spectral convergence for orthogonal polynomials on triangles we study the behavior of orthogonal polynomials on triangles and their coefficients in the context of spectral approximations of partial differential equations in these spectral approximations one studies series expansions u sum k 0 infty hat u k phi k where the phi k are orthogonal polynomials our results show that for any function u in c infty the series expansion converges faster than with polynomial order

combined direct and indirect approach to adaptive control an approach to adaptive control is introduced using a combination of both direct and indirect methods on the basis of estimates of the plant parameters and the current values of the control parameters closed loop estimation errors epsilon sub theta t and epsilon sub k t are defined these in turn are used in the adaptive laws for updating both identification and control parameters the global uniform stability of the overall system is shown by constructing a lyapunov function only the control of a first order plant is treated in detail to get across the principal concepts involved in particular the dynamic adjustment of the control parameters based on the estimates of the plant parameters is introduced as an indirect method and is the precursor of the combined method

robust exact differentiation via sliding mode technique abstract the main problem in differentiator design is to combine differentiation exactness with robustness in respect to possible measurement errors and input noises the proposed differentiator provides for proportionality of the maximal differentiation error to the square root of the maximal deviation of the measured input signal from the base signal such an order of the differentiation error is shown to be the best possible one when the only information known on the base signal is an upper bound for lipschitz s constant of the derivative

a mesh free numerical method for solution of the family of kuramoto sivashinsky equations in this paper radial basis function rbfs based mesh free method is implemented to find numerical solution of the kuramoto sivashinsky equations this approach has an edge over traditional methods such as finite difference and finite element methods because it does not require a mesh to discretize the problem domain and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method the accuracy of the method is assessed in terms of the error norms l 2 l number of nodes in the domain of influence free parameter dependent parameter rbfs and time step length numerical experiments demonstrate accuracy and robustness of the method for solving a class of nonlinear partial differential equations

numerical study of pseudospectral methods in shock wave applications abstract in this paper we use shock capturing spectral methods to simulate compressible flows in the presence of shock waves three applications are considered the first is an interaction of a one dimensional shock with an entropy wave the second case deals with interactions between shock wave and an entropy wave in two space dimensions and the third case is two dimensional shock vortex interactions the first two applications are found in the study of turbulence in high speed flow the last application is a key element in understanding the acoustic dynamics in aeroacoustic and in design of supersonic jet in aerodynamics the purpose of this study is to show the feasibility of simulating shocked flow with spectral methods the numerical methods involved are the chebyshev and fourier collocation methods the euler equations of gas dynamics is discretized by pseudospectral collocation methods in space and a nonlinearly stable third order runge kutta methods in time the fluxes are evaluated pointwise directly and not by the cell averaging technique a significant reduction in cpu time and storage usage are achieved by incorporating several well established numerical techniques such as grid transformation and filtering into the spectral algorithm the results of this study indicate that spectral method is well suited not only for smooth problems but also for those with discontinuity

convergence results for pseudospectral approximations of hyperbolic systems by a penalty type boundary treatment a new method of imposing boundary conditions in the pseudospectral approximation of hyperbolic systems of equations is proposed it is suggested to collocate the equations not only at the inner grid points but also at the boundary points and use the boundary conditions as penalty terms in the pseudo spectral legrendre method with the new boundary treatment a stability analysis for the case of a constant coefficient hyperbolic system is presented and error estimates are derived

on the gibbs phenomenon i recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function it is well known that the fourier series of an analytic and periodic function truncated after 2n 1 terms converges exponentially with n even in the maximum norm it is also known that if the function is not periodic the rate of convergence deteriorates in particular there is no convergence in the maximum norm although the function is still analytic this is known as the gibbs phenomenon in this paper we show that the first 2n 1 fourier coefficients contain enough information about the function so that an exponentially convergent approximation in the maximum norm can be constructed the proof is a constructive one and makes use of the gegenbauer polynomials c n x it consists of two steps in the first step we show that the first m coefficients of the gegenbauer expansion based on c n x for 0 n m of any l2 function can be obtained within exponential accuracy provided that both and m are proportional to but smaller than n in the second step we construct the gegenbauer expansion based on c n 0 n m from the coefficients found in the first step we show that this series converges exponentially with n provided that the original function is analytic though nonperiodic thus we prove that the gibbs phenomenon can be completely overcome

chebyshev legendre super spectral viscosity method for nonlinear conservation laws in this paper a super spectral viscosity method using the chebyshev differential operator of high order d s sqrt 1 x 2 opx s is developed for nonlinear conservation laws the boundary conditions are treated by a penalty method compared with the second order spectral viscosity method the super one is much weaker while still guaranteeing the convergence of the bounded solution of the chebyshev galerkin chebyshev collocation or legendre galerkin approximations to nonlinear conservation laws which is proved by compensated compactness arguments

the interrelation between stochastic differential inclusions and set valued stochastic differential equations abstract in this paper we connect the well established theory of stochastic differential inclusions with a new theory of set valued stochastic differential equations solutions to the latter equations are understood as continuous mappings taking on their values in the hyperspace of nonempty bounded convex and closed subsets of the space l 2 consisting of square integrable random vectors we show that for the solution x to a set valued stochastic differential equation corresponding to a stochastic differential inclusion there exists a solution x for this inclusion that is a l 2 continuous selection of x this result enables us to draw inferences about the reachable sets of solutions for stochastic differential inclusions as well as to consider the viability problem for stochastic differential inclusions

spectral simulation of supersonic reactive flows we present in this paper numerical simulations of reactive flows interacting with shock waves we argue that spectral methods are suitable for these problems and review the recent developments in spectral methods that have made them a powerful numerical tool appropriate for long term integrations of complicated flows even in the presence of shock waves a spectral code is described in detail and the theory that leads to each of its components is explained results of interactions of hydrogen jets with shock waves are presented and analyzed and comparisons with eno finite difference schemes are carried out

numerical solution of complex modified korteweg de vries equation by petrov galerkin method abstract in this paper a petrov galerkin method is used to derive a numerical scheme for the complex modified korteweg de vries equation cmkdv where we have used cubic b splines as test functions and linear b splines as trial functions an implicit midpoint rule is used to discretize in time a block non linear pentadiagonal system is obtained we solve this system by newton s method the resulting scheme has a fourth order accuracy in space direction and second order in time direction it is unconditionally stable by von neumann method the exact soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness of the scheme the interaction of two solitary waves for different parameters is discussed

polynomial interpolation results in sobolev spaces abstract in this paper we derive new results related to the lagrange polynomial interpolation on the nodes of gauss type quadrature formulas in particular we obtain asymptotic error estimates of the same order as the distance to the best fit in the polynomial space

brief paper adaptive model predictive control for a class of constrained linear systems based on the comparison model this paper proposes an adaptive model predictive control mpc algorithm for a class of constrained linear systems which estimates system parameters on line and produces the control input satisfying input state constraints for possible parameter estimation errors the key idea is to combine the robust mpc method based on the comparison model with an adaptive parameter estimation method suitable for mpc to this end first a new parameter update method based on the moving horizon estimation is proposed which allows to predict an estimation error bound over the prediction horizon second an adaptive mpc algorithm is developed by combining the on line parameter estimation with an mpc method based on the comparison model suitably modified to cope with the time varying case this method guarantees feasibility and stability of the closed loop system in the presence of state input constraints a numerical example is given to demonstrate its effectiveness

martingale problem to stratonovich stochastic inclusion abstract in the paper we consider a martingale problem approach to stratonovich stochastic inclusion the proposed method enables us to investigate both the existence and topological properties of the set of weak solutions to such inclusions

a legendre petrov galerkin method for fourth order differential equations in this paper we present a legendre petrov galekin method for one dimensional linear fourth order differential equations a legendre petrov galerkin and chebyshev collocation method is developed for the nonlinear kuramoto sivashinsky equation numerical results are presented to demonstrate the efficiency of the proposed schemes and optimal rates of convergence in the l 2 norm are rigorously derived

optimal error estimates of the legendre petrov galerkin method for the korteweg de vries equation in this paper the legendre petrov galerkin method for the korteweg de vries equation with nonperiodic boundary conditions is analyzed the nonlinear term is computed with the legendre spectral method and some pseudospectral methods respectively optimal error estimates in l2 norm are obtained for both semidiscrete and fully discrete schemes the method is also applicable to some 2m 1 th order differential equations

convergence analysis of spectral collocation methods for a singular differential equation solutions of partial differential equations with coordinate singularities often have special behavior near the singularities which forces them to be smooth special treatment for these coordinate singularities is necessary in spectral approximations in order to avoid degradation of accuracy and efficiency it has been observed numerically in the past that for a scheme to attain high accuracy it is unnecessary to impose all the pole conditions the constraints representing the special solution behavior near singularities in this paper we provide a theoretical justification for this observation specifically we consider an existing approach which uses a pole condition as the boundary condition at a singularity and solves the reformulated boundary value problem with a commonly used gauss lobatto collocation scheme spectral convergence of the legendre and chebyshev collocation methods is obtained for a singular differential equation arising from polar and cylindrical geometries

crank nicolson quasi wavelets method for solving fourth order partial integro differential equation with a weakly singular kernel in this paper we study a novel numerical scheme for the fourth order partial integro differential equation with a weakly singular kernel in the time direction a crank nicolson time stepping is used to approximate the differential term and the product trapezoidal method is employed to treat the integral term and the quasi wavelets numerical method for space discretization our interest in the present paper is a continuation of the investigation in yang et al 33 where we study discretization in time by using the forward euler scheme the comparisons of present results with the previous ones show that the present scheme is more stable and efficient for numerically solving the fourth order partial integro differential equation with a weakly singular kernel we also tested the method proposed on several one and two dimensional problems with very promising results besides in order to demonstrate the power of the quasi wavelets method in comparison with standard discretization methods we also consider the high frequency oscillation problems with the integro differential term

legendre pseudospectral viscosity method for nonlinear conservation laws in this paper the legendre spectral viscosity sv method for the approximate solution of initial boundary value problems associated with nonlinear conservation laws is studied the authors prove that by adding a small amount of sv bounded solutions of the legendre sv method converge to the exact scalar entropy solution the convergence proof is based on compensated compactness arguments and therefore applies to certain 2 times 2 systems finally numerical experiments for scalar as well as the one dimensional system of gas dynamics equations are presented which confirm the convergence of the legendre sv method moreover these numerical experiments indicate that by post processing the sv approximation one can recover the entropy solution within spectral accuracy

wavelet based system identification for nonlinear control compactly supported orthogonal wavelets have certain properties that are useful for system identification and learning control drawing on the rich theory of wavelets we propose a system identification scheme based on orthogonal wavelets better accuracy of estimation can be obtained by adding more terms to the wavelet based identifier and these terms do not alter the coefficients of the existing terms these terms can be selectively added depending upon the region of interest for example we may require more terms in regions where the identified functions vary rapidly we illustrate the concepts by applying the procedure to determine a speed controller for an electric vehicle

spectral viscosity approximations to hamilton jacobi solutions the spectral viscosity approximate solution of convex hamilton jacobi equations with periodic boundary conditions is studied it is proved in this paper that the approximation and its gradient remain uniformly bounded formally spectral accurate and converge to the unique viscosity solution the l1 convergence rate of the order 1 varepsilon forall varepsilon 0 is obtained

concurrent learning for parameter estimation using dynamic state derivative estimators a concurrent learning cl based parameter estimator is developed to identify the unknown parameters in a nonlinear system unlike state of the art cl techniques that assume knowledge of the state derivative or rely on numerical smoothing cl is implemented using a dynamic state derivative estimator a novel purging algorithm is introduced to discard possibly erroneous data recorded during the transient phase for cl asymptotic convergence of the error states to the origin is established under a persistent excitation condition and the error states are shown to be uniformly ultimately bounded under a finite excitation condition

some results about the pseudospectral approximation of one dimensional fourth order problems we analyze the pseudospectral approximation of fourth order problems we give convergence results in the one dimensional case numerical experiments are shown in two dimensions for the approximation of the rhombic plate bending problem eigenvalues and preconditioning are also investigated

the chebyshev legendre method implementing legendre methods on chebyshev points a new collocation method for the numerical solution of partial differential equations is presented this method uses the chebyshev collocation points but because of the way the boundary conditions are implemented it has all the advantages of the legendre methods in particular l 2 estimates can be easily obtained for hyperbolic and parabolic problems

model based reinforcement learning for approximate optimal regulation reinforcement learning rl based online approximate optimal control methods applied to deterministic systems typically require a restrictive persistence of excitation pe condition for convergence this paper develops a concurrent learning cl based implementation of model based rl to solve approximate optimal regulation problems online under a pe like rank condition the development is based on the observation that given a model of the system rl can be implemented by evaluating the bellman error at any number of desired points in the state space in this result a parametric system model is considered and a cl based parameter identifier is developed to compensate for uncertainty in the parameters uniformly ultimately bounded regulation of the system states to a neighborhood of the origin and convergence of the developed policy to a neighborhood of the optimal policy are established using a lyapunov based analysis and simulation results are presented to demonstrate the performance of the developed controller

nonlinear variation of parameters formula for set differential equations in a metric space nonlinear variation of parameters formula for set differential equations in a metric space is developed investigating necessary tools such as differentiability of solutions relative to initial values other needed results are also proven

on the gibbs phenomenon iv recovering exponential accuracy in a subinterval from a gegenbauer partial sum of a piecewise analytic function we continue the investigation of overcoming gibbs phenomenon i e obtaining exponential accuracy at all points including at the discontinuities themselves from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function we show that if we are given the first n expansion coefficients of an l 2 function f x in terms of either the trigonometrical polynomials or the chebyshev or legendre polynomials we can construct an exponentially convergent approximation to the point values of f x in any sub interval in which it is analytic

set valued stochastic integral equations driven by martingales abstract we consider a notion of set valued stochastic lebesgue stieltjes trajectory integral and a notion of set valued stochastic trajectory integral with respect to martingale then we use these integrals in a formulation of set valued stochastic integral equations the existence and uniqueness of the solution to such the equations is proven as a generalization of set valued case results we consider the fuzzy stochastic trajectory integrals and investigate the fuzzy stochastic integral equations driven by bounded variation processes and martingales

on the gibbs phenomenon v recovering exponential accuracy from collocation point values of a piecewise analytic function this paper presents a method to recover exponential accuracy at all points including at the discontinuities themselves from the knowledge of an approximation to the interpolation polynomial or trigonometrical polynomial we show that if we are given the collocation point values or a highly accurate approximation at the gauss or gauss lobatto points we can reconstruct an uniform exponentially convergent approximation to the function f x in any sub interval of analyticity the proof covers the cases of fourier chebyshev legendre and more general gegenbauer collocation methods a numerical example is also provided

a new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations a new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested this method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions stability and convergence results are proven for the chebyshev approximation of linear scalar hyperbolic equations the eigenvalues of this method applied to parabolic equations are shown to be real and negative

a legendre petrov galerkin and chebyshev collocation method for third order differential equations a legendre petrov galerkin lpg method for the third order differential equation is developed by choosing appropriate base functions the method can be implemented efficiently also this new approach enables us to derive an optimal rate of convergence in l2 norm the method is applied to some nonlinear problems such as the korteweg de vries kdv equation with the chebyshev collocation treatment for the nonlinear term it is a legendre petrov galerkin and chebyshev collocation lpg cc method numerical experiments are given to confirm the theoretical result

spectral viscosity approximations to multidimensional scalar conservation laws the authors study the spectral viscosity sv method in the context of multidimensional scalar conservation laws with periodic boundary conditions they show that the spectral viscosity which is sufficiently small to retain the formal spectral accuracy of the underlying fourier approximation is large enough to enforce the correct amount of entropy dissipation which is otherwise missing in the standard fourier method moreover they prove that because of the presence of the spectral viscosity the truncation error in this case becomes spectrally small independent of whether the underlying solution is smooth or not consequently the sv approximation remains uniformly bounded and converges to a measure valued solution satisfying the entropy condition that is the unique entropy solution they also show that the sv solution has a bounded total variation provided that the total variation of the initial data is bounded thus confirming its strong convergence to the entropy solution they obtain an l sup 1 convergence rate of the usual optimal order one half 22 refs

efficient spectral galerkin method ii direct solvers of second and fourth order equations using chebyshev polynomials efficient direct solvers based on the chebyshev galerkin methods for second and fourth order equations are presented they are based on appropriate base functions for the galerkin formulation that lead to discrete systems with special structured matrices that can be efficiently inverted numerical results indicate that the direct solvers presented in this paper are significantly more accurate and efficient than those based on the chebyshev tau method

a class of methods based on non polynomial spline functions for the solution of a special fourth order boundary value problems with engineering applications we use a quintic non polynomial spline functions to develop a numerical method for computing approximations to the solution of a system of fourth order boundary value problems associated with plate deflection theory we show that the present family of methods gives better approximations and generalize all the existing finite difference and spline functions based methods up to order six convergence of the methods is shown through standard convergence analysis numerical exampls are given to illustrate the applicability and efficiency of the new method

sparse spectral approximations of high dimensional problems based on hyperbolic cross hyperbolic cross approximations by some classical orthogonal polynomials functions in both bounded and unbounded domains are considered in this paper optimal error estimates in proper anisotropic weighted korobov spaces for both regular hyperbolic cross approximations and optimized hyperbolic cross approximations are established these fundamental approximation results indicate that spectral methods based on hyperbolic cross approximations can be effective for treating certain high dimensional problems and will serve as basic tools for analyzing sparse spectral methods in high dimensions

differentiable selections of multifunctions and their applications abstract in the paper we present some selection properties of differentiable multifunctions next we introduce the definition of a set valued stratonovich stochastic integral finally using selection properties of differentiable multifunctions we discuss the existence of weak solutions to stochastic inclusions with respect to such an integral

analysis of the spectral vanishing viscosity method for periodic conservation laws the convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid burgers equation is analyzed it is proved that this kind of vanishing viscosity is responsible for spectral decay of those fourier coefficients located toward the end of the computed spectrum consequently the discretization error is shown to be spectrally small independently of whether or not the underlying solution is smooth this in turn implies that the numerical solution remains uniformly bounded and convergence follows by compensated compactness arguments

on one sided filters for spectral fourier approximations of discontinuous functions in this paper the existence of one sided filters is proved for spectral fourier ap proximations of discontinuous functions which can recover spectral accuracy up to the discontinuity from one side a least square procedure is used to construct such a filter and is tested on several discontinuous functions numerically

adaptive model predictive control for constrained nonlinear systems in this paper a method is proposed for the adaptive model predictive control of constrained nonlinear system rather than relying on the inherent robustness properties of standard nmpc the developed technique explicitly account for the transient effect of parametric estimation error by combining a parameter adjustment mechanism with robust mpc algorithms the parameter estimation routine employed guarantees non increase of the estimation error vector this means that the controller employs a process model which approaches that of the true system over time these estimates are used to update the parameter uncertainty set at every time step resulting in a gradual reduction in the conservative and or computational effects of the incorporated robust features

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

a legendre spectral galerkin method for the biharmonic dirichlet problem a legendre spectral galerkin method is presented for the solution of the biharmonic dirichlet problem on a square the solution and its laplacian are approximated using the set of basis functions suggested by shen which are linear combinations of two legendre polynomials a schur complement approach is used to reduce the resulting linear system to one involving the approximation of the laplacian of the solution on the two vertical sides of the square the schur complement system is solved by a preconditioned conjugate gradient method or the cholesky method the total cost of the algorithm is o n3 numerical results demonstrate the spectral convergence of the method

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

convergence of spectral methods for nonlinear conservation laws we discuss the convergence of fourier methods for scalar nonlinear conservation laws that exhibit spontaneous shock discontinuities numerical tests indicate that the convergence may and in fact in some cases we prove it must fail with or without post processing of the numerical solution instead we introduce here a new kind of spectrally accurate vanishing viscosity to augment the fourier approximation of such nonlinear conservation laws using compensated compactness arguments augmented by the assumption of l infty stability we show for the inviscid burgers model equation that this spectral viscosity method prevents oscillations and convergence to the unique entropy solution follows