polynomials for infinite domain spectral elements in this note it is found that laguerre type elements may be used in conjunction with legendre type elements to resolve more accurately regions of outflow and infinite boundaries

asymptotic coefficients of hermite function series abstract by using complex variable methods steepest descent and residues to asymptotically evaluate the coefficient integrals the numerical analysis of hermite function series is discussed there are striking similarities and differences with the author s earlier work on chebyshev polynomial methods j comp phys 45 1982 45 49 for infinite or semi infinite domains like chebyshev series the hermite coefficients are asymptotically given by the sum of two types of terms i stationary point steepest descent contributions and ii residues at the poles of f z the function being expanded as a hermite series the stationary point term is determined solely by the asymptotic behavior of f z i e how rapidly f z decays as z along the real axis unlike chebyshev series however it is necessary to perform a separate analysis for functions which decay faster or slower than the gaussian function exp az2 singular functions too fall into two categories those that decay rapidly with z have asymptotic hermite coefficients which are dominated by the singularity but functions which decay as slowly as sech z or slower have hermite coefficients dominated by the stationary point terms and the singularity is irrelevant the end product of the analysis is the same as in the earlier work simple explicit formulas to optimize the efficiency of hermite methods and estimate a priori how many degrees of freedom are needed provided one knowns at least crudely i the asymptotic behavior of f z and ii its singularity nearest the real axis rather surprisingly one finds hermite functions superior to chebyshev polynomials for some classes of functions when the computational domain is infinite

two dimensional riemann problem for a 2 2 system of hyperbolic conservation laws involving three constant states abstract zhang and zheng 1990 conjectured on the structure of a solution for a two dimensional riemann problem for euler equation to resolve this illuminating conjecture many researchers have studied the simplified 2 2 systems in this paper 3 pieces riemann problem for two dimensional 2 2 hyperbolic system is considered without the restriction that each jump of the initial data projects one planar elementary wave we classify twelve topologically distinct solutions and construct analytical and numerical solutions the computed numerical solutions clearly confirm the constructed analytic solutions

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

global structure of riemann solutions to a system of two dimensional hyperbolic conservation laws abstract the riemann problem for a two dimensional nonstrictly hyperbolic system of conservation laws is considered without the restriction that each jump of the initial data projects one planar elementary wave ten topologically distinct solutions are obtained by applying the method of generalized characteristic analysis some of these solutions involve the nonclassical waves i e the delta shock wave and the delta contact discontinuity for which we explicitly give the expressions of their strengths locations and propagation speeds moreover we demonstrate that the nature of our solutions is identical with that of solutions to the corresponding one dimensional cauchy problem which provides a verification that our construction produces the correct unique global solutions

matrices determined by a linear recurrence relation among entries abstract a matrix a a ij is called a 7 matrix if its entries satisfy the recurrence relation a i 1 j 1 a i 1 j a ij where are fixed numbers a 7 matrix is completely determined by its first row and first column in this paper we determine the structure of 7 matrices and investigate the sequences represented by columns of infinite 7 matrices

johann faulhaber and sums of powers early 1 7th century mathematical publications of johann faulhaber contain some remarkable theorems such as the fact that the r fold summation of lm 2m nm is a polynomial in n n r when m is a positive odd number the present paper explores a computation based approach by which faulhaber may well have discovered such results and solves a 360 year old riddle that faulhaber presented to his readers it also shows that similar results hold when we express the sums in terms of central factorial powers instead of ordinary powers faulhaber s coefficients can moreover be generalized to noninteger exponents obtaining asymptotic series for 1 2 na in powers of n 1 n 1 1

two dimensional riemann problem involving three j s for a hyperbolic system of nonlinear conservation laws in this paper we study the riemann problem for a two dimensional hyperbolic system of conservation laws with initial data projecting exactly three contact discontinuities we use the generalized characteristic analysis method to analyze interactions of contact discontinuities and obtain six analytical solutions and corresponding criteria delta shock waves and spiral structures appear in some solutions we also conduct numerical experiments and numerical results confirm the geometric structures of the constructed solutions

a formula to construct all involutions in riordan matrix groups abstract we get a formula for all finite riordan involutions using the natural approximation induced by the inverse limit construction we also describe all infinite riordan matrices which are involutions in the riordan group finally we get some consequences

double parameter recurrences for polynomials in bi infinite riordan matrices and some derived identities abstract we present some double parameter recurrence relations for families of polynomials in bi infinite riordan matrices from this point of view we treat some riordan families of generalized fibonacci and lucas polynomials we use these to obtain some identities related mainly with binomial coefficients and fibonacci numbers

riordan group involutions we study involutions in the riordan group especially those with combinatorial meaning we give a new determinantal criterion for a matrix to be a riordan involution and examine several classes of examples a complete characterization of involutions in the appell subgroup is developed in another direction we find several examples that generalize the rna matrix but are of independent interest

simple proofs of open problems about the structure of involutions in the riordan group abstract we prove that if d g x f x is an element of order 2 in the riordan group then g x exp x xf x for some antisymmetric function x z also we prove that every element of order 2 in the riordan group can be written as bmb 1 for some element b and m 1 1 in the riordan group these proofs provide solutions to two open problems presented by l shapiro l w shapiro some open questions about random walks involutions limiting distributions and generating functions adv appl math 27 2001 585 596

the asymptotic chebyshev coefficients for functions with logarithmic endpoint singularities mappings and singular basis functions abstract when a function is singular at the ends of its expansion interval its chebyshev coefficients a n converge very poorly we analyze three numerical strategies for coping with such singularities of the form 1 x k log 1 x and in the process make some modest additions to the theory of chebyshev expansions the first two numerical methods are the convergence improving changes of coordinate x sin 2 y and x tanh ly 1 y 2 1 2 we derive the asymptotic chebyshev coefficients in the limit n for both mappings and for the original untransformed chebyshev series for the original function the asymptotic approximation for general k is augmented by the exact chebyshev coefficients for integer k numerical tests show that the sine mapping is excellent for k 1 increasing the rate of convergence to b n o 1 n 4 k 1 although the tanh transformation is guaranteed to be better for sufficiently large n we offer both theoretical and numerical evidence to explain why the sine mapping is usually better in practice sufficiently large n is usually huge instead of mapping one may use a third strategy supplementing the chebyshev polynomials with singular basis functions simple experiments show that this approach is also successful

laguerre approximation of stable manifolds with application to connecting orbits we present an algorithm based on approximation by laguerre polynomials for computing a point on the stable manifold of a stationary solution of an autonomous system a superconvergence phenomenon means that the accuracy of our results is much higher than the usual spectral accuracy both the theory and the implementation of the method are considered finally as an application of the algorithm we describe a fully spectral approximation of homo and heteroclinic orbits

some inverse limit approaches to the riordan group abstract using the inverse limit tool we obtain the riordan group in its infinite and bi infinite representations from groups of finite riordan matrices we define two different reflections for bi infinite riordan matrices employing these definitions we give answers to the problems d d and d d in the riordan group so we describe the self complementary and self dual riordan matrices

the optimization of convergence for chebyshev polynomial methods in an unbounded domain by using the method of steepest descent i have compared the suitability of three different methods for solving problems in a semi infinite or infinite domain using chebyshev polynomials exponential mappings are uniformyly bad domain truncation and algebraic mapping both work well but each is superior for a different category of problems when the solution is an entire function then domain truncation is best it is always simpler to apply than algebraic mapping and always at least as accurate much more accurate for functions which decay very rapidly for singular functions on the other hand algebraic mapping is better because it is less sensitive to the mapping scale factor l permitting a better compromise in resolving both the singularity and the exponential decay for both types of problems i give simple explicit estimates of both the optimum choise of domain size or mapping factor and of the attainable accuracy for the model entire function exp azk on a semi infinite interval 0 the optimum domain size is l 0 896 na 1k and the smallest attainable error is roughly e n 0 896 k where n is the number of chebyshev polynomals used similar formulas for singular functions handled via algebraic mapping are given in 3 41 to 3 45 below

two dimensional riemann problem for a hyperbolic system of nonlinear conservation laws i four j cases abstract the riemann problem for a two dimensional 2 2 hyperbolic system of nonlinear conservation laws has been solved analytically the solutions reveal a variety of geometric structures 57 exact solutions and the corresponding criteria have been obtained a kind of new nonlinear hyperbolic wave which we call delta shock appears in some solutions the delta shock is a discontinuity surface which is the support of a dirac function

recurrence relations for polynomial sequences via riordan matrices we give recurrence relations for any family of generalized appell polynomials unifying so some known recurrences for many classical sequences of polynomials our main tool to get our goal is the riordan group we use the product of riordan matrices to interpret some relationships between different polynomial families moreover using the hadamard product of series we get a general recurrence relation for the polynomial sequences associated to the so called generalized umbral calculus

bivariate delta evolution equations and convolution polynomials computing polynomial expansions of solutions abstract this paper describes an application of rota and collaborator s ideas about the foundation on combinatorial theory to the computing of solutions of some linear functional partial differential equations we give a dynamical interpretation of the convolution families of polynomials concretely we interpret them as entries in the matrix representation of the exponentials of certain contractive linear operators in the ring of formal power series this is the starting point to get symbolic solutions for some functional partial differential equations we introduce the bivariate convolution product of convolution families to obtain symbolic solutions for natural extensions of functional evolution equations related to delta operators we put some examples to show how these symbolic methods allow us to get closed formulas for solutions of genuine partial differential equations we create an adequate framework to base theoretically some of the performed constructions and to get some existence and uniqueness results

error estimation of hermite spectral method for nonlinear partial differential equations hermite approximation is investigated some inverse inequalities imbedding inequalities and approximation results are obtained a hermite spectral scheme is constructed for burgers equation the stability and convergence of the proposed scheme are proved strictly the techniques used in this paper are also applicable to other nonlinear problems in unbounded domains

new developments of delta shock waves and its applications in systems of conservation laws a theory of delta shock waves with dirac delta functions developing in both state variables for a class of nonstrictly hyperbolic systems of conservation laws is established in this paper we solve constructively the riemann problems for the system under consideration in solutions we find another kind of delta shock waves on which both state variables simultaneously contain the dirac delta functions it is quite different from the previous ones on which only one state variable contains the dirac delta function we strictly prove it satisfies the system in the sense of distributions the generalized rankine hugoniot relation and entropy condition are proposed to solve the delta shock waves furthermore we show all of the existence and stability of solutions including the delta shock waves to reasonable viscous perturbations the generalized rankine hugoniot relation is also confirmed in particular our theory on the delta shock waves possesses the generality and practicability which can be conveniently and successfully applied to those systems investigated by korchinski 1977 tan zhang and zheng 1991 ercole 2000 cheng and yang 2011 etc and we also give a simplified approach to solve a 2 d riemann problem for the system studied by tan and zhang 1990 for the case 2j 2j and obtain the explicit formulae of the delta shock waves finally the numerical simulations completely coinciding with theoretical analysis are presented

spectral methods for the navier stokes equations with one infinite and two periodic directions abstract two numerical methods were designed to solve the time dependent three dimensional incompressible navier stokes equations in boundary layers method a semi infinite domain and mixing layers or wakes method b fully infinite domain their originality lies in the use of rapidly decaying spectral basis functions to approximate the vertical dependence of the solutions combined with one method a or two method b slowly decaying extra functions for each wave vector that exactly represent the irrotational component of the solution at large distances both methods eliminate the pressure term as part of the formulation thus avoiding fractional step time integration they yield rapid convergence and are free of spurious modes in the orr sommerfeld spectra they are also efficient although the operation count is of order n 2 n is the number of modes in the infinite direction these methods have been used for extensive direct numerical simulations of transition and turbulence a new time integration scheme with low storage requirements and good stability properties is also described

quadrature based vector fitting for discretized mathcal h _2 approximation vector fitting is a popular method of constructing rational approximants designed to fit given frequency response measurements the original method which we refer to as vf is based on a least squares fit to the measurements by a rational function using an iterative reallocation of the poles of the approximant we show that one can improve the performance of vf significantly by using a particular choice of frequency sampling points and properly weighting their contribution based on quadrature rules that connect the least squares objective with an mathcal h 2 error measure our modified approach designated here as quadvf helps recover the original transfer function with better global fidelity as measured with respect to the mathcal h 2 norm than the localized least squares approximation implicit in vf we extend the new framework also to incorporate derivative information leading to rational approximants that minimize system error with respect to a discrete sobolev norm we consider the

the fourier transform of the quartic gaussian exp ax4 hypergeometric functions power series steepest descent asymptotics and hyperasymptotics and extensions to exp ax2n the fourier transform of a quartic gaussian f k exp ikx exp x 4 dx is important in the theory of radial basis functions with f r exp r 4 we show that the transform is the exact sum of two generalized hypergeometric functions 0f 2 this implies that f k is an entire function and also gives an analytical form for the power series coefficients of arbitrary degree using the method of steepest descent for integrals we derive the first five terms in an asymptotic series in powers of k 4 3 through a simple hyperasymptotic analysis we show that the magnitude of the error of the optimally truncated series superasymptotic error is oexp 0 818k 4 3 the lowest order approximation f k 2 7 6 p 3k 1 3exp 2 1 33 16 k 4 3 cos 3 3 22 1 3 16 k 4 3 p 6 shows that the transform f k is an exponentially decaying oscillation we show that the steepest descent method 2 27 bender and orszag 1978 miller 2006 26 33 lauwerier 1966 ursell 1965 can be applied to the transform of exp ax 2 n for general integer n and give the first two terms another theme is that modern computer technology has simplified the steepest descent method the steepest descent paths can be plotted by merely displaying the contours of the imaginary part of the phase function j t where z j t is the logarithm of the integrand and z is the large parameter here proportional to k 4 3 the task of inverting w t to obtain a series in powers of w for the metric factor dt dw after the usual change of integration variable can now be done to high order by a few lines of an algebraic manipulation language as illustrated by a short table with the complete maple code

spectral collocation methods for stokes flow in contraction geometries and unbounded domains abstract a spectral element method is described which enables stokes flow in contraction geometries and unbounded domains to be solved as a set of coupled problems over semi infinite rectangular subregions expansions in terms of the eigenfunctions of singular sturm liouville problems are used to compute solutions to the governing biharmonic equation for the stream function the coefficients in these expansions are determined by collocating the differential equation and boundary conditions and imposing c 3 continuity across the subregion interface the suitability of domain truncation and algebraic mapping techniques are compared as well as the choice of trial functions

two dimensional riemann problem for a hyperbolic system of conservation laws in three pieces abstract this paper studies the two dimensional riemann problem for a hyperbolic system of conservation laws with three constant initial data separated by x positive y positive and x negative axes using the generalized characteristic analysis method by considering the interactions of rarefaction waves shock waves and contact discontinuities 13 exact solutions and the corresponding criteria are obtained delta shock waves appear in some solutions moreover the numerical illustrations agree completely with the geometric structures of the exact solutions

numerical and perturbative computations of solitary waves of the benjamin ono equation with higher order nonlinearity using christov rational basis functions computation of solitons of the cubically nonlinear benjamin ono equation is challenging first the equation contains the hilbert transform a nonlocal integral operator second its solitary waves decay only as o 1 x 2 to solve the integro differential equation for waves traveling at a phase speed c we introduced the artificial homotopy h u x x c u 1 d u 2 du 3 0 d 0 1 and solved it in two ways the first was continuation in the homotopy parameter d marching from the known benjamin ono soliton for d 0 to the cubically nonlinear soliton at d 1 the second strategy was to bypass continuation by numerically computing perturbation series in d and forming pade approximants to obtain a very accurate approximation at d 1 to further minimize computations we derived an elementary theorem to reduce the two parameter soliton family to a parameter free function the soliton symmetric about the origin with unit phase speed solitons for higher order benjamin ono equations are also computed and compared to their korteweg devries counterparts all computations applied the pseudospectral method with a basis of rational orthogonal functions invented by christov which are eigenfunctions of the hilbert transform

identities induced by riordan arrays historically there exist two versions of the riordan array concept the older one better known as recursive matrix consists of bi infinite matrices d n k n k is an element of z k n implies d n k 0 deals with formal laurent series and has been mainly used to study algebraic properties of such matrices the more recent version consists of infinite lower triangular arrays d n k n k is an element of n k n implies d n k 0 deals with formal power series and has been used to study combinatorial problems here we show that every riordan array induces two characteristic combinatorial sums in three parameters n k m is an element of z these parameters can he specialized and generate an indefinite number of other combinatorial identities which are valid in the hi infinite realm of recursive matrices

some algebraic structure of the riordan group abstract in this paper we highlight some of the algebraic properties of the riordan group we present some new properties and provide proofs for some known properties for instance proofs of properties of the power bell and derivative subgroups are presented we also give a new property for generating stochastic riordan matrices which are elements of the stochastic riordan subgroup centralizers stabilizers and certain riordan group isomorphisms are also given and proved we conclude by proving some properties connecting similar riordan matrices and pseudo involutions in the riordan group

pseudospectral methods on a semi infinite interval with application to the hydrogen atom a comparison of the mapped fourier sine method with laguerre series and rational chebyshev expansions the fourier sine with mapping pseudospectral algorithm of fattal et al phys rev e 53 1996 1217 has been applied in several quantum physics problems here we compare it with pseudospectral methods using laguerre functions and rational chebyshev functions we show that laguerre and chebyshev expansions are better suited for solving problems in the interval r 0 for example the coulomb schrodinger equation than the fourier sinemapping scheme all three methods give similar accuracy for the hydrogen atom when the scaling parameter l is optimum but the laguerre and chebyshev methods are less sensitive to variations in l we introduce a new variant of rational chebyshev functions which has a more uniform spacing of grid points for large r and gives somewhat better results than the rational chebyshev functions of boyd j comp phys 70 1987 63

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

two dimensional riemann problem involving three contact discontinuities for 2 2 hyperbolic conservation laws in anisotropic media abstract in this study we consider the two dimensional riemann problem for a system of conservation laws which models polymer flooding in an anisotropic oil reservoir the initial data are constants in three sectors centered at the origin which only involve three contact discontinuities by the generalized characteristic analysis method and the phase plane analysis method we find that the solutions are not unique for certain values of the initial data we propose an additional stability condition for the interaction of the contact discontinuities by this stability condition all of the exact solutions and their corresponding criteria can be obtained the solutions exhibit various geometrical structures in particular envelope rarefaction waves develop in some solutions

orthogonal rational functions on a semi infinite interval abstract by applying a mapping to the chebyshev polynomials we define a new spectral basis the rational chebyshev functions on the semi infinite interval denoted by tl n y continuing earlier work by the author and by grosch and orszag we show that these rational functions inherit most of the good numerical characteristics of the chebyshev polynomials orthogonality completeness exponential or infinite order convergence matrix sparsity for equations with polynomial coefficients and simplicity seven numerical examples illustrate their versatility the charney stability problem of meteorology for example is solved to show the feasibility of applying spectral methods to a semi infinite atmosphere for functions that are singular at both endpoints such as k 1 y one may combine rational chebyshev functions with a preliminary mapping to obtain a single exponentially convergent expansion on y 0 finally we successfully generalize the wkb method to obtain for the j 0 bessel function an amplitude phase approximation which is convergent rather than asymptotic and is accurate not merely for large y but for all y even the origin

the rate of convergence of chebyshev polynomials for functions which have asymptotic power series about one endpoint the theorem proved here extends chebyshev theory into what has previously been no man s land functions which have an infinite number of bounded derivatives on the expansion interval a b but which are singular at one endpoint the chebyshev series in l x for all the familiar special functions fall into this category so this class of functions is very important indeed in words the theorem shows that the more slowly the asymptotic power series about the singular point converges the slower the convergence of the corresponding chebyshev series must be more formally if f x analytic on a b is singular at x b in such a way that it has an asymptotic power seriesf x e a x b about that endpoint then if loglanl nr logn r it is proved that the coefficients of the convergent chebyshev polynomial series on a bj f x i bn tn y wherey 2 x 0 5 b a l b a satisfy the inequality lim logj logjbn1 0 when a function has a singularity which is not on 1 1 it is known that bn ae for some a and x which depend received march 21 1980 revised august 21 1980 and october 29 1980 ams mos subject classifications 1970 primary 42a56 secondary 33a65 41a10

the arctan tan and kepler burgers mappings for periodic solutions with a shock front or internal boundary layer abstract many periodic solutions have internal regions of rapid change internal boundary layers shock waves and geophysical fronts are one class of examples a second class is composed of functions which decay rapidly away from a central peak or peaks spherical harmonics mathieu eigenfunctions prolate spheroidal wave functions and geophysical hough functions may all be locally approximated by hermite functions in the appropriate parameter range and decay exponentially fast outside a narrow subinterval similarly the large amplitude cnoidal waves of the korteweg devries equation are narrow isolated peaks which are well approximated by the sech 2 y form of the solitary wave in this article we show that a change of coordinate is a powerful tool for resolving such internal boundary layers in the first part we develop a general theory of mappings for the spherical harmonic cnoidal wave class of examples which decay rapidly away towards the edges of the spatial period the particular map y arc tan l tan x is a particularly effective choice four numerical examples show that this map and the fourier pseudospectral method are a good team in the second part we generalize the earlier theory to describe mappings which asymptote to a constant but non zero resolution at the ends of the periodicity interval we explain why the kepler burgers mapping is particularly suitable for shock and fronts

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

finite and infinite dimensional lie group structures on riordan groups abstract we introduce a frechet lie group structure on the riordan group we give a description of the corresponding lie algebra as a vector space of infinite lower triangular matrices we describe a natural linear action induced on the frechet space k n by any element in the lie algebra we relate this to a certain family of bivariate linear partial differential equations we obtain the solutions of such equations using one parameter groups in the riordan group we show how a particular semidirect product decomposition in the riordan group is reflected in the lie algebra we study the stabilizer of a formal power series under the action induced by riordan matrices we get stabilizers in the fraction field k x using bi infinite representations we provide some examples the main tool to get our results is the paper 18 where the riordan group was described using inverse sequences of groups of finite matrices

chebyshev spectral collocation methods for laminar flow through a channel contraction abstract a chebyshev spectral element method is described for solving the navier stokes equations in a channel contraction the flow region is divided into two semi infinite elements the governing equation for the stream function is solved using a newton linearization the semi infinite elements are treated by means of algebraic type mappings to transform them onto finite domains the stream function is represented by a double chebyshev expansion in each element the coefficients are determined by collocating the linearized equation at each newton step and imposing c3 continuity conditions across the element interface in a collocation sense efficient direct methods based on capacitance matrix ideas are described which take advantage of the structure of the spectral element matrix results are presented for reynolds numbers in the range 0 200 which are in good agreement with previously published work but requiring fewer degrees of freedom only several steps of the newton process are required to achieve a converged solution for re 120 the method converges from a zero initial approximation and thereafter continuation in the reynolds number is used with increments of 10

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

bijections and the riordan group one of the cornerstone ideas in mathematics is to take a problem and to look at it in a bigger space in this paper we examine combinatorial sequences in the context of the riordan group various subgroups of the riordan group each give us a different view of the original sequence in many cases this leads to both a combinatorial interpretation and to eco rewriting rules in this paper we will concentrate on just four of the subgroups of the riordan group to demonstrate some of the possibilities of this approach

structural properties of riordan matrices and extending the matrices abstract we consider an infinite lower triangular matrix l l n k n k n 0 and a sequence n n n 0 called the a b sequence such that every element l n 1 k 1 except lying in column 0 can be expressed as l n 1 k 1 i 0 n k m i l n ai k bi 0 0 where a and b are integers with a b m 0 and b 0 this concept generalizes the a sequence of a riordan matrix as a result we explore several structural properties of riordan matrices by means of a b sequences in particular if b

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

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

riordan matrices in the reciprocation of quadratic polynomials we iterate contractive one degree polynomials with coefficients in the ring k x of formal power series to calculate the reciprocal in k x of a quadratic polynomial doing this we meet the structure of riordan array we interpret certain changes of variable as a riordan array we finish the paper by using our techniques to find new ways to get known formulas for the sum of powers of natural numbers involving stirling and eulerian numbers

hybrid chebyshev function bases for sparse spectral methods in parity mixed pdes on an infinite domain abstract we present a numerical spectral method to solve systems of differential equations on an infinite interval y in presence of linear differential operators of the form q y y b where q y is a rational fraction and b a positive integer even when these operators are not parity preserving we demonstrate how a mixed expansion in interleaved chebyshev rational functions t b n y and s b n y preserves the sparsity of their discretization this paves the way for fast o n ln n and spectrally accurate mixed implicit explicit time marching of sets of linear and nonlinear equations in unbounded geometries

spectral elements on infinite domain a spectral element technique to approximate partial differential equations pdes on an infinite domain is examined the method is based on boyd s mapping of a semiinfinite interval to a finite interval and it is extended to a variational setting which allows for an implementation using a spectral element method by extending the method to a variational form a straightforward implementation allows for high order approximations over an infinite computational domain methods to find a high order approximation have been proposed by funaro maday mavriplis and many others the proposed methods all have drawbacks including restrictions on the behavior of the approximation at infinity because the proposed method is based on a mapping to a finite domain a much wider variety of boundary conditions can be enforced on the approximation

exponentially accurate nonconforming least squares spectral element method for elliptic problems on unbounded domain abstract non conforming approximation methods are becoming increasingly popular because of the potential to apply to multi material and multi model analysis for both bounded and unbounded domains in this paper we present a least square approximation based method to solve the one or two dimensional elliptic problems on an unbounded domain the method gives exponential accuracy and shows superior performance when compared to other numerical methods differentiability estimates and the main stability estimate theorem using a non conforming spectral element method are also discussed the exponential convergence rate of the proposed method is also shown through rigorous error estimate and specific numerical examples

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

gegenbauer approximation and its applications to differential equations on the whole line abstract a gegenbauer approximation is discussed several imbedding inequalities and inverse inequalities are obtained some approximation results are given by variable transformation differential equations on the whole line are changed to certain equations on a finite interval gegenbauer polynomials are used for their numerical solutions the stabilities and convergences of proposed schemes are proved the main idea and techniques used in this paper are also applicable to other multiple dimensional problems in unbounded domains

two dimensional riemann problem for a hyperbolic system of nonlinear conservation laws ii initial data involving some rarefaction waves abstract this paper is a continuation of our first paper j differential equations in press in this paper we solve the 2 d riemann problem with the initial data projecting some contact discontinuities and rarefaction waves the solutions reveal a variety of geometric structures for the interaction of elementary waves such as contact discontinuities rarefaction waves shocks and or shocks defined in our first paper

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

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