler method for nonlinear parabolic integro differential equations npides on bounded domains is consideblue by approximating the local effects implicitly and nonlocal effects explicitly based on nakagawa s criteria a suitable adaptive time stepping strategy is introduced by the discrete energy instead of the infinite norm with the help of lower discrete energies the finite blow up behaviors are replicated for any positive solution numerical simulations are carried out to examine the effectiveness of our blowup analysis which also motivate furthermore to prove that a global numerical solution exists for certain npides with a weakly singular kernel

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

convergence analysis for h1 galerkin mixed finite element approximation of one nonlinear integro differential model in this paper we investigate h 1 galerkin mixed finite element approximation of one nonlinear integro differential equation this method possesses some advantages such as approximating the unknown function and its gradient simultaneously as well as the finite element spaces being free of lbb condition a priori error estimates of the unknown function and its gradient are derived for both semi discrete and fully discrete schemes a numerical example is presented to illustrate the theoretical findings

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

fractal weyl law for open quantum chaotic maps we study a semiclassical quantization of poincar e maps arising in scattering problems with fractal hyperbolic trapped sets the main application is the proof of a fractal weyl upper bound for the number of resonances scattering poles in small domains near the real axis this result includes the case of several convex hard obstacles satisfying a no eclipse condition

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

some new error estimates of a semidiscrete finite volume element method for a parabolic integro differential equation with nonsmooth initial data a semidiscrete finite volume element fve approximation to a parabolic integro differential equation pide is analyzed in a two dimensional convex polygonal domain an optimal order l 2 error estimate for smooth initial data and nearly the same optimal order l 2 error estimate for nonsmooth initial data are obtained more precisely for homogeneous equations an elementary energy technique and a duality argument are used to derive an error estimate of order o left t 1 h 2 ln h right in the l 2 norm for positive time when the given initial function is only in l 2

large time behavior of solutions to a nonlinear integro differential system abstract asymptotic behavior of solutions as t to the nonlinear integro differential system associated with the penetration of a magnetic field into a substance is studied initial boundary value problems with two kinds of boundary data are considered the first with homogeneous conditions on whole boundary and the second with non homogeneous boundary data on one side of lateral boundary the rates of convergence are given too

a new collocation approach for solving systems of high order linear volterra integro differential equations with variable coefficients this paper contributes an efficient numerical approach for solving the systems of high order linear volterra integro differential equations with variable coefficients under the mixed conditions the method we have used consists of reducing the problem to a matrix equation which corresponds to a system of linear algebraic equations the obtained matrix equation is based on the matrix forms of fibonacci polynomials and their derivatives by means of collocations in addition the method is presented with error numerical results with comparisons are given to demonstrate the applicability efficiency and accuracy of the proposed method the results of the examples indicated that the method is simple and effective and could provide an approximate solution with high accuracy or exact solution of the system of high order linear volterra integro differential equations

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

galerkin finite element method for one nonlinear integro differential model galerkin finite element method for the approximation of a nonlinear integro differential equation associated with the penetration of a magnetic field into a substance is studied first type initial boundary value problem is investigated the convergence of the finite element scheme is proved the rate of convergence is given too the decay of the numerical solution is compared with the analytical results

periodic solutions of a nonlinear parabolic equation associated with the penetration of a magnetic field into a substance abstract we prove the existence and uniqueness of a t periodic weak solution of the nonlinear parabolic equation u t 1 r r a 0 t u r r s 2 ds r u r f u 0 r t 0 1 x o t u r 0 t u r 1 t h t u 1 t 0 in a sobolev space with weight in the proof the galerkin method is employed numerical results are given

finite difference scheme for one system of nonlinear partial integro differential equations abstract system of maxwell equations is considered reduction to the integro differential form is given existence uniqueness and large time behavior of solutions of the initial boundary value problem for integro differential model with two component and one dimensional case are studied finite difference scheme is investigated wider class of nonlinearity is studied than one has been investigated before freefem realization code and results of numerical experiments are given

non fickian convection diffusion models in porous media in this paper we propose a numerical scheme to approximate the solution of a non fickian coupled model that describes e g miscible transport in porous media the model is defined by a system of a quasilinear elliptic equation which governs the fluid pressure and a quasilinear integro differential equation which models the convection diffusion transport process the numerical scheme is based on a conforming piecewise linear finite element method for the discretization in space the fully discrete approximations is obtained with an implicit explicit method estimates for the continuous in time and the fully discrete methods are derived showing that the numerical approximation for the concentrations and the pressure are second order convergent in a discrete l 2 norm and in a discrete h 1 norm respectively

a parallel method for the numerical solution of integro differential equation with positive memory abstract an efficient parallel numerical method is proposed for an integro differential equation with positive memory instead of solving the equation in classical time marching methods which require massive storage of solutions of previous time steps in order to advance to a next time step the fourier laplace transformation in time is applied to obtain a set of complex valued elliptic problems parameterized by points on a contour in the complex plane using the independence of an elliptic problem corresponding to one contour point is independent of those elliptic problems corresponding to other contour points all elliptic problems can be solved in parallel essentially without data communications then the time domain solution can be obtained by the fourier laplace inversion formula an error analysis and the numerical implementation of this parallel method is presented

fractal weyl law behavior in an open hamiltonian system we numerically show fractal weyl law behavior in an open hamiltonian system that is described by a smooth potential and which supports numerous above barrier resonances this behavior holds even relatively far away from the classical limit the complex resonance wave functions are found to be localized on the fractal classical repeller

on the lti properties of adaptive feedforward systems with tap delay line regressors it is shown that an adaptive system whose regressor is formed by tap delay line tdl filtering of a multitone sinusoidal signal is representable as a parallel connection of a linear time invariant lti block and a linear time varying ltv block a norm bound induced 2 norm is computed explicitly on the ltv block and is shown to decrease as n sup 1 where n is the number of taps hence the adaptive system becomes lti in the limit as the number of taps goes to infinity in the more realistic case where the number of taps n is finite the new lti plus norm bounded perturbation representation renders for the first time the adaptive system analyzable by standard robust control methods

distribution of resonances for open quantum maps we analyze a simple model of quantum chaotic scattering system namely the quantized open baker s map this model provides a numerical confirmation of the fractal weyl law for the semiclassical density of quantum resonances the fractal exponent is related to the dimension of the classical repeller we also consider a variant of this model for which the full resonance spectrum can be rigorously computed and satisfies the fractal weyl law for that model we also compute the shot noise of the conductance through the system and obtain a value close to the prediction of random matrix theory

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

a general theory of linear time invariant adaptive feedforward systems with harmonic regressors establishes necessary and sufficient conditions for an adaptive system with a harmonic regressor i e a regressor comprised exclusively of sinusoidal signals to admit an exact linear time invariant lti representation these conditions are important because a large number of adaptive systems used in practice have sinusoidal regressors and the stability and performance of such systems having lti representations can be completely analyzed by well known methods the theory is extended to applications where the lti conditions do not hold in which case the harmonic adaptive system can be written as the parallel connection of a purely lti subsystem and a linear time varying ltv subsystem an explicit upper bound is established on the induced two norm of the ltv block which allows systematic treatment using emerging robust control methods applicable to lti systems with norm bounded ltv perturbations

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

galerkin methods for nonlinear parabolic integrodifferential equations with nonlinear boundary conditions galerkin methods are analyzed for the nonlinear parabolic integrodiflerential equation u t nabla cdot left a u nabla u int 0 t b t u x s nabla u x s ds right f u in omega times 0 t t 0 omega subset r d d leqq 3 subject to the nonlinear boundary condition a u frac partial u partial v int 0 t b t u x s frac partial u partial v x s ds g u on s t partial omega times 0 t and the usual initial condition optimal order error estimates are derived in l 2 omega norm for the continuous crank nicolson and modified crank nicolson galerkin approximations as usual the latter will yield a system of linear algebraic equations to be solved at each level

on a class of volterra nonlinear equations of parabolic type in this paper we show the backward uniqueness in time of solution for a class of volterra nonlinear equations of parabolic type we also give reasonable physical interpretation for our conclusion

mixed finite element approximations of parabolic integro differential equations with nonsmooth initial data we analyze the semidiscrete mixed finite element methods for parabolic integro differential equations that arise in the modeling of nonlocal reactive flows in porous media a priori l 2 error estimates for pressure and velocity are obtained with both smooth and nonsmooth initial data more precisely a mixed ritz volterra projection introduced earlier by ewing et al in siam j numer anal 40 2002 pp 1538 1560 is used to derive optimal l 2 error estimates for problems with initial data in h 2 cap h 0 1 in addition for homogeneous equations we derive optimal l 2 error estimates for initial data just in l 2 here we use an elementary energy technique and duality argument

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

global existence of solutions of volterra integrodifferential equations of parabolic type abstract sufficient conditions for the global existence of a strong solution of the equation u t t x t 0 k t s u x s x x ds f t x are given the kernel k satisfies r k z i k z and is increasing with sup p

finite differences versus finite elements for solving nonlinear integro differential equations consider the nonlinear integro differential equation ut x t 0t a t x ux x d f x t 0 x 0 t t with appropriate initial and boundary conditions this problem serves as a model for one dimensional heat flow in materials with memory the numerical solution via finite elements was discussed in b neta j math anal appl 89 1982 598 611 in this paper we compare the results obtained there with finite difference approximation from the point of view of accuracy and computer storage it turns out that the finite difference method yields comparable results for the same mesh spacing using less computer storage

laplace transform finite element method for non fickian flows abstract in this paper we consider numerical methods for integro differential problems based on time discretization via laplace transformation we focus our attention in models arising in the context of non fickian solute transport phenomena in porous media the mathematical models which describe the evolution of the solute concentrations are characterized by volterra equations we present and analyze an hybrid method which combines the laplace transformation with respect to the time variable with the finite element discretization in the spatial variables numerical results illustrate the performance of the method

large time behavior of solutions and finite difference scheme to a nonlinear integro differential equation the large time behavior of solutions and finite difference approximations of the nonlinear integro differential equation associated with the penetration of a magnetic field into a substance are studied asymptotic properties of solutions for the initial boundary value problem with homogeneous dirichlet boundary conditions is considered the rates of convergence are given too the convergence of the semidiscrete and the finite difference schemes are also proved

an integro differential formulation for magnetic induction in bounded domains boundary element finite volume method simulations of magnetohydrodynamic mhd flows in bounded domains using spectral methods suffer from a number of serious limitations alternative methods based on local discretization raise the problem of how to implement non local boundary conditions for the magnetic field we have developed a new strategy for the numerical solution of mhd problems in bounded domains which combines the flexibility of a local discretization with a rigorous formulation of magnetic boundary conditions next to an insulator in arbitrary geometries in accordance with the character of underlying equations we apply a global integral approach at the boundary and a differential approach inside the conducting domain the formulation of the boundary problem in terms of primitive variables allows us to combine these approaches and propose a mixed finite volume and boundary element method we illustrate its efficiency on magnetic diffusion problems in a sphere and in a finite cylinder

the finite element numerical analysis for a class nonlinear evolution equations a complete stability and convergence analysis is given for two level scheme for the approximate solution of a class generalized nerve conduction equation defining a kind of truncation error and using a kind of technique optimal order estimates in l 2 is constructively obtained

comparison and extensions of control methods for narrow band disturbance rejection techniques for designing and implementing algorithms for the control of periodic narrowband disturbances are discussed and the strong similarities between the different methodologies are shown the analysis of linear time invariant feedback systems results in a suggestion of how to extend two of the lq based multiple input multiple output methodologies to achieve improved performance and stability robustness properties in the analysis of the adaptive feedforward methods it is concluded that the popular filtered x lms algorithm is useful for implementation but is best analyzed from a classical linear time invariant feedback perspective this perspective results in a suggestion of how to extend the multiple error lms algorithm to achieve improved performance and stability robustness stability bounds of the adaptive feedforward approaches are derived in terms of allowable model error

numerical solution of a nonlinear integro differential equation abstract galerkin s method with appropriate discretization in time is considered for approximating the solution of the nonlinear integro differential equation u t x t 0 t a t x u x x d f x t 0 an error estimate in a suitable norm will be derived for the difference u uh between the exact solution u and the approximant uh it turns out that the rate of convergence of uh to u as h 0 is optimal this result was confirmed by the numerical experiments

numerical solution of semilinear integrodifferential equations of parabolic type with nonsmooth data error estimates are derived for the discretization in space by finite elements and in time by the backward euler method of a parabolic equation with a semilinear memory term special attention is paid to the case of nonsmooth data and the computation of the memory term by quadrature

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

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

finite element methods of the two nonlinear integro differential equations abstract transformation of dependent variables such as the kirchhoff transformation is a classical tool for solving nonlinear partial differential equations in larsson et al this approach was used in connection with the finite element method zlamal used it in galerkin approximation in space the euler method in time and applied it to the solution of nonlinear heat equations here we give justification of the methods for the two nonlinear integro heat equations in the case that the discretization is carried through by piecewise linear polynomials in space and by the implicit euler method in time with quadrature rules which require the storage of relatively few values of the solution the estimate of the discretization error in the maximum norm is introduced the results depend on superconvergence in the gradient for a related elliptic problem which is shown to hold only for very special meshes

periodic disturbance cancellation with uncertain frequency this paper presents a new algorithm to cancel periodic disturbances with uncertain frequency the disturbances are cancelled using an internal model structure with adaptive frequency in parallel with a stabilizing controller the time varying internal model controller s states in steady state can be mapped to two time invariant variables the magnitude or energy of the internal model and frequency of the disturbance an additional integral controller then can be used to reduce the difference between the internal model controller imc and disturbance frequency to zero the stability of the feedback control system with this algorithm and convergence of the algorithm to the correct frequency with exact disturbance cancellation are justified by singular perturbation and averaging theories the algorithm is locally exponentially stable rather than asymptotically stable simulations demonstrate the performance of the algorithm the ability of this algorithm to identify the frequency of periodic disturbances and to reject periodic disturbances with uncertain frequency

a recursive least squares based control algorithm for the suppression of tonal disturbances mitigation of the effects of tonal noise and vibration has importance across a wide range of application domains for low frequency disturbances active control methodologies have proven to be highly effective when coupled with harmonic algorithms that target specific problematic frequencies in this paper a novel harmonic control approach is presented that combines a classical exponential forgetting recursive least squares exf rls estimator with an optimal gradient descent update law the paper provides a rigorous proof that with a sinusoidal reference the exf rls estimator can be approximated by a fixed parameter filter and that the accuracy of the approximation is dependent upon the values of both the forgetting factor and the sampling frequency with this approximation it is also proven that the complete iterative frequency domain harmonic control algorithm when used with an instantaneous update can be accurately represented by a linear time invariant lti feedback compensator such representations have clear importance since the stability performance convergence and robustness properties of the adaptive system can thereby be examined using well established lti analysis tools the algorithm and the analytical results described above are illustrated in the paper using an experimental active vibration isolation mount that has been developed for marine systems application