comparison of systems with complex behavior abstract we present a formalism for comparing the asymptotic dynamics of dynamical systems with physical systems that they model based on the spectral properties of the koopman operator we first compare invariant measures and discuss this in terms of a statistical takens theorem proved here we also identify the need to go beyond comparing only invariant ergodic measures of systems and introduce an ergodic theoretic treatment of a class of spectral functionals that allow for this the formalism is extended for a class of stochastic systems discrete random dynamical systems the ideas introduced in this paper can be used for parameter identification and model validation of driven nonlinear models with complicated behavior as an illustration we provide an example in which we compare the asymptotic behavior of a combustion system measured experimentally with the asymptotic behavior of a class of models that have the form of a random dynamical system

complexity of a quasiperiodically driven spin system a quasiperiodically forced purely quantum spin system can exhibit rather irregular behaviour with a continuous power spectrum we have shown analytically and numerically that in these regimes both the topological entropy of a corresponding symbolic sequence and the maximal lyapunov exponent vanish

adjoint complement to viscous finite volume pressure correction methods abstract a hybrid adjoint navier stokes method for the pressure based computation of hydrodynamic objective functional derivatives with respect to the shape is systematically derived in three steps the underlying adjoint partial differential equations and boundary conditions for the frozen turbulence reynolds averaged navier stokes equations are considered in the first step in step two the adjoint discretisation is developed from the primal unstructured finite volume discretisation such that adjoint consistent approximations to the adjoint partial differential equations are obtained following a so called hybrid adjoint approach a unified discrete boundary description is outlined that supports high and low reynolds number turbulent wall boundary treatments for both the adjoint boundary condition and the boundary based gradient formula the third component focused in the development of the industrial adjoint cfd method is the adjoint counterpart to the primal pressure correction algorithm the approach is verified against the direct differentiation method and an application to internal flow problems is presented

numerical study of the turbulent flow past an airfoil with trailing edge separation presentation d une methode numerique par volume fini pour la resolution des equations de navier stokes bidimensionnelles incompressibles et stationnaires en coordonnees generales curvilignes application de la methode aux ecoulements turbulents sur des profils avec et sans separation au bord de sortie posterieur comparaison des calculs avec des donnees experimentales

hermite spectral methods with a time dependent scaling for parabolic equations in unbounded domains hermite spectral methods are investigated for linear diffusion equations and nonlinear convection diffusion equations in unbounded domains when the solution domain is unbounded the diffusion operator no longer has a compact resolvent which makes the hermite spectral methods unstable to overcome this difficulty a time dependent scaling factor is employed in the hermite expansions which yields a positive bilinear form as a consequence stability and spectral convergence can be established for this approach the present method plays a similar role in the stability of the similarity transformation technique proposed by funaro and kavian math comp 57 1991 pp 597 619 however since coordinate transformations are not required the present approach is more efficient and is easier to implement in fact with the time dependent scaling the resulting discretization system is of the same form as that associated with the classical straightforward but unstable hermite spectral method numerical experiments are carried out to support the theoretical stability and convergence results

a new generalized laguerre spectral approximation and its applications a new family of generalized laguerre polynomials is introduced various orthogonal projections are investigated some approximation results are established as an example of their important applications the mixed spherical harmonic generalized laguerre approximation is developed a mixed spectral scheme is proposed for a three dimensional model problem its convergence is proved numerical results demonstrate the high accuracy of this new spectral method

on the proper treatment of grid sensitivities in continuous adjoint methods for shape optimization abstract the continuous adjoint method for shape optimization problems in flows governed by the navier stokes equations can be formulated in two different ways each of which leads to a different expression for the sensitivity derivatives of the objective function with respect to the control variables the first formulation leads to an expression including only boundary integrals it thus has low computational cost but when used with coarse grids its accuracy becomes questionable the second formulation comprises a sum of boundary and field integrals due to the field integrals it has noticeably higher computational cost obtaining though higher accuracy in this paper the equivalence of the two formulations is revisited from the mathematical and particularly the numerical point of view internal and external aerodynamics cases in which the objective function is either the total pressure losses or the force exerted on a solid body are examined and differences in the computed gradients are discussed after identifying the reason behind these discrepancies the adjoint formulation is enhanced by the adjoint to a hypothetical grid displacement model and the new approach is proved to reproduce the accuracy of the second adjoint formulation while maintaining the low cost of the first one

rational legendre pseudospectral approach for solving nonlinear differential equations of lane emden type lane emden equation is a nonlinear singular equation in the astrophysics that corresponds to the polytropic models in this paper a pseudospectral technique is proposed to solve the lane emden type equations on a semi infinite domain the method is based on rational legendre functions and gauss radau integration the method reduces solving the nonlinear ordinary differential equation to solve a system of nonlinear algebraic equations the comparison of the results with the other numerical methods shows the efficiency and accuracy of this method

a generalization of the wiener rational basis functions on infinite intervals part ii numerical investigation in part i we introduced the generalized wiener rational basis functions and here in part ii we continue our investigation with numerical experiments wiener s generalized basis can utilize the fast fourier transform for integer values of the decay parameter s we outline two algorithms for doing so in addition the issue of galerkin representations for polynomial nonlinearities of expansions is addressed the wiener basis set is compared against domain truncation methods fourier and chebyshev polynomials hermite functions sinc interpolations and mapped chebyshev expansions and we show that for both exponentially and algebraically decaying functions the wiener approximation is as good as or superior to these alternatives in addition we carry out preliminary investigations regarding tuning of the decay parameter s numerical simulations of korteweg de vries type equations show the effectiveness of the wiener expansion we also explore the practical use of the wiener basis functions on the semi infinite interval which is compared against laguerre function methods and other jacobi polynomial mappings

quasi genericity of bifurcations to high dimensional invariant tori for maps we consider a family of maps in a banach spacee near the situation when the derivative at the fixed point has two pairs of complex eigenvalues lying on the unit circle the other part of the spectrum being strictly inside the unit disc we focus our attention on the region of the parameter space where the truncated normal form of the maps shows a bifurcation of a family of invariantt 1 circles into a family of invariantt 2 tori we show that this problem needs a 3 dimensional parameter unfolding and that for the complete maps bifurcation occurs at points where is the rotation number on the non normally hyperbolict 1 circle ande 2i are the eigenvalues of the constant matrix conjugated to the non contracting part of the linearization on the normal fiber bundle overt 1 making some non resonance and diophantine assumptions on leading to a positive measure cantor set int 2 we show that in paraboloidal regions of the 3 dim parameter space we have clean bifurcations as for the truncated normal form the complement of these regions forms a set of bubbles such as the ones obtained by chenciner in chen for a codimension 2 problem for maps in 2 the main tool here is a generalization for a matrix function ont 1 close to a constant of the quasi conjugacy to a constant modulo a minimum of additional parameters moved quasi conjugacy for the infinite dimensional case we use ac decoupling result on the angular dependent linear parts into a contraction still angular dependent and another part quasi conjugated to a constant matrix this type of analysis applies for a wide range of problems where truncated normal forms of the maps give bifurcations fromt n tot n 1 tori and this needs a n 1 dimensional parameter unfolding

laguerre pseudospectral approximation to the thomas fermi equation we propose an iterative method to solve the nonlinear thomas fermi equation based on laguerre pseudospectral approximation different from the direct use of pseudospectral methods for solving the problem in the literature we represent the solution of the thomas fermi equation as sum of two parts due to its singularity at the origin both parts regard the initial slope as a parameter unknown to be determined one singular part is a power series expansion the other smooth part satisfies a nonlinear two point boundary value problem we use a laguerre pseudospectral method to discretize and solve it numerically by the newton iteration moreover an introduced new boundary condition is employed innovatively to construct an outer iteration to determine the initial slope we show numerically this separation of solution as two parts is necessary for improvement of accuracy by direct laguerre approximations of the thomas fermi equation comparisons of our numerical results with those obtained by other methods show effectiveness and accuracy of our approach

reduction of the adjoint gradient formula in the continuous limit we present a new continuous adjoint method for aerodynamic shape optimization using the euler equations which reduces the computational cost of the gradients by reducing the volume integral part of the adjoint gradient formula to a surface integral the savings are particularly significant for three dimensional aerodynamic shape optimization problems on general unstructured and overset meshes in order to validate the concept the new gradient equations have been tested for various aerodynamic shape optimization problems including an inverse problem for threedimensional wing configurations and drag minimization problems of a single element airfoil and a three dimensional wing fuselage configuration in order to assess their accuracy the results are compared with finite difference gradients complex step gradients and gradients calculated by the previous adjoint method which includes a volume integral thomas v jones professor of engineering department of aeronautics and astronautics stanford university aiaa fellow postdoctoral fellow department of aeronautics and astronautics stanford university aiaa member

symbolic dynamics behind the singular continuous power spectra of continuous flows abstract we investigate a class of dissipative flows which possess singular continuous fractal power spectra the attractor for these flows contains a saddle point so that the times between the returns of an orbit onto a secant plane are unbounded as a result the poincare map does not reflect precisely the qualitative dynamics of the underlying flow the power spectrum of this map can be discrete we introduce the construction which takes into account that the time intervals between the intersections of the poincare plane can vary and construct a symbolic representation for the continuous process with logarithmic divergence in the distribution of such intervals spectral properties of the resulting symbolic sequences are shown to reproduce rather well those of the original flow

adjoint based error estimation and adaptive mesh refinement for the rans and k ω turbulence model equations abstract in this article we present the extension of the a posteriori error estimation and goal oriented mesh refinement approach from laminar to turbulent flows which are governed by the reynolds averaged navier stokes and k turbulence model rans k equations in particular we consider a discontinuous galerkin discretization of the rans k equations and use it within an adjoint based error estimation and adaptive mesh refinement algorithm that targets the reduction of the discretization error in single as well as in multiple aerodynamic force coefficients the accuracy of the error estimation and the performance of the goal oriented mesh refinement algorithm is demonstrated for various test cases including a two dimensional turbulent flow around a three element high lift configuration and a three dimensional turbulent flow around a wing body configuration

numerical integration based on laguerre gauss interpolation abstract in this paper we propose two efficient numerical integrators for ordinary differential equations based on modified laguerre gauss interpolations the global convergence of proposed algorithms is proved numerical results demonstrate the spectral accuracy of these new schemes and agree well with the theoretical analysis

discrete adjoint based approach for optimization problems on three dimensional unstructured meshes a comprehensive strategy for developing and implementing discrete adjoint methods for aerodynamic shape optimization problems is presented by linearizing each procedure in the entire optimization problem transposing each linearization and reversing the sequential order of operations the adjoint of the complete optimization problem including flow equations and mesh motion equations is constructed in a modular and verifiable fashion this construction is also shown to produce minimal memory overheads and retain the same convergence characteristics of the original analysis problem in the sensitivity analysis these techniques are implemented in a three dimensional unstructured multigrid navierstokes solver and demonstrated on a transonic drag reduction problem for a wing body configuration

modified laguerre pseudospectral method refined by multidomain legendre pseudospectral approximation a modified laguerre pseudospectral method is proposed for differential equations on the half line the numerical solutions are refined by multidomain legendre pseudospectral approximation numerical results show the spectral accuracy of this approach some approximation results on the modified laguerre and legendre interpolations are established the convergence of proposed method is proved

integration processes of ordinary differential equations based on laguerre radau interpolations in this paper we propose two integration processes for ordinary differential equations based on modified laguerre radau interpolations which are very efficient for long time numerical simulations of dynamical systems the global convergence of proposed algorithms are proved numerical results demonstrate the spectral accuracy of these new approaches and coincide well with theoretical analysis

the spectral method for high order problems with proper simulations of asymptotic behaviors at infinity in this paper we investigate new spectral and multidomain spectral methods for high order problems we introduce a family of new generalized laguerre functions which are mutually orthogonal with the weight function x a d x c d 0 a and c being arbitrary real numbers the corresponding quasi orthogonal approximation and laguerre gauss radau type interpolation are proposed the spectral and multidomain spectral schemes are provided for several model problems which not only fit the mixed inhomogeneous boundary conditions on the fixed boundary exactly but also match the asymptotic behaviors at infinity reasonably numerical results demonstrate the efficiency of suggested algorithms and confirm the analysis well

mixed laguerre legendre pseudospectral method for incompressible fluid flow in an infinite strip in this paper we investigate the mixed laguerre legendre interpolation approximation and its application some approximation results are established a mixed laguerre legendre pseudospectral scheme is constructed for incompressible fluid flow in an infinite strip its stability and convergence are proved numerical results show the efficiency of this new approach

topology optimization of creeping fluid flows using a darcy stokes finite element a new methodology is proposed for the topology optimization of fluid in stokes flow the binary design variable and no slip condition along the solid fluid interface are regularized to allow for the use of continuous mathematical programming techniques the regularization is achieved by treating the solid phase of the topology as a porous medium with flow governed by darcy s law fluid flow throughout the design domain is then expressed as a single system of equations created by combining and scaling the stokes and darcy equations the mixed formulation of the new darcy stokes system is solved numerically using existing stabilized finite element methods for the individual flow problems convergence to the no slip condition is demonstrated by assigning a low permeability to solid phase and results suggest that auxiliary boundary conditions along the solid fluid interface are not needed the optimization objective considered is to minimize dissipated power and the technique is used to solve examples previously examined in literature the advantages of the darcy stokes approach include that it uses existing stabilization techniques to solve the finite element problem it produces 0 1 void solid topologies i e there are no regions of artificial material and that it can potentially be used to optimize the layout of a microscopically porous material copyright 2005 john wiley sons ltd

time dependent hermite galerkin spectral method and its applications a time dependent hermite galerkin spectral method thgsm is investigated in this paper for the nonlinear convection diffusion equations in the unbounded domains the time dependent scaling factor and translating factor are introduced in the definition of the generalized hermite functions ghf as a consequence the thgsm based on these ghf has many advantages not only in theoretical proofs but also in numerical implementations the stability and spectral convergence of our proposed method have been established in this paper the korteweg de vries burgers kdvb equation and its special cases including the heat equation and the burgers equation as the examples have been numerically solved by our method the numerical results are presented and it surpasses the existing methods in accuracy our theoretical proof of the spectral convergence has been supported by the numerical results

effect of approximations of the discrete adjoint on gradient based optimization an exact discrete adjoint of an unstructured nite volume solver for the rans equations has been developed the adjoint is exact in the sense of being based on the full linearization of all terms in the solver including all turbulence model contributions from this starting point various approximations to the adjoint are derived with the intention of simplifying the development and memory requirements of the method considered are many approximations already seen in the literature the eect of these approximations on the accuracy of the resulting design gradients and the convergence and nal solution of optimizations is studied as it applies to a two dimensional high lift conguration

algorithm developments for discrete adjoint methods this paper presents a number of algorithm developments for adjoint methods using the discrete approach in which the discretisation of the non linear equations is linearised and the resulting matrix is then transposed with a new iterative procedure for solving the adjoint equations exact numerical equivalence is maintained between the linear and adjoint discretisations the incorporation of strong boundary conditions within the discrete approach is discussed as well as a new application of adjoint methods to linear unsteady flow in turbomachinery

a rational spectral method for the kdv equation on the half line a petrov galerkin method using orthogonal rational functions is proposed for the korteweg de vries kdv equation on the half line with initial boundary values the nonlinear term and the right hand side term are treated by chebyshev rational interpolation explicitly and the linear terms are computed with the galerkin method implicitly such an approach is applicable using fast algorithms numerical results are presented for problems with both exponentially and algebraically decaying solutions respectively highlighting the performance of the proposed method

strange non chaotic attractor in a quasiperiodically forced circle map abstract we show that in the quasiperiodically forced circle map strange non chaotic attractors can appear for non linearities far from the border of chaos the destruction of a two frequency quasiperiodic torus connected with the appearance of a strange non chaotic attractor is described in detail this strange non chaotic attractor is characterized by logarithmically slow diffusion of the phase it is shown that in this regime the high order phase locking states disappear and the rotation number varies rather smoothly with the parameters

quasi periodic stability of subfamilies of an unfolded skew hopf bifurcation in the skew hopf bifurcation a quasi periodic attractor with nontrivial normal linear dynamics loses hyperbolicity the simplest setting concerns rotationally symmetric diffeomorphisms of s1 r2 their dynamics involve periodicity quasi periodicity and chaos including mixed spectrum the present paper deals with the persistence under symmetry breaking of quasi periodic invariant circles in this bifurcation it turns out that when adding sufficiently many unfolding parameters the invariant circle persists for a large hausdorff measure subset of a submanifold in parameter space

jacobi and laguerre quasi orthogonal approximations and related interpolations in this paper we investigate jacobi quasi orthogonal approximation and generalized jacobi gauss lobatto interpolation we also propose laguerre quasi orthogonal approximation and generalized laguerre gauss radau interpolation a series of sharp results on these approximations are established which are applicable to spectral and pseudospectral methods for mixed nonhomogeneous boundary value problems of high order

accurate solution of the thomasfermi equation using the fractional order of rational chebyshev functions in this paper the nonlinear singular thomasfermi differential equation for neutral atoms is solved using the fractional order of rational chebyshev orthogonal functions frcs of the first kind ftn t l on a semi infinite domain where l is an arbitrary numerical parameter first using the quasilinearization method the equation be converted into a sequence of linear ordinary differential equations ldes and then these ldes are solved using the frcs collocation method using 300 collocation points we have obtained a very good approximation solution and the value of the initial slope y 0 1 5880710226113753127186845094239501095 highly accurate to 37 decimal places

on spectral methods for volterra type integro differential equations this paper considers the spectral methods for a volterra type integro differential equation firstly the volterra type integro differential equation is equivalently restated as two integral equations of the second kind secondly a legendre collocation method is used to solve them then the error analysis is conducted based on the l norm in addition numerical results are presented to confirm our analysis

an adjoint view on flux consistency and strong wall boundary conditions to the navier stokes equations inconsistent discrete expressions in the boundary treatment of navier stokes solvers and in the definition of force objective functionals can lead to discrete adjoint boundary treatments that are not a valid representation of the boundary conditions to the corresponding adjoint partial differential equations the underlying problem is studied for an elementary 1d advection diffusion problem first using a node centred finite volume discretisation the defect of the boundary operators in the inconsistently defined discrete adjoint problem leads to oscillations and becomes evident with the additional insight of the continuous adjoint approach a homogenisation of the discretisations for the primal boundary treatment and the force objective functional yields second order functional accuracy and eliminates the defect in the discrete adjoint boundary treatment subsequently the issue is studied for aerodynamic reynolds averaged navier stokes problems in conjunction with a standard finite volume discretisation on median dual grids and a strong implementation of noslip walls found in many unstructured general purpose flow solvers going out from a base line discretisation of force objective functionals which is independent of the boundary treatment in the flow solver two improved flux consistent schemes are presented based on either body wall defined or farfield defined control volumes they resolve the dual inconsistency the behaviour of the schemes is investigated on a sequence of grids in 2d and 3d

continuous adjoint method for unstructured grids adjoint based shape optimization methods have proven to be computationally efficient for aerodynamic problems the majority of the studies on adjoint methods have used structured grids to discretize the computational domain because of the potential advantages of unstructured grids for complex configurations in this study we have developed and validated a continuous adjoint formulation for unstructured grids the hurdles posed in the computation of the gradient for unstructured grids are resolved by using a reduced gradient formulation the methods to impose thickness constraints on unstructured grids are also discussed the results for two and three dimensional simulations of airfoils and wings in inviscid transonic flow are used to validate the design procedure finally the design procedure is applied to redesign the shape of a transonic business jet configuration we were able to reduce the inviscid drag of the aircraft from 235 to 216 counts resulting in a shock free wing although the euler equations are the focus of the study in this paper of the adjoint based approach the solution of the adjoint system and gradient formulation can be conceptually extended to viscous flows the approach presented in this study has been successfully used by the first and third authors for viscous flows using structured grids however particular aspects of the design process such as the robustness of the mesh deformation process for unstructured grids need more attention for viscous flows and are therefore the subject of ongoing research

mixed spectral method for three dimensional exterior problems using spherical harmonic and generalized laguerre functions in this paper we develop the mixed spectral method for three dimensional exterior problems using spherical harmonic and generalized laguerre functions some basic approximation results are established the mixed spectral schemes are proposed for two model problems their convergences are proved numerical results demonstrate the efficiency of this new approach

rational scaled generalized laguerre function collocation method for solving the blasius equation in this paper we propose a collocation method for solving the blasius equation the blasius equation is a third order nonlinear ordinary differential equation this approach is based on a rational scaled generalized laguerre function collocation method we also present the comparison of this work with some well known results and show that the present solution is accurate

the laguerre spectral method for solving neumann boundary value problems in this paper we propose a laguerre spectral method for solving neumann boundary value problems this approach differs from the classical spectral method in that the homogeneous boundary condition is satisfied exactly moreover a tridiagonal matrix is employed instead of the full stiffness matrix encountered in the classical variational formulation of such problems for analyzing the numerical errors some basic results on laguerre approximations are established the convergence is proved the numerical results demonstrate the efficiency of this approach

stable and efficient spectral methods in unbounded domains using laguerre functions stable and efficient spectral methods using laguerre functions are proposed and analyzed for model elliptic equations on regular unbounded domains it is shown that spectral galerkin approximations based on laguerre functions are stable and convergent with spectral accuracy in the usual not weighted sobolev spaces efficient accurate and well conditioned algorithms using laguerre functions are developed and implemented numerical results indicating the spectral convergence rate and effectiveness of these algorithms are presented

generalized laguerre interpolation and pseudospectral method for unbounded domains in this paper error estimates for generalized laguerre gauss type interpolations are derived in nonuniformly weighted sobolev spaces weighted with omega alpha beta x x alpha e beta x alpha 1 beta 0 generalized laguerre pseudospectral methods are analyzed and implemented two model problems are considered the proposed schemes keep spectral accuracy and with suitable choice of basis functions lead to sparse and symmetric linear systems

laguerre galerkin method for nonlinear partial differential equations on a semi infinite interval summary a laguerre galerkin method is proposed and analyzed for the burgers equation and benjamin bona mahony bbm equation on a semi infinite interval by reformulating these equations with suitable functional transforms it is shown that the laguerre galerkin approximations are convergent on a semi infinite interval with spectral accuracy an efficient and accurate algorithm based on the laguerre galerkin approximations to the transformed equations is developed and implemented numerical results indicating the high accuracy and effectiveness of this algorithm are presented

a general non orthogonal collocated finite volume algorithm for turbulent flow at all speeds incorporating second moment turbulence transport closure part 1 computational implementation abstract a computational procedure has been developed for predicting separated turbulent flows in complex two dimensional and three dimensional geometries the procedure is based on the fully conservative structured finite volume framework within which the volumes are non orthogonal and collocated such that all flow variables are stored at one and the same set of nodes to ease the task of discretization and to enhance the conservative property of the scheme a cartesian or datum line adapted decomposition of the velocity field has been used the solution algorithm is iterative in nature approaching the steady solution with the aid of a pressure correction scheme convection is approximated with a range of schemes among them higher order upstream weighted approximations and a tvd type muscl form the last applied principally to the transport equations governing turbulence properties effects of turbulence are represented either by two equation eddy viscosity models or by a full reynolds stress transport closure the former category includes both high and low reynolds number variants in two as well as three dimensional conditions to achieve a stable implementation of the reynolds stress equations a special interpolation practice analogous to that of rhie and chow for momentum 1 has been introduced within the general framework the procedure has been formulated so as to apply to both incompressible and compressible flows the latter may contain shocks and highly supersonic regions to achieve this range of applicability the retarded density concept has been combined with the basic pressure based algorithm to capture shock waves a full approximation multigrid scheme for convergence acceleration has been incorporated and applied in conjunction with all turbulence models including second moment closure the present first part of a twin paper focuses on numerical and turbulence modelling issues in part 2 computational results are presented for six representative applications out of fifteen recently predicted with the algorithm within an extensive validation exercise

jacobi spectral method for differential equations with rough asymptotic behaviors at infinity abstract in this paper we change differential equations on the half line to certain singular problems on finite interval and then use a specific jacobi approximation for solving the resulting problems numerically theoretical analysis and numerical results demonstrate the efficiency of this method

a generalized laguerre hermite pseudospectral method for computing symmetric and central vortex states in bose einstein condensates a generalized laguerre hermite pseudospectral method is proposed for computing symmetric and central vortex states in bose einstein condensates becs in three dimensions with cylindrical symmetry the new method is based on the properly scaled generalized laguerre hermite functions and a normalized gradient flow it enjoys three important advantages i it reduces a three dimensional 3d problem with cylindrical symmetry into an effective two dimensional 2d problem ii it solves the problem in the whole space instead of in a truncated artificial computational domain and iii it is spectrally accurate extensive numerical results for computing symmetric and central vortex states in becs are presented for one dimensional 1d bec 2d bec with radial symmetry and 3d bec with cylindrical symmetry

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

chebyshev rational spectral and pseudospectral methods on a semi infinite interval a weighted orthogonal system on the half line based on the chebyshev rational functions is introduced basic results on chebyshev rational approximations of several orthogonal projections and interpolations are established to illustrate the potential of the chebyshev rational spectral method a model problem is considered both theoretically and numerically error estimates for the chebyshev rational spectral and pseudospectral methods are established preliminary numerical results agree well with the theoretical estimates and demonstrate the effectiveness of this approach

efficient spectral collocation methods for a class of linear fredholm integro differential equations on the half line abstract in this paper an extension of the legendre spectral collocation method has been proposed for the numerical solution of a class of linear fredholm integro differential equation on the half line the properties of mapped legendre functions are first presented these properties together with the legendre gauss points are then utilized to reform the fredholm integro differential equation in semi infinite interval into a singular equation in finite interval and to reduce it to the solution of a simple matrix equation besides in order to show the efficiency and accuracy of the proposed method some numerical examples are considered and solved through a survey of three approaches namely exponential rational and logarithmic legendre functions collocation methods furthermore a comparison of the results shows that using exponential functions leads to more accurate results and faster convergence

combined hermite spectral finite difference method for the fokker planck equation the convergence of a class of combined spectral finite difference methods using hermite basis applied to the fokker planck equation is studied it is shown that the hermite based spectral methods are convergent with spectral accuracy in weighted sobolev space numerical results indicating the spectral convergence rate are presented a velocity scaling factor is used in the hermite basis and is shown to improve the accuracy and effectiveness of the hermite spectral approximation with no increase in workload some basic analysis for the selection of the scaling factors is also presented

bifurcations de points fixes elliptiques ii orbites periodiques et ensembles de cantor invariants formes normales orbites bien ordonnees estimation a priori sur c estimations lipschitziennes sur les orbites bien ordonnees elimination des problemes de bord existence d ensembles invariants d aubry mather ensembles translates d aubry mather

the spectral methods for parabolic volterra integro differential equations in this paper we study the numerical solutions to parabolic volterra integro differential equations in one dimensional bounded and unbounded spatial domains in a bounded domain the given parabolic volterra integro differential equation is converted to two equivalent equations then a legendre collocation method is used to solve them and finally a linear algebraic system is obtained for an unbounded case we use the algebraic mapping to transfer the problem on a bounded domain and then apply the same presented approach for the bounded domain in both cases some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method

the sinc collocation method for solving the thomas fermi equation a numerical technique for solving nonlinear ordinary differential equations on a semi infinite interval is presented we solve the thomas fermi equation by the sinc collocation method that converges to the solution at an exponential rate this method is utilized to reduce the nonlinear ordinary differential equation to some algebraic equations this method is easy to implement and yields very accurate results

legendre laguerre coupled spectral element methods for second and fourth order equations on the half line some legendre spectral element laguerre spectral coupled methods are proposed to numerically solve second and fourth order equations on the half line the proposed methods are based on splitting the infinite domain into two parts then using the legendre spectral element method in the finite subdomain and laguerre method in the infinite subdomain c 0 or c 1 continuity according to the problem under consideration is imposed to couple the two methods rigorous error analysis is carried out to establish the convergence of the method more importantly an efficient computational process is introduced to solve the discrete system several numerical examples are provided to confirm the theoretical results and the efficiency of the method

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 laguerre legendre spectral method for the stokes problem in a semi infinite channel a mixed spectral method is proposed and analyzed for the stokes problem in a semi infinite channel the method is based on a generalized galerkin approximation with laguerre functions in the x direction and legendre polynomials in the y direction the well posedness of this method is established by deriving a lower bound on the inf sup constant numerical results indicate that the derived lower bound is sharp rigorous error analysis is also carried out

extraction and prediction of coherent patterns in incompressible flows through space time koopman analysis abstract we develop methods for detecting and predicting the evolution of coherent spatiotemporal patterns in incompressible time dependent fluid flows driven by ergodic dynamical systems our approach is based on representations of the generators of the koopman and perron frobenius groups of operators governing the evolution of observables and probability measures on lagrangian tracers respectively in a smooth orthonormal basis learned from velocity field snapshots through the diffusion maps algorithm these operators are defined on the product space between the state space of the fluid flow and the spatial domain in which the flow takes place and as a result their eigenfunctions correspond to global space time coherent patterns under a skew product dynamical system moreover using this data driven representation of the generators in conjunction with leja interpolation for matrix exponentiation we construct model free prediction schemes for the evolution of observables and probability densities defined on the tracers we present applications to periodic gaussian vortex flows and aperiodic flows generated by lorenz 96 systems

on the rational second kind chebyshev pseudospectral method for the solution of the thomas fermi equation over an infinite interval in this paper we propose a pseudospectral method for solving the thomas fermi equation which is a nonlinear singular ordinary differential equation on a semi infinite interval this approach is based on the rational second kind chebyshev pseudospectral method that is indeed a combination of tau and collocation methods this method reduces the solution of this problem to the solution of a system of algebraic equations the slope at origin is provided with high accuracy comparison with some numerical solutions shows that the present solution is effective and highly accurate