strategy iteration is strongly polynomial for 2 player turn based stochastic games with a constant discount factor ye 2011 showed recently that the simplex method with dantzig s pivoting rule as well as howard s policy iteration algorithm solve discounted markov decision processes mdps with a constant discount factor in strongly polynomial time more precisely ye showed that both algorithms terminate after at most o mn1 log n1 iterations where n is the number of states m is the total number of actions in the mdp and 0

on the coefficients of integrated expansions of bessel polynomials a new formula expressing explicitly the integrals of bessel polynomials of any degree and for any order in terms of the bessel polynomials themselves is proved another new explicit formula relating the bessel coefficients of an expansion for infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is also established an application of these formulae for solving ordinary differential equations with varying coefficients is discussed

the complexity of mean payoff games on graphs abstract we study the complexity of finding the values and optimal strategies of mean payoff games on graphs a family of perfect information games introduced by ehrenfeucht and mycielski and considered by gurvich karzanov and khachiyan we describe a pseudo polynomial time algorithm for the solution of such games the decision problem for which is in np conp finally we describe a polynomial reduction from mean payoff games to the simple stochastic games studied by condon these games are also known to be in np conp but no polynomial or pseudo polynomial time algorithm is known for them

direct minimization of the least squares spectral element functional part i direct solver this paper describes an equivalent but improved least squares formulation for the numerical approximation of the solution of partial differential equations instead of using variational analysis to impose the conditions for minimizing the residual the residuals are minimized directly thus leading to a method we will denote by direct minimization dm dm circumvents setting up the normal equations which consists of matrix matrix multiplications matrix matrix multiplications are expensive may lead to loss of accuracy and destroy the sparsity pattern present in the original system the condition number of the dm formulation is the square root of the condition number which would be obtained if variational analysis was employed an element by element procedure will be presented which allows for parallelization of dm a computational comparison between dm and the conventional least squares formulation based on variational analysis will be presented

a computational method for solving the branched channel flow problem in the present work a method is suggested to find the solution of poisson s equation with dirichlet boundary conditions in polar coordinates this method is expressed over the unit circle and consequently over the upper half plane from which geometry we find the solution to poisson s equation over the branched channel domain the results seems very reasonable and the derived transformation from unit circle to y shape channel is accurate poisson s equation can be solved similarly over much other geometry the solution looks very encouraging and resembles that is experimentally known

an adaptive algorithm for spectral computations on unbounded domains abstract we consider spectral methods for solving differential equations on unbounded domains in particular we consider a spectral rational function method and a method based on domain truncation combined with fourier series both methods contain a free parameter which determines the accuracy of the spectral method when solving evolution equations the optimal parameter may change with time resulting in a significant loss of accuracy if the parameter is kept fixed we develop an automatic algorithm which adapts the parameter at regular time levels to maintain optimum accuracy in numerical tests we found that the proposed method is robust and efficient

a mortar spectral finite element method for complex 2d and 3d elastodynamic problems abstract we present a hybrid spectral element finite element domain decomposition method for solving elastic wave propagation problems aim of the method is to exploit both the enhanced accuracy of spectral elements allowing significant reductions of the computational load and the flexibility of finite elements in treating irregular geometries and non linear media interfaces between finite and spectral elements are managed by the mortar method which enjoys optimal accuracy and allows to discretize the different regions independently several test cases illustrate the robustness of this method and its tolerance to a significant range of geometrical models characterized by different grid refinements

low order polynomial bounds on the expected performance of local improvement algorithms we present a general abstract model of local improvement applicable to such diverse cases as principal pivoting methods for the linear complementarity problem and hill climbing in artificial intelligence the model accurately predicts the behavior of the algorithms and allows for a variety of probabilistic assumptions that permit degeneracy simulation indicates an approximately linear average number of iterations under a variety of probability assumptions we derive theoretical bounds of 2en logn and en2 for different distributions respectively as well as polynomial bounds for a broad class of probability distributions we conclude with a discussion of the applications of the model to lcp and linear programming

on the topological structure of spaces of fuzzy compacta we prove that the space of fuzzy compacta in a peano continuum is homeomorphic to the hilbert space 2 as a corollary we find that the spaces of fuzzy compacta in r n and 2 are also homeomorphic to hilbert space

combinatorial structure and randomized subexponential algorithms for infinite games the complexity of solving infinite games including parity mean payoff and simple stochastic is an important open problem in verification automata and complexity theory in this paper we develop an abstract setting for studying and solving such games based on function optimization over certain discrete structures we introduce new classes of recursively local global rlg and partial recursively local global prlg functions and show that strategy evaluation functions for simple stochastic mean payoff and parity games belong to these classes in this setting we suggest randomized subexponential algorithms appropriate for rlg and prlg function optimization we show that the subexponential algorithms for combinatorial linear programming due to kalai and matousek sharir welzl can be adapted for optimizing the rlg and prlg functions

an adaptive multidomain chebyshev method for nonlinear eigenvalue problems application to self similar solutions of gas dynamics equations with nonlinear heat conduction an adaptive multidomain chebyshev collocation method for handling nonlinear eigenvalue problems such as those arising in the computation of self similar solutions is presented the algorithm is made of two stages the second one using an iterative method the method is applied to the time dependent one dimensional self similar solutions of the euler equations with nonlinear heat conduction in which a thermal front follows a shock wave it leads to highly accurate solutions for solutions stiffness ranging from 1 to 10 7 the method is quite general and may be applied to a large class of problems

a pseudo polynomial algorithm for mean payoff stochastic games with perfect information and few random positions abstract we consider two person zero sum stochastic mean payoff games with perfect information or bwr games given by a digraph g v e with local rewards r e z and three types of positions black v b white v w and random v r forming a partition of v it is a long standing open question whether a polynomial time algorithm for bwr games exists or not even when v r 0 in fact a pseudo polynomial algorithm for bwr games would already imply their polynomial solvability in this paper 1 we show that bwr games with a constant number of random positions can be solved in pseudo polynomial time more precisely in any bwr game with v r o 1 a saddle point in uniformly optimal pure stationary strategies can be found in time polynomial in v w v b the maximum absolute local reward and the common denominator of the transition probabilities

fractional order legendre operational matrix of fractional integration for solving the riccati equation with fractional order this paper is devoted to both theoretical and numerical study of riccati equation with fractional order a formulation to the fractional order legendre operational matrix of fractional integration is constructed existence and uniqueness results for the considered problem are provided and proved the fractional derivative is described in the caputo sense some numerical examples are discussed to demonstrate the efficiency and the accuracy of the proposed algorithm

the topological structure of fuzzy sets with endograph metric for a non degenerate convex subset y of the n dimensional euclidean space r n let k y be the family of all fuzzy sets of r n which are upper semicontinuous fuzzy convex and normal with compact supports contained in y we show that the space k y with the topology of endograph metric is homeomorphic to the hilbert cube q 1 1 w iff y is compact and the space k y is homeomorphic to x n q sup x n 1 iff y is non compact and locally compact

a note on stability of pseudospectral methods for wave propagation in this paper we deal with the effects on stability of subtle differences in formulations of pseudospectral methods for solution of the acoustic wave equation we suppose that spatial derivatives are approximated by chebyshev pseudospectral discretizations through reformulation of the equations as first order hyperbolic systems any appropriate ordinary differential equation solver can be used to integrate in time however the resulting stability and hence efficiency properties of the numerical algorithms are drastically impacted by the manner in which the absorbing boundary conditions are incorporated specifically mathematically equivalent well posed approaches are not equivalent numerically an analysis of the spectrum of the resultant system operator predicts these properties

numerical solution of linear differential algebraic equations in appl math comput 140 2003 77 and appl math comput in press a reducing and modified reducing index methods have been proposed for some cases of semi explicit daes differential algebraic equations in this paper these methods are generalized to more cases also numerical implementation of generalized methods will be presented through pseudospectral method with domain decomposition in addition aforementioned methods will be considered by some examples

memoryless determinacy of parity and mean payoff games a simple proof we give a simple direct and constructive proof of memoryless determinacy for parity and mean payoff games first we prove by induction that the finite duration versions of these games played until some vertex is repeated are determined and both players have memoryless winning strategies in contrast to the proof of ehrenfeucht and mycielski internat j game theory 8 1979 109 113 our proof does not refer to the infinite duration versions second we show that memoryless determinacy straightforwardly generalizes to infinite duration versions of parity and mean payoff games

iterative methods based on polynomial interpolation filters to detect discontinuities and recover point values from fourier data in previous papers we proposed new filters based on polynomial interpolation to approximate the point values of a piecewise smooth function f on 0 1 from its fourier coefficients and derived error estimates we proved that if all the discontinuity points of f are nodes we can reconstruct point values of f accurately even close to the discontinuities we use the new filters to develop iterative methods for detecting the discontinuity points and therefore accurately approximate the point values of the function from its fourier coefficients

spectral chebyshev fourier collocation for the helmholtz and variable coefficient equations in a disk the paper is concerned with the spectral collocation solution of the helmholtz equation in a disk in the polar coordinates r and q we use spectral chebyshev collocation in r spectral fourier collocation in q and a simple integral condition to specify the value of the approximate solution at the center of the disk the scheme is solved at a quasi optimal cost using the idea of superposition a matrix decomposition algorithm and fast fourier transforms both the dirichlet and neumann boundary conditions are considered and extensions to equations with variable coefficients are discussed numerical results confirm the spectral convergence of the method

free and forced vibrations of trusses by fourier decomposition and homotopy methods for nonlinear matrix eigenvalue problems i methods abstract vibrations of elastic frames or trusses are investigated by use of fourier frequency domain methods using classical solutions of linear hyperbolic pdes with constant coefficients and angle preserving connections of beams by clamping separation of variables is used e g a x e t for free vibrations eigensolutions in particular their values i are determined homotopically and individually employing systems of quasilinear odes usually explicit additionally to the methods in part i properties and supplements are presented in part ii

recurrences and explicit formulae for the expansion and connection coefficients in series of ordinary bessel polynomials abstract a formula expressing explicitly the derivatives of ordinary bessel polynomials of any degree and for any order in terms of the ordinary bessel polynomials themselves is proved another explicit formula which expresses the ordinary bessel expansion coefficients of a general order derivative of an infinitely differentiable function in terms of its original ordinary bessel coefficients is also given a formula for the ordinary bessel coefficients of the moments of one single ordinary bessel polynomial of certain degree is proved formula for the ordinary bessel coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its ordinary bessel coefficients is also obtained application of these formulae for solving ordinary differential equations with varying coefficients by reducing them to recurrence relations in the expansion coefficients of the solution is explained an algebraic symbolic approach using mathematica in order to build and solve recursively for the connection coefficients between generalized bessel and ordinary bessel polynomials is described explicit formula for these coefficients between jacobi and ordinary bessel polynomials is given of which the ultraspherical polynomials and its consequences are important special cases two analytical formulae for the connection coefficients between laguerre ordinary bessel and hermite ordinary bessel are also developed

a chebyshev collocation method for computing the eigenvalues of the laplacian chebyshev collocation techniques are developed in this paper to compute the eigenvalues of the laplacian based on a boundary integral formulation for two dimensional domains with piecewise smooth boundaries unlike the traditional domain methods for example the finite element method which discretizes the eigenfunctions on the two dimensional domain only a one dimensional function defined on the boundary is discretized global expansions in terms of chebyshev polynomials are used in each smooth piece of the boundary to solve the integral equation comparing with the boundary element method this method obtains higher accuracy for a smaller discretized matrix finally an efficient algorithm for generating the discretized matrix say n n is developed that requires only o n2 log n operations

generalized spectral decomposition for stochastic nonlinear problems we present an extension of the generalized spectral decomposition method for the resolution of nonlinear stochastic problems the method consists in the construction of a reduced basis approximation of the galerkin solution and is independent of the stochastic discretization selected polynomial chaos stochastic multi element or multi wavelets two algorithms are proposed for the sequential construction of the successive generalized spectral modes they involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems compared to the classical galerkin method the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes the methodology is detailed and tested on two model problems the one dimensional steady viscous burgers equation and a two dimensional nonlinear diffusion problem these examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space

an accurate solution of the poisson equation by the chebyshev collocation method abstract a new chebyshev collocation method is presented for the 2d poisson equation the resolution of the mixed collocation equations leads to two quasi tridiagonal systems which can be solved by standard techniques comparison of the results for the test problem u x y sin 4 x sin 4 y j comput phys 30 167 1979 with those computed by haidvogel and zang using the matrix diagonalization method indicates that our method would be more accurate at large n values

rooted tree analysis of runge kutta methods with exact treatment of linear terms we investigate a class of time discretization schemes called etd runge kutta methods where the linear terms of an ordinary differential equation are treated exactly while the other terms are numerically integrated by a one step method these schemes proposed by previous authors can be regarded as modified runge kutta methods whose coefficients are matrices instead of scalars from this viewpoint we reexamine the notion of consistency convergence and order to provide a mathematical foundation for new methods applying the rooted tree analysis expansion theorems of both the strict and numerical solutions are proved and two types of order conditions are defined several classes of formulas with up to four stages that satisfy the conditions are constructed and it is shown that the power series of matrices employed as their coefficients can be determined using the order conditions

a subexponential randomized algorithm for the simple stochastic game problem we describe a randomized algorithm for the simple stochastic game problem that requires 2o n expected operations for games with n vertices this is the first subexponential time algorithm for this problem

delaminating quadrature method for multi dimensional highly oscillatory integrals a delaminating quadrature method for multi dimensional highly oscillatory integrals is put forward in comparison with the levin method the new method has the benefits of being numerically stable and having a high computational speed numerical experiments testified the advantages of this method

adaptive multiquadric collocation for boundary layer problems an adaptive collocation method based upon radial basis functions is presented for the solution of singularly perturbed two point boundary value problems using a multiquadric integral formulation the second derivative of the solution is approximated by multiquadric radial basis functions this approach is combined with a coordinate stretching technique the required variable transformation is accomplished by a conformal mapping an iterated sine transformation a new error indicator function accurately captures the regions of the interval with insufficient resolution this indicator is used to adaptively add data centres and collocation points the method resolves extremely thin layers accurately with fairly few basis functions the proposed adaptive scheme is very robust and reaches high accuracy even when parameters in our coordinate stretching technique are not chosen optimally the effectiveness of our new method is demonstrated on two examples with boundary layers and one example featuring an interior layer it is shown in detail how the adaptive method refines the resolution

the complexity of stochastic games abstract we consider the complexity of stochastic games simple games of chance played by two players we show that the problem of deciding which player has the greatest chance of winning the game is in the class np co np

some remarks about the collocation method on a modified legendre grid abstract we compare the results obtained by applying the standard collocation method at the legendre gauss lobatto nodes for a model problem simulating a steady advection diffusion equation with those obtained by collocating at a new set of nodes these nodes are derived from a suitable modification of the legendre grid according to the magnitude of the ratio between the advective and the diffusive parts of the differential operator

an error minimized pseudospectral penalty direct poisson solver this paper presents a direct poisson solver based on an error minimized chebyshev pseudospectral penalty formulation for problems defined on rectangular domains in this study the penalty parameters are determined analytically such that the discrete l 2 error is minimized numerical experiments are conducted and the results show that the penalty scheme computes numerical solutions with better accuracy compared to the traditional approach with boundary conditions enforced strongly

on the coefficients of differentiated expansions and derivatives of jacobi polynomials a formula expressing explicitly the derivatives of jacobi polynomials of any degree and for any order in terms of the jacobi polynomials themselves is proved another explicit formula which expresses the jacobi expansion coefficients of a general order derivative of an infinitely differentiable function in terms of its original jacobi coefficients is also given the results for the special case of ultraspherical polynomials are considered the results for chebyshev polynomials of the first and second kinds and for legendre polynomials are also noted an application of how to use jacobi polynomials for solving ordinary and partial differential equations is described

a direct spectral collocation poisson solver in polar and cylindrical coordinates in this paper we present a direct spectral collocation method for the solution of the poisson equation in polar and cylindrical coordinates the solver is applied to the poisson equations for several different domains including a part of a disk an annulus a unit disk and a cylinder unlike other poisson solvers for geometries such as unit disks and cylinders no pole condition is involved for the present solver the method is easy to implement fast and gives spectral accuracy we also use the weighted interpolation technique and nonclassical collocation points to improve the convergence

on the construction of recurrence relations for the expansion and connection coefficients in series of jacobi polynomials formulae expressing explicitly the jacobi coefficients of a general order derivative integral of an infinitely differentiable function in terms of its original expansion coefficients and formulae for the derivatives integrals of jacobi polynomials in terms of jacobi polynomials themselves are stated a formula for the jacobi coefficients of the moments of one single jacobi polynomial of certain degree is proved another formula for the jacobi coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its original expanded coefficients is also given a simple approach in order to construct and solve recursively for the connection coefficients between jacobi jacobi polynomials is described explicit formulae for these coefficients between ultraspherical and jacobi polynomials are deduced of which the chebyshev polynomials of the first and second kinds and legendre polynomials are important special cases two analytical formulae for the connection coefficients between laguerre jacobi and hermite jacobi are developed

hopf bifurcation and strange attractors in chebyshev spectral solutions of the burgers equation abstract in this paper we apply the chebyshev pseudospectral collocation and spectral methods to the solution of the burgers equation the results provide evidence for the existence of strange attractors it is shown that there is a critical viscosity v c above which the solution is stable and below which the motion becomes highly unstable in favor of motion on a strange attractor

technique for evaluation of profiles of a composite chiral slab through inversion and pseudospectral approximation in this paper a technique is developed to simultaneously evaluate the permeability permittivity and chirality parameter of a composite chiral slab where they are functions of depth of the slab additionally the chirality parameter is a function of frequency the developed technique is based on the use of the riccati equation for the reflection coefficient associated with electromagnetic propagation through the slab and the green functions the technique also makes use of nth degree interpolation polynomials to approximate the unknowns using legendre gauss lobatto collocation nodes a numerical example is given through which the application of the technique is utilized

the topological structure of the space of fuzzy compacta abstract a fuzzy compactum in a space x is an upper semi continuous map f a 0 1 from a nonempty compact subspace a x to the unit closed interval 0 1 such that x a f x 0 is dense in a in this paper we investigate the topological structure of the space k e x of fuzzy compacta of a separable metric space x with the hausdorff metric in particular we prove that k e x l 2 if x is an infinite locally compact locally connected connected and separable metric space

accurate reconstructions of functions of finite regularity from truncated fourier series expansions kowledge of a truncated fourier series expansion for a 2 periodic function of finite regularity which is assumed to be piecewise smooth in each period is used to accurately reconstruct the corresponding function an algebraic equation of degree m is constructed for the m singularity locations in each period for the function in question the m coefficients in this algebraic equation are obtained by solving an algebraic system of m equations determined by the coefficients in the known truncated expansion if discontinuities in the derivatives of the function are considered in addition to discontinuities in the function itself that algebraic system will be nonlinear with respect to the m unknown coefficients the degree of the algebraic system will depend on the desired order of accuracy for the reconstruction i e a higher degree will normally lead to a more accurate determination of the singularity locations by solving an additional linear algebraic system for the jumps of the function and its derivatives up to the arbitrarily specified order at the calculated singularity locations we are able to reconstruct the 2 periodic function of finite regularity as the sum of a piecewise polynomial function and a function which is continuously differentiab1e up to the specified order

spectral element feti dp and bddc preconditioners with multi element subdomains abstract the discrete systems generated by spectral or hp version finite elements are much more ill conditioned than the ones generated by standard low order finite elements or finite differences this paper focuses on spectral elements based on gauss lobatto legendre gll quadrature and the construction of primal and dual non overlapping domain decomposition methods belonging to the family of balancing domain decomposition methods by constraints bddc and dual primal finite element tearing and interconnecting feti dp algorithms new results are presented for the spectral multi element case and also for inexact feti dp methods for spectral elements in the plane theoretical convergence estimates show that these methods have a convergence rate independent of the number of subdomains and coefficient jumps of the elliptic operator while there is only a polylogarithmic dependence on the spectral degree p and the ratio h h of subdomain and element sizes parallel numerical experiments on a linux cluster confirm these results for tests with spectral degree up to p 32 thousands of subdomains and coefficient jumps up to 8 orders of magnitude

viscous drag by cellular automata a simple method to compute the drag coefficient of two dimensional bodies with arbitrary shapes is presented the procedure is based on cellular automata as an extreme idealization of the molecular dynamics of a viscous fluid we verify the algorithm by examples and obtain results in quantitative agreement with experiments even when eddies behind the obstacle are formed

the shallow water equations in lagrangian coordinates recent advances in the collection of lagrangian data from the ocean and results about the well posedness of the primitive equations have led to a renewed interest in solving flow equations in lagrangian coordinates we do not take the view that solving in lagrangian coordinates equates to solving on a moving grid that can become twisted or distorted rather the grid in lagrangian coordinates represents the initial position of particles and it does not change with time we apply numerical methods traditionally used to solve differential equations in eulerian coordinates to solve the shallow water equations in lagrangian coordinates the difficulty with solving in lagrangian coordinates is that the transformation from eulerian coordinates results in solving a highly nonlinear partial differential equation the non linearity is mainly due to the jacobian of the coordinate transformation which is a precise record of how the particles are rotated and stretched the inverse jacobian must be calculated thus lagrangian coordinates cannot be used in instances where the jacobian vanishes for linear spatial flows we give an explicit formula for the jacobian and describe the two situations where the lagrangian shallow water equations cannot be used because either the jacobian vanishes or the shallow water assumption is violated we also prove that linear in space steady state solutions of the lagrangian shallow water equations have jacobian equal to one in the situations where the shallow water equations can be solved in lagrangian coordinates accurate numerical solutions are found with finite differences the chebyshev pseudospectral method and the fourth order runge kutta method the numerical results shown here emphasize the need for high order temporal approximations for long time integrations

mean payoff games and propositional proofs we associate a cnf formula to every instance of the mean payoff game problem in such a way that if the value of the game is non negative the formula is satisfiable and if the value of the game is negative the formula has a polynomial size refutation in s 2 frege i e dnf resolution this reduces mean payoff games to the weak automatizability of s 2 frege and to the interpolation problem for s 2 2 frege since the interpolation problem for s 1 frege i e resolution is solvable in polynomial time our result is close to optimal up to the computational complexity of solving mean payoff games the proof of the main result requires building low depth formulas that compute the bits of the sum of a constant number of integers in binary notation and low complexity proofs of the required arithmetic properties

a spectral collocation method to solve helmholtz problems with boundary conditions involving mixed tangential and normal derivatives abstract the performing spectral method developed by haldenwang et al j comput phys 55 1984 115 to solve multi dimensional helmholtz equations associated to mixed boundary conditions with constant coefficients is extended to boundary conditions mixing a first order normal derivative with a second order tangential derivative the accuracy of the proposed algorithm is evaluated on two test cases for which analytical solutions exist an academic problem and a physical configuration including an interface with shear viscosity the procedure is also applied to the research of the rayleigh benard instability thresholds in closed cavities with thin diffusive walls

analytical behavior of two dimensional incompressible flow in porous media in this paper we study the analytic structure of a two dimensional mass balance equation of an incompressible fluid in a porous medium given by darcy s law we obtain local existence and uniqueness by the particle trajectory method and we present different global existence criterions these analytical results with numerical simulations are used to indicate nonformation of singularities moreover blow up results are shown in a class of solutions with infinite energy

parameter estimation by spectral approximation in this paper chebyshev and legendre approximations are proposed for estimating parameters in differential equations which are easy to be performed the convergence and the spectral accuracy are proved even without some conditions as imposed in other papers the numerical results show the advantages of this new approach

a numerical scheme for periodic travelling wave simulations in some nonlinear dispersive wave models a numerical method for simulating periodic travelling wave solutions of some nonlinear dispersive wave equations is proposed the construction of the scheme is based on an efficient computation of the elements that characterize these solutions the initial profile and the velocity of the wave

recurrences and explicit formulae for the expansion and connection coefficients in series of bessel polynomials a formula expressing explicitly the derivatives of bessel polynomials of any degree and for any order in terms of the bessel polynomials themselves is proved another explicit formula which expresses the bessel expansion coefficients of a general order derivative of an infinitely differentiable function in terms of its original bessel coefficients is also given a formula for the bessel coefficients of the moments of one single bessel polynomial of certain degree is proved a formula for the bessel coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its bessel coefficients is also obtained application of these formulae for solving ordinary differential equations with varying coefficients by reducing them to recurrence relations in the expansion coefficients of the solution is explained an algebraic symbolic approach using mathematica in order to build and solve recursively for the connection coefficients between bessel bessel polynomials is described an explicit formula for these coefficients between jacobi and bessel polynomials is given of which the ultraspherical polynomial and its consequences are important special cases two analytical formulae for the connection coefficients between laguerre bessel and hermite bessel are also developed

a stable penalty method for the compressible navier stokes equations i open boundary conditions the purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible navier stokes equations in three spatial dimensions the treatment uses the conservation form of the navier stokes equations and utilizes linearization and localization at the boundaries based on these variables the proposed boundary conditions are applied through a penalty procedure thus ensuring correct behavior of the scheme as the reynolds number tends to infinity the versatility of this method is demonstrated for the problem of a compressible flow past a circular cylinder

linear programming the simplex algorithm and simple polytopes in the first part of the paper we survey some far reaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes in the second part we discuss some recent developments concerning the simplex algorithm we describe subexponential randomized pivot rules and upper bounds on the diameter of graphs of polytopes 1997 the mathematical programming society inc published by elsevier science b v

fuzzy dynamical systems a fuzzy dynamical system on an underlying complete locally compact metric state space x is defined axiomatically in terms of a fuzzy attainability set mapping on x this definition includes as special cases crisp single and multivalued dynamical systems on x it is shown that the support of such a fuzzy dynamical system on x is a crisp multivalued dynamical system on x and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of x in addition fuzzy trajectories are defined their existence established and various properties investigated

immersed boundary method with non uniform distribution of lagrangian markers for a non uniform eulerian mesh this study presents a technique to incorporate spheres in a channel flow that uses a non uniform eulerian grid using immersed boundary methods with direct forcing an efficient algorithm is presented which distributes the lagrangian markers non uniformly to match the fluid grid and keep the number of markers optimized also a novel method to calculate the area weights of the lagrangian markers is given it is observed that even the best available algorithms for uniform distribution of markers on a sphere result in a finite error using vector spherical harmonics this error is quantified and reduced to machine precision a series of simulations of a stationary and moving sphere in a periodic channel at reynolds number range of 1 100 are presented results for a sphere in an ambient shear flow in close proximity of a wall are also shown where the present non uniform distribution offers an order of magnitude reduction over uniform distribution of lagrangian markers simulations of a random cluster of 640 monodisperse spherical particles show a 77 reduction in lagrangian markers with an error of 0 135 in computing the total drag

chebyshev spectral collocation methods for nonlinear isothermal magnetostatic atmospheres abstract the equations of magnetohydrodynamic mhd equilibria for a plasma in gravitational field are investigated for equilibria with one ignorable spatial coordinate the mhd equations are reduced to a single nonlinear elliptic equation nlpde for the magnetic flux u known as the grad shafranov equation the chebyshev spectral collocation methods cscms are described and applied to obtain numerical solutions for nonlinear boundary value problems modelling two classes of isothermal magnetostatic atmospheres in which the current density j is proportional to the exponential of the magnetic flux and moreover falls off exponentially with distance vertical to the base with an e folding distance equal to the gravitational scale height for the first class and proportional to the sinh u for the second class the accuracy and efficiency of this chebyshev approach are compared favorably with those of the standard finite difference methods

mean cost cyclical games we study the mean cost cyclical game in a more general setting than that in gurvitch et al 1988 and karzanow and lebedev 1993 the game is played on a directed graph and generalizes the single source shortest path problem the minimum mean cycle problem see karp 1978 and the ergodic extension of matrix games moulin 1976 we prove the existence of a solution in uniform stationary strategies and present an algorithm for finding such optimal strategies in fact our algorithm is an extension of the algorithms due to gurvitch et al 1988 and karzanow and lebedev 1993 which were proved to be finite but exponential in the worst case we prove that all these algorithms are pseudopolynomial

computation of frequency responses for linear time invariant pdes on a compact interval we develop mathematical framework and computational tools for calculating frequency responses of linear time invariant pdes in which an independent spatial variable belongs to a compact interval in conventional studies this computation is done numerically using spatial discretization of differential operators in the evolution equation in this paper we introduce an alternative method that avoids the need for finite dimensional approximation of the underlying operators in the evolution model this method recasts the frequency response operator as a two point boundary value problem and uses state of the art automatic spectral collocation techniques for solving integral representations of the resulting boundary value problems with accuracy comparable to machine precision our approach has two advantages over currently available schemes first it avoids numerical instabilities encountered in systems with differential operators of high order and second it alleviates difficulty in implementing boundary conditions we provide examples from newtonian and viscoelastic fluid dynamics to illustrate utility of the proposed method

efficient multi dimensional solution of pdes using chebyshev spectral methods abstract a robust methodology is presented for efficiently solving partial differential equations using chebyshev spectral techniques it is well known that differential equations in one dimension can be solved efficiently with chebyshev discretizations o n operations for n unknowns however this efficiency is lost in higher dimensions due to the coupling between modes this paper presents the quasi inverse technique qit which combines optimizations of one dimensional spectral differentiation matrices with kronecker matrix products to build efficient multi dimensional operators this strategy results in o n 2 d 1 operations for n d unknowns independent of the form of the differential operators qit is compared to the matrix diagonalization technique mdt of haidvogel and zang d b haidvogel t zang the accurate solution of poisson s equation by expansion in chebyshev polynomials j comput phys 30 1979 167 180 and shen j shen efficient spectral galerkin method ii direct solvers of second and fourth order equations using chebyshev polynomials siam j sci comp 16 1 1995 74 87 while the cost for mdt and qit are the same in two dimensions there are significant differences mdt utilizes an eigenvalue eigenvector decomposition and can only be used for relatively simple differential equations qit is based upon intrinsic properties of the chebyshev polynomials and is adaptable to linear pdes with constant coefficients in simple domains we present results for a standard suite of test problems and discuss of the adaptability of qit to more complicated problems

spectral analysis of hermite cubic spline collocation systems in this paper we present an eigenvalue analysis of the first order and second order hermite cubic spline collocation differentiation matrices with arbitrary collocation points some important features are explored and compared with some other discrete methods such as finite difference methods

chebyshev pseudospectral method for computing numerical solution of convection diffusion equation abstract a method for computing highly accurate numerical solutions of 1d convection diffusion equations is proposed in this method the equation is first discretized with respect to the spatial variable transforming the original problem into a set of ordinary differential equations and then the resulting system is integrated in time by the fourth order runge kutta method spatial discretization is done by using the chebyshev pseudospectral collocation method before describing the method we review a finite difference based method by salkuyeh d khojasteh salkuyeh on the finite difference approximation to the convection diffusion equation appl math comput 179 2006 79 86 and contrary to the proposal of the author we show that this method is not suitable for problems involving time dependent boundary conditions which calls for revision stability analysis based on pseudoeigenvalues to determine the maximum time step for the proposed method is also carried out superiority of the proposed method over a revised version of salkuyeh s method is verified by numerical examples

pseudospectral method for the good boussinesq equation we prove the nonlinear stability and convergence of a fully discrete pseudospectral scheme for the good boussinesq equation utt uxxxx uxx u2 xx numerical comparisons with finite difference schemes are also reported

an accurate solution of the poisson equation by the chebyshev tau method abstract a new tau method is presented for the two dimensional poisson equation comparison of the results for the test problem u x y sin 4 x sin 4 y with those computed by haidvogel and zang using the matrix diagonalization method and dang vu and delcarte using the chebyshev collocation method indicates that our method would be more accurate

layer methods for stochastic navier stokes equations using simplest characteristics we propose and study a layer method for stochastic navier stokes equations snse with spatial periodic boundary conditions and additive noise the method is constructed using conditional probabilistic representations of solutions to snse and exploiting ideas of the weak sense numerical integration of stochastic differential equations we prove some convergence results for the proposed method including its first mean square order results of numerical experiments on two model problems are presented

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

numerical methods for the generalized zakharov system we present two numerical methods for the approximation of the generalized zakharov system zs the first one is the time splitting spectral tssp method which is explicit time reversible and time transverse in variant if the generalized zs is keeps the same decay rate of the wave energy as that in the generalized zs gives exact results for the plane wave solution and is of spectral order accuracy in space and second order accuracy in time the second one is to use a local spectral method the discrete singular convolution dsc for spatial derivatives and the fourth order runge kutta rk4 for time integration which is of high the same as spectral order accuracy in space and can be applied to deal with general boundary conditions in order to test accuracy and stability we compare these two methods with other existing methods fourier pseudospectral method fps and wavelet galerkin method wg for spatial derivatives combining with the rk4 for time integration as well as the standard finite difference method fd for solving the zs with a solitary wave solution furthermore extensive numerical tests are presented for plane waves solitary wave collisions in 1d as well as a 2d problem of the generalized zs numerical results show that tssp and dsc are spectralorder accuracy in space and much more accurate than fd and for stability tssp requires k o h dsc rk4 requires k o h2 for fixed acoustic speed where k is the time step and h is the spatial mesh size

on shifted jacobi spectral approximations for solving fractional differential equations in this paper a new formula of caputo fractional order derivatives of shifted jacobi polynomials of any degree in terms of shifted jacobi polynomials themselves is proved we discuss a direct solution technique for linear multi order fractional differential equations fdes subject to nonhomogeneous initial conditions using a shifted jacobi tau approximation a quadrature shifted jacobi tau q sjt approximation is introduced for the solution of linear multi order fdes with variable coefficients we also propose a shifted jacobi collocation technique for solving nonlinear multi order fractional initial value problems the advantages of using the proposed techniques are discussed and we compare them with other existing methods we investigate some illustrative examples of fdes including linear and nonlinear terms we demonstrate the high accuracy and the efficiency of the proposed techniques

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis