postprocessing the galerkin method the finite element case a postprocessing technique developed earlier for spectral methods is extended here to galerkin finite element methods for dissipative evolution partial differential equations the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed this technique increases the convergence rate of the finite element method to which it is applied and this is done at almost no additional computational cost the numerical experiments presented here show that the resulting postprocessed method is computationally more efficient than the method to which it is applied say quadratic finite elements as well as standard methods of similar order of convergence as the postprocessed one say cubic finite elements the error analysis of the new method is performed in l2 and in l infty norms

a new method based on haar wavelet for the numerical solution of two dimensional nonlinear integral equations a new numerical method based on haar wavelet is proposed for two dimensional nonlinear fredholm volterra and volterra fredholm integral equations of first and second kind the proposed method is an extension of the haar wavelet method aziz and siraj ul islam 2013 siraj ul islam et al 2013 and siraj ul islam et al 2014 from one dimensional nonlinear integral equations fredholm and volterra to two dimensional nonlinear integral equations fredholm volterra and volterra fredholm the main characteristic of the method is that unlike several other methods it does not involve numerical integration which results in an improved accuracy of the method in order to show the effectiveness of the method it is applied to several benchmark problems the numerical results are compared with other methods existing in the recent literature

some practical experience with the time integration of dissipative equations different methods for the numerical integration of evolution dissipative partial differential equations are tested with the kuramoto sivashinsky equation discretizations in space include galerkin and nonlinear galerkin methods for integration in time three different codes are used including standard stiff ode methods numerical tests show that standard codes for stiff ode render a gain of computing time of several orders of magnitude with respect problem tailored methods

positive periodic solutions of a single species model with feedback regulation and distributed time delay in this paper by using the continuation theorem of coincidence degree theory a criterion of the existence of positive periodic solutions is obtained for a single species model with feedback regulation and distributed time delay

multiple positive periodic solutions of nonlinear functional differential system with feedback control abstract in this paper we employ avery henderson fixed point theorem to study the existence of positive periodic solutions to the following nonlinear nonautonomous functional differential system with feedback control dx dt r t x t f t x t u t t du dt h t u t g t x t t we show that the system above has at least two positive periodic solutions under certain growth condition imposed on f

multi level spectral galerkin method for the navier stokes problem i spatial discretization a multi level spectral galerkin method for the two dimensional non stationary navier stokes equations is presented the method proposed here is a multiscale method in which the fully nonlinear navier stokes equations are solved only on a low dimensional space subsequent approximations are generated on a succession of higher dimensional spaces j 2 j by solving a linearized navier stokes problem around the solution on the previous level error estimates depending on the kinematic viscosity 0 1 are also presented for the j level spectral galerkin method the optimal accuracy is achieved when we demonstrate theoretically that the j level spectral galerkin method is much more efficient than the standard one level spectral galerkin method on the highest dimensional space

the coiflet galerkin method for linear volterra integral equations this paper deals with the application of the wavelet galerkin method based on coiflets as a basis for solving linear volterra integral equations vies the main contribution of this work is that some new connection coefficients are introduced and a suitable algorithm is developed for their solution once they have been computed they can be stored and applied to any linear vie the convergence properties of the coiflet galerkin method are analyzed some test examples are presented to illustrate the performance of the method with respect to the error norms and cpu time

evolutionary computational intelligence in solving a class of nonlinear volterra fredholm integro differential equations in this paper a stochastic computational intelligence technique for solving a class of nonlinear volterra fredholm integro differential equations with mixed conditions is presented the strength of feed forward artificial neural networks is used to accurately model the integro equation comparisons with the exact solution and other numerical techniques are presented to show the efficiency of the proposed method theoretical and numerical results are presented analysis for the presented method is given

periodic solutions for a kind of second order neutral differential systems with deviating arguments by means of the generalized continuation theorem of coincidence degree theory a kind of second order neutral differential systems with deviating arguments as followsd 2dt 2 x t cx t r ddtgradf x t gradg x email protected t p t is studied a new result on the existence of periodic solutions is obtained which relates to the deviating arguments t t and r

galerkin and subspace decomposition methods in space and time for the navier stokes equations abstract the galerkin method and the subspace decomposition method in space and time for the two dimensional incompressible navier stokes equations with the h 2 initial data are considered the subspace decomposition method consists of splitting the approximate solution as the sum of a low frequency component discretized by the small time step t and a high frequency one discretized by the large time step p t with p 1 the h 2 stability and l 2 error analysis for the subspace decomposition method are obtained finally some numerical tests to confirm the theoretical results are provided

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

the postprocessed mixed finite element method for the navier stokes equations refined error bounds a postprocessing technique for mixed finite element methods for the incompressible navier stokes equations is analyzed the postprocess which amounts to solving a linear stokes problem is shown to increase the order of convergence of the method to which it is applied by one unit times the logarithm of the mesh diameter in proving the error bounds some superconvergence results are also obtained contrary to previous analysis of the postprocessing technique in the present paper we take into account the loss of regularity suffered by the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data as in h g heywood and r rannacher siam j numer anal 25 1988 pp 489 512 where the same hypothesis is assumed no better than fifth order convergence is achieved

rationalized haar functions method for solving fredholm and volterra integral equations this paper presents a computational technique for fredholm integral equation of the second kind and volterra integral equation of the second kind the method is based upon haar functions approximation properties of rationalized haar functions are first presented the operational matrix of integration together with product operational matrix and newton cotes nodes are utilized to reduce the computation of integral equations into some algebraic equations the method is computationally attractive and applications are demonstrated through illustrative examples

numerical solution of nonlinear volterra fredholm hammerstein integral equations via tau collocation method with convergence analysis in this paper we consider the nonlinear volterra fredholm hammerstein integral equations the approximate solution for the nonlinear volterra fredholm hammerstein integral equations is obtained by using the tau collocation method to do this the nonlinear volterra fredholm hammerstein integral equations is transformed into a system of nonlinear algebraic equations in matrix form thus by solving this system unknown coefficients are obtained the spectral rate of convergence for the proposed method is established in the l 2 norm the numerical results obtained with minimum amount of computation are compared with the exact solutions to show the efficiency of the method the results show that the tau collocation method is of high accuracy more convenient and efficient for solving nonlinear volterra fredholm hammerstein integral equations

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

on approximate inertial manifolds to the navier stokes equations abstract recently the theory of inertial manifolds has shown that the long time behavior the dynamics of certain dissipative partial differential equations can be fully discribed by that of a finite ordinary differential system although we are still unable to prove existence of inertial manifolds to the navier stokes equations we present here a nonlinear finite dimensional analytic manifold that approximates closely the global attractor in the two dimensional case and certain bounded invariant sets in the three dimensional case this approximate manifold and others allow us to introduce modified galerkin approximations

the postprocessing galerkin and nonlinear galerkin methods a truncation analysis point of view we revisit the postprocessing algorithm and give a justification from a classical truncation analysis point of view we assume a perturbation expansion for the high frequency mode component of solutions to the underlying equation keeping terms to certain orders we then generate approximate systems which correspond to numerical schemes we show that the first two leading order methods are in fact the postprocessed galerkin and postprocessed nonlinear galerkin methods respectively hence postprocessed galerkin is a natural leading order method more natural than the standard galerkin method for approximating solutions of parabolic dissipative pdes the analysis is presented in the framework of the two dimensional navier stokes equation nse however similar analysis may be done for any parabolic dissipative nonlinear pde the truncation analysis is based on asymptotic estimates in time for the low and high mode components we also introduce and investigate an alternative postprocessing scheme whic

a posteriori error estimations for mixed finite element approximations to the navier stokes equations a posteriori estimates for mixed finite element discretizations of the navier stokes equations are derived we show that the task of estimating the error in the evolutionary navier stokes equations can be reduced to the estimation of the error in a steady stokes problem as a consequence any available procedure to estimate the error in a stokes problem can be used to estimate the error in the nonlinear evolutionary problem a practical procedure to estimate the error based on the so called postprocessed approximation is also considered both the semidiscrete in space and the fully discrete cases are analyzed some numerical experiments are provided

postprocessing for stochastic parabolic partial differential equations we investigate the strong approximation of stochastic parabolic partial differential equations with additive noise we introduce postprocessing in the context of a standard galerkin approximation although other spatial discretizations are possible in time we follow g j lord and j rougemont ima j numer anal 24 2004 pp 587 604 and use an exponential integrator we prove strong error estimates and discuss the best number of postprocessing terms to take numerically we evaluate the efficiency of the methods and observe rates of convergence some experiments with the implicit euler maruyama method are described

a semi analytical approach to solve integro differential equations in this work we present a semi analytical method for solving integro differential equations under multi point or mixed boundary conditions the proposed method solves linear and nonlinear fredholmvolterra integro differential equations a convergence analysis of the proposed method is directly examined numerical examples are worked out to demonstrate the main results moreover proper graphs are provided to confirm the efficiency and the accuracy of the proposed scheme we show that with a few number of obtained approximating terms we achieve a high accuracy level of the obtained results however increasing the number of approximating terms yields a significant decrease of the error of the approximation the proposed method is very useful reliable and flexible for solving different kinds of integro differential equations

efficient methods using high accuracy approximate inertial manifolds summary we extend the idea of the post processing galerkin method in the context of dissipative evolution equations to the nonlinear galerkin the filtered galerkin and the filtered nonlinear galerkin methods in general the post processing algorithm takes advantage of the fact that the error committed in the lower modes of the nonlinear galerkin method and galerkin method for approximating smooth bounded solutions is much smaller than the total error of the method in each case an improvement in accuracy is obtained by post processing these more accurate lower modes with an appropriately chosen highly accurate approximate inertial manifold aim we present numerical experiments that support the theoretical improvements in accuracy both the theory and computations are presented in the framework of a two dimensional reaction diffusion system with polynomial nonlinearity however the algorithm is very general and can be implemented for other dissipative evolution systems the computations clearly show the post processed filtered galerkin method to be the most efficient method

positive periodic solutions for a neutral differential system with feedback control applying a fixed point theorem of strict set contractions some new criteria are established for the existence of positive periodic solutions of the following neutral differential system with feedback controldx t dt x t r t a 0 t x t a 1 t x email protected 1 t a 2 t x email protected 2 t email protected t u email protected 1 t du t dt a t u t b t x email protected 2 t

bessel polynomial solutions of high order linear volterra integro differential equations abstract in this study a practical matrix method which is based on collocation points is presented to find approximate solutions of high order linear volterra integro differential equations vides under the mixed conditions in terms of bessel polynomials numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with the existing results the results show the efficiency and accuracy of the present work all of the numerical computations have been performed on the computer using a program written in matlab v7 6 0 r2008a

existence and global stability of positive periodic solution of periodic integrodifferential systems with feedback controls abstract sufficient conditions are obtained for the existence and global stability of a positive periodic solution in some periodic integrodifferential systems with feedback controls by using the technique of coincidence degree and lyapunov functional

haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations haar wavelet operational matrix has been widely applied in system analysis system identification optimal control and numerical solution of integral and differential equations in the present paper we derive the haar wavelet operational matrix of the fractional order integration and use it to solve the fractional order differential equations including the bagley torvik ricatti and composite fractional oscillation equations the results obtained are in good agreement with the existing ones in open literatures and it is shown that the technique introduced here is robust and easy to apply

a class of numerical algorithms for large time integration the nonlinear galerkin methods the nonlinear galerkin methods are numerical schemes well adapted to the long term integration of evolution partial differential equations they consist of looking for approximate solutions lying in nonlinear manifolds that are close in some sense to the attractor the aim of this paper is to extend these schemes to a large class of manifolds convergence results are derived for the schemes introduced

a postprocessed galerkin method with chebyshev or legendre polynomials we present an approximate inertial manifold based postprocess to enhance chebyshev or legendre spectral galerkin methods we prove that the postprocess improves the order of convergence of the galerkin solution yielding the same accuracy as the nonlinear galerkin method numerical experiments show that the new method is computationally more efficient than galerkin and nonlinear galerkin methods new approximation results for chebyshev polynomials are presented

on the rate of convergence of the nonlinear galerkin methods in this paper we provide estimates to the rate of convergence of the nonlinear galerkin approximation method in particular and by means of an illustrative example we show that the nonlinear galerkin method converges faster than the usual galerkin method

hybrid function method for solving fredholm and volterra integral equations of the second kind numerical solutions of fredholm and volterra integral equations of the second kind via hybrid functions are proposed in this paper based upon some useful properties of hybrid functions integration of the cross product a special product matrix and a related coefficient matrix with optimal order are applied to solve these integral equations the main characteristic of this technique is to convert an integral equation into an algebraic hence the solution procedures are either reduced or simplified accordingly the advantages of hybrid functions are that the values of n and m are adjustable as well as being able to yield more accurate numerical solutions than the piecewise constant orthogonal function for the solutions of integral equations we propose that the available optimal values of n and m can minimize the relative errors of the numerical solutions the high accuracy and the wide applicability of the hybrid function approach will be demonstrated with numerical examples the hybrid function method is superior to other piecewise constant orthogonal functions w f blyth r l may p widyaningsih volterra integral equations solved in fredholm form using walsh functions anziam j 45 e 2004 c269 c282 m h reihani z abadi rationalized haar functions method for solving fredholm and volterra integral equations j comp appl math 200 2007 12 20 for these problems

on a criterion for locating stable stationary solutions to the navier stokes equations on donne une estimation superieure uniforme explicite a la borne de constantin foias ternam et on donne une forme applicable explicite aux conditions suffisantes on considere seulement le cas a 2 dimensions

postprocessing the galerkin method a novel approach to approximate inertial manifolds a postprocess of the standard galerkin method for the discretization of dissipative equations is presented the postprocessed galerkin method uses the same approximate inertial manifold phi app to approximate the high wave number modes of the solution as in the nonlinear galerkin nlg method however in this postprocessed galerkin method the value of phi app is calculated only once after the time integration of the standard galerkin method is completed contrary to the nlg in which phi app evolves with time and affects the time evolution of the lower wave number modes the postprocessed galerkin method which is much cheaper to implement computationally than the nlg is shown in the case of fourier modes to possess the same rate of convergence accuracy as the simplest version of the nlg which is based on either the foias manley temam approximate inertial manifold or the euler galerkin approximate inertial manifold this is proved for some problems in one and two spatial dimensions including the navier stokes equations under periodic boundary conditions the advantages of postprocessing that we present here apply not only to the standard galerkin method but also to the computationally more efficient pseudospectral method

an approximate inertial manifolds approach to postprocessing the galerkin method for the navier stokes equation in a recent paper we have introduced a postprocessing procedure for the galerkin method for dissipative evolution partial differential equations with periodic boundary conditions the postprocessing technique uses approximate inertial manifolds to approximate the high modes the small scale components in the exact solutions in terms of the galerkin approximations which in this case play the role of the lower modes large scale components this procedure can be seen as a defect correction technique but contrary to standard procedures the correction is computed only when the time evolution is completed here we extend these results to more realistic boundary conditions specifically we study in detail the two dimensional navier stokes equations subject to homogeneous nonslip dirichlet boundary conditions we also discuss other equations such as reaction diffusion systems and the cahn hilliard equations

new algorithms for the numerical solution of nonlinear fredholm and volterra integral equations using haar wavelets two new algorithms based on haar wavelets are proposed the first algorithm is proposed for the numerical solution of nonlinear fredholm integral equations of the second kind and the second for the numerical solution of nonlinear volterra integral equations of the second kind these methods are designed to exploit the special characteristics of haar wavelets in both one and two dimensions formulae for calculating haar coefficients without solving the system of equations have been derived these formulae are then used in the proposed numerical methods in contrast to other numerical methods the advantage of our method is that it does not involve any intermediate numerical technique for evaluation of the integral present in integral equations the methods are validated on test problems and numerical results are compared with those from existing methods in the literature

positive solutions of a nonlinear four point boundary value problems in banach spaces abstract in this paper by using the fixed points of strict set contractions we study the existence of at least one or two positive solutions to the four point boundary value problem y t a t f y t 0 t 1 y 0 y y 1 y in banach space e where is zero element of e 0 1 0 1 1 0 1 and 1 1 1 0 as an application we also give one example to demonstrate our results

a new algorithm for the solution of nonlinear two dimensional volterra integro differential equations of high order abstract in this manuscript a new numerical approach is developed for approximating the solution of the one dimensional and two dimensional volterra integro differential equations 2d vides of high order new operational matrices of integration based on shifted jacobi polynomials are presented and used in combination with the collocation method to convert the 2d vide into a system of algebraic equations in addition the convergence analysis of the suggested approach is investigated the efficiency of the algorithm is demonstrated by means of several examples and the results are compared with those given using other numerical schemes

positive periodic solutions of a class of functional differential systems with feedback controls abstract in this paper by using the upper and lower solutions method we investigate the existence and nonexistence of positive periodic solutions of infinite delay functional differential system with a parameter and feedback controls x t a t x t f t x t x g t x t u t x t u t b t u t e t x t x h t x t

performance sensitivity of helicopter global and local optimal harmonic vibration controller a comparison of local and global controllers is conducted for helicopter vibration reduction using actively controlled single dual trailing edge flaps it is found that for high speed flight conditions both local and global controllers perform equally well however for low speed flight local controller gives a marginally better optimal solution compared to the global controller but at the cost of much greater computation time an objective function containing the hub vibratory loads is minimized using the global controller the control effort is weighted such that each flap is used to complete authority a reduction of 73 and 75 in hub vibration is obtained at high speed flight using single and dual flap configurations respectively at low speed flight single and dual flap configurations gave 52 and 63 reduction in vibration respectively numerical studies using an aeroelastic simulation show that the vibration control system performs well up to 5 degrees error in flap phasing from optimal phase the control law is insensitive to small perturbations up to 0 3 deg in the optimal flap harmonics and is fairly insensitive to stiffness and mass changes of the rotor blade as well as noise in the measured hub load data dual flap configuration was found to be more insensitive to errors in optimal flap control harmonics than single flap configuration

nonlinear galerkin methods this article presents a new method of integrating evolution differential equations the non linear galerkin method that is well adapted to the long term integration of such equations while the usual galerkin method can be interpreted as a projection of the considered equation on a linear space the methods considered here are related to the projection of the equation on a nonlinear manifold from the practical point of view some terms have been identified as small and sometimes but not always disregarded