vandermonde matrices on integer nodes the rectangular case we consider a rectangular vandermonde matrix v on integer nodes using combinatorial identities we give an explicit cholesky factorization of the matrix b v rm t v a factorization of the pseudo inverse of v and asymptotic estimates of the extremal eigenvalues of b we also discuss the numerical properties of the proposed formulas

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

vandermonde matrices on integer nodes this paper deals with vandermonde matrices widetilde v whose nodes are the first n integer numbers we give an analytic factorization of such matrices and explicit formulas for the entries of their inverses and explore their computational issues we also give asymptotic estimates of the frobenius norm of both widetilde v and its inverse

a 2d chebyshev differential operator for the elastic wave equation abstract this work analyses the performance of a two dimensional chebyshev differential operator for solving the elastic wave equation the technique allows the implementation of non periodic boundary conditions at the four boundaries of the numerical mesh which requires a special treatment of these conditions based on one dimensional characteristics in addition spatial grid adaptation by appropriate one dimensional coordinate mappings allows a more accurate modeling of complex media and reduction of the computational cost by controlling the minium grid spacing the examples illustrate the ability of the method to simulate rayleigh waves around a corner and adapt the mesh to the model geometry in addition a domain decomposition example shows how the boundary treatment handles wave propagation from one mesh to another mesh

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

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

on the inversion of the vandermonde matrix abstract the inversion of the vandermonde matrix has received much attention for its role in the solution of some problems of numerical analysis and control theory this work deals with the problem of getting an explicit formula for the generic element of the inverse we derive two algorithms in o n2 and o n3 and compare them with the parker traub and the bjorck pereyra algorithms

a dynamical pseudo spectral domain decomposition technique a dynamical spectral domain decomposition method is presented in each subdomain a transformation of coordinate is used both the locations of the interfaces and the parameters of the mappings are dynamically adapted by minimizing theh 2 norm of the calculated solution we show on some functions that the total norm of the chebyshev series depends on the location of the interfaces moreover there exists a minimum that defines the best location of the interface this defines a dynamical generation of chebyshev collocation points this numerical method is applied on partial differential equations and it is shown that both the overall accuracy and the matching at the interfaces are improved with respect to a fixed interface calculation this algorithm is then used for the numerical solution of the time dependent full navier stokes equations the solution technique consists in a fourier chebyshev collocation method combined with a matching method the computational domain is decomposed into subdomains in the vertical direction in each subdomain a coordinate transform is used and the locations of the interfaces are dynamically determined the elliptic problems coming from the viscous terms are solved by means of the chebyshev acceleration method density is matched with an upwind procedure whereas the velocities the temperature and the concentration are handled with the influence matrix method numerical examples are carried out on the compressible kelvin helmholtz and rayleigh taylor flows

boundary and interface conditions within a finite element preconditioner for spectral methods abstract the performances of a finite element preconditioner in the iterative solution of spectral collocation schemes for elliptic boundary value problems is investigated it is shown how to make the preconditioner cheap by adi iterations and how to take advantage of the finite element properties in enforcing neumann and interface conditions in the spectral schemes

exact solutions of the saturable discrete nonlinear schrodinger equation exact solutions to a nonlinear schrodinger lattice with a saturable nonlinearity are reported for finite lattices we find two different standing wave like solutions and for an infinite lattice we find a localized soliton like solution the existence requirements and stability of these solutions are discussed and we find that our solutions are linearly stable in most cases we also show that the effective peierls nabarro barrier potential is nonzero thereby indicating that this discrete model is quite likely nonintegrable

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

spectral theory for the periodic sine gordon equation a concrete viewpoint a summary of the spectral theory for quasiperiodic sine and sinh gordon equations is given analogies with whole line solitons and scattering theory motivates the discussion the relation between the ingredients in the inverse spectral solution of the periodic sine gordon equation and physical characteristics of sine gordon waves is emphasized the explicit topics covered are summarized in the table of contents in the introduction

bubble stabilization of spectral legendre methods for the advection diffusion equation abstract an advection diffusion model problem is discretized by a spectral legendre method augmented by trial test functions with local support bubbles the numerical analysis shows that the correction is equivalent to adding a spectrally accurate streamline upwind stabilization term the performances of two strategies for tuning the bubble stabilization are investigated

a fast algorithm for chebyshev fourier and sinc interpolation onto an irregular grid a chebyshev or fourier series may be evaluated on the standard collocation grid by the fast fourier transform fft unfortunately the fft does not apply when one needs to sum a spectral series at n points which are spaced irregularly the cost becomes o n2 operations instead of the ffts o n log n this sort of off grid interpolation is needed by codes which dynamically readjust the grid every few time steps to resolve a shock wave or other narrow features it is even more crucial to semi lagrangian spectral algorithms for solving convection diffusion and navier stokes problems because off grid interpolation must be performed several times per time step in this work we describe an alternative algorithm the first step is to pad the set of spectral coefficients an with zeros and then take an fft of length 3n to interpolate the chebyshev series to a very fine grid the second step is to apply either the mth order euler sum acceleration or 2m 1 point lagrangian interpolation to approximate the sum of the series on the irregular grid we show that both methods yield full precision with m n allowing an order of magnitude reduction in cost with no loss of accuracy

group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations a complete group classification of a class of variable coefficient 1 1 dimensional telegraph equations f x utt h u ux x k u ux is given by using a compatibility method and additional equivalence transformations a number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained furthermore the possible additional equivalence transformations between equations from the class under consideration are investigated exact solutions of special forms of these equations are also constructed via classical lie method and generalized conditional transformations local conservation laws with characteristics of order 0 of the class under consideration are classified with respect to the group of equivalence transformations a complete group classification of a class of variable coefficient 1 1 dimensional telegraph equations f x utt h u ux x k u ux is given by using a compatibility method and additional equivalence transformations a number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained furthermore the possible additional equivalence transformations between equations from the class under consideration are investigated exact solutions of special forms of these equations are also constructed via classical lie method and generalized conditional transformations local conservation laws with characteristics of order 0 of the class under consideration are classified with respect to the group of equivalence transformations

pseudospectral solution of near singular problems using numerical coordinate transformations based on adaptivity the work presented here describes a method of coordinate transformation that enables spectral methods to be applied efficiently to differential problems with steep solutions the approach makes use of the adaptive finite difference method presented by huang and sloan siam j sci comput 15 1994 pp 776 797 this method is applied on a coarse grid to obtain a rough approximation of the solution and a suitable adapted mesh the adaptive finite difference solution permits the construction of a smooth coordinate transformation that relates the computational space to the physical space the map between the spaces is based on chebyshev polynomial interpolation finally the standard pseudospectral ps method is applied to the transformed differential problem to obtain highly accurate nonoscillatory numerical solutions numerical results are presented for steady problems in one and two space dimensions

symmetric functions and the vandermonde matrix this work deduces the lower and the upper triangular factors of the inverse of the vandermonde matrix using symmetric functions and combinatorial identities the l and u matrices are in turn factored as bidiagonal matrices the elements of the upper triangular matrices in both the vandermonde matrix and its inverse are obtained recursively the particular value x i 1 q qi 1 in the indeterminates of the vandermonde matrix is investigated and it leads to q binomial and q stirling matrices it is also shown that q stirling matrices may be obtained from the pascal matrix

error analysis for mapped legendre spectral and pseudospectral methods a general framework is introduced to analyze the approximation properties of mapped legendre polynomials and of interpolations based on mapped legendre gauss lobatto points optimal error estimates featuring explicit expressions on the mapping parameters for several popular mappings are derived these results not only play an important role in numerical analysis of mapped legendre spectral and pseudospectral methods for differential equations but also provide quantitative criteria for the choice of parameters in these mappings

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

mappings and accuracy for chebyshev pseudo spectral approximations abstract the effect of mappings on the approximation by chebyshev collocation of functions which exhibit localized regions of rapid variation is studied a general strategy is introduced whereby mappings are adaptively constructed which map specified classes of rapidly varying functions into low order polynomials a particular family of mappings constructed in this way is tested on a variety of rapidly varying functions similar to those occurring in applications it is shown that the mapped function can be approximated much more accurately by chebyshev polynomial approximations than the original function the effect on the approximation of introducing subdomains is also studied the accuracy of the pseudo spectral approximation is very sensitive to the location of the interface although this sensitivity is reduced when mappings are employed within the subdomains

approximations by orthonormal mapped chebyshev functions for higher dimensional problems in unbounded domains this paper is concerned with approximation properties of orthonormal mapped chebyshev functions omcfs in unbounded domains unlike the usual mapped chebyshev functions which are associated with weighted sobolev spaces the omcfs are associated with the usual non weighted sobolev spaces this leads to particularly simple stiffness and mass matrices for higher dimensional problems the approximation results for both the usual tensor product space and hyperbolic cross space are established with the latter particularly suitable for higher dimensional problems

structure and dynamics of modulated traveling waves in cellular flames we describe the formation and evolution of spatial and temporal patterns in cylindrical premixed flames we consider the cellular regime le 1 where the lewis number le is the ratio of thermal to mass diffusivity of a deficient component of the combustible mixture a transition from stationary axisymmetric flames to stationary cellular flames is predicted analytically if le is decreased below a critical value we present the results of numerical computations to show that as le is further decreased with all other parameters fixed traveling waves tws along the flame front arise via an infinite period bifurcation which breaks the reflection symmetry of the cellular array upon further decreasing le we find the development of different kinds of periodically modulated traveling waves mtws as well as a branch of quasiperiodically modulated traveling waves qpmtws these transitions are accompanied by the development of different spatial and temporal symmetries including period doublings and period halvings in appropriate coordinate systems we also observe the apparently chaotic temporal behavior of a disordered cellular pattern involving creation and annihilation of cells we analytically describe the stability of the tw solution near its onset using suitable phase amplitude equations within this framework one of the mtws can be identified as a localized wave traveling through an underlying stationary spatially periodic structure

construction of lighthill s unitary functions the imbricate series of unity the poisson sum theorem implies that every spatially periodic function f x can be written as a so called imbricate series that is as the superposition of an infinite number of evenly space copies of a pattern function a x the pattern is the fourier transform of a function a n that specifies the usual fourier coefficients because there are many a n which coincide with a given set of fourier coefficients aj when n is an integer the pattern function a x is not unique in this work we show how to construct pattern functions of the form a x f x w x where f x is the periodic function itself the imbrication of a x and where w x is what lighthill called a unitary function we derive an explicit formula to construct all possible unitary functions and tabulate eight examples we also prove that all unitary functions are pattern functions of unity that is for the imbricate series that sums to f x 1 x

complex mapped matrix methods in hydrodynamic stability problems the ordinary differential equations governing the linear stability of inviscid flows contain singularities at real or complex points called critical latitudes which degrade the accuracy of standard numerical schemes however the use of a complex mapping prior to the numerical attack offers some respite this mapping shifts the computational domain to a contour in the complex plane to avoid the critical latitudes both quadratic and cubic complex maps are considered in some detail an analytic result for choosing the optimum quadratic complex map in the case of a single critical latitude is presented numerical results are given for two test problems and a barotropic vortex model a comparison is made between methods with and without these mappings the results show that the use of complex maps can lead to remarkably accurate solutions

preliminary group classification for the nonlinear wave equation utt f x u uxx g x u abstract a classification is given for the nonlinear wave equation u t t f x u u x x g x u admitting an extension by one dimension of the principal lie algebra of the equation under consideration the paper is one of the few applications of an algebraic approach to the problem of group classification the method of preliminary group classification a number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained the result of the work is a wide class of equations summarized in table form

on invariance properties of the wave equation a complete group classification is given of both the wave equation c2 x uxx utt 0 i and its equivalent system vt ux c2 x vx ut ii when the wave speed c x const equations i and ii admit either a two or four parameter group for the exceptional case c x ax b 2 equation i admits an infinite group equations i and ii do not always admit the same group for a given c x the group for i can have more parameters or fewer parameters than that for ii moreover the groups can be different with the same number of parameters separately for i and ii all possible c x that admit a four parameter group are found explicitly the corresponding invariant similarity solutions are considered some of these wave speeds have realistic physical properties c x varies monotonically from one positive constant to another positive constant as x goes from to

solitons from sine waves analytical and numerical methods for non integrable solitary and cnoidal waves abstract the fkdv equation u t uu x u xxxxx 0 is used as a testbed for a variety of analytical and numerical methods that can be applied to solitary waves and cnoidal waves of non integrable differential equations that is to say to equations which cannot be solved by the inverse scattering transform the basic tools are i pade approximants formed from power series in the amplitude ii a newton kantorovich pseudospectral fourier continuation numerical method iii singular perturbation theory for two interacting solitons of almost identical phase speed iv bifurcation and branch switching methods v the imbricate soliton series a number of new results for the fkdv equation are obtained including extensive numerical calculations of the spatially periodic solutions with one peak cnoidal wave and two peaks bicnoidal wave per period an analytical expression for the double peaked soliton bion calculation of both the limit and bifurcation points for the bicnoidal wave and finally the computation of accurate analytical approximations to the cnoidal wave for all amplitudes more important all of these analytical and numerical tools are highly effective for this equation in spite of the fact that it cannot be solved by the inverse scattering transform work now in progress will apply these methods to non integrable equations in two space dimensions

an analytic approximation to the cardinal functions of gaussian radial basis functions on an infinite lattice gaussian radial basis functions rbfs have been very useful in computer graphics and for numerical solutions of partial differential equations where these rbfs are defined on a grid with uniform spacing h as translates of the master function f x a h exp a 2 h 2 x 2 where a is a user choosable constant unfortunately computing the coefficients of f x jh a h requires solving a linear system with a dense matrix it would be much more efficient to rearrange the basis functions into the equivalent lagrangian or cardinal basis because the interpolation matrix in the new basis is the identity matrix the cardinal basis c j x a h is defined by the set of linear combinations of the gaussians such that c j kh 1 when k j and c j kh 0 for all integers k j we show that the cardinal functions for the uniform grid are c j x h a c x h j a where c x a a 2 p sin px sinh a 2x the relative error is only about 4exp email protected 2 a 2 as demonstrated by the explicit second order approximation it has long been known that the error in a series of gaussian rbfs does not converge to zero for fixed a as h 0 but only to an error saturation proportional to exp email protected 2 a 2 because the error in our approximation to the master cardinal function c x a is the square of the error saturation there is no penalty for using our new approximations to obtain matrix free interpolating rbf approximations to an arbitrary function f x the master cardinal function on a uniform grid in d dimensions is just the direct product of the one dimensional cardinal functions thus in two dimensions c x y a a 4 p 2 sin px sin py sinh a 2x sinh a 2y we show that the matrix free interpolation can be extended to non uniform grids by a smooth change of coordinates

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

new series for the cosine lemniscate function and the polynomialization of the lemniscate integral we discuss the numerical computation of the cosine lemniscate function and its inverse the lemniscate integral these were previously studied by bernoulli euler gauss abel jacobi and ramanujan we review general elliptic formulas for this special case and provide some novelties we show that a fourier series by ramanujan converges twice as fast as the standard elliptic cosine fourier series specialized to this case the so called imbricate series however converges geometrically fast over the entire complex plane we derive two new expansions the rational plus fourier series converges much faster than ramanujan s for real z each term is asymptotically 12 400 times smaller than its immediate predecessor coslem z 4b q 1 q cos bz 1 q 2 4qcos 2 bz n 1 q n 1 2 1 1 q 2 n 1 1 cos 2n 1 bz where q exp p is the elliptic nome k 1 85 is the complete elliptic integral of the first kind for a modulus m 1 2 and b p k2 the rational imbricate series is uniformly valid over the complex plane but converges twice as fast as the sech imbricate series coslem z 4bq 1 q j q 1 q cos bz 1 q 2 4qcos 2 b z jpi where p 4 2 k is the period in both the real and imaginary directions we devise a new approximation for the lemniscate integral for real argument as the arccosine of a chebyshev series and show that 17 terms yield about 15 digits of accuracy for complex argument we show that the lemniscate integral can be found to near machine precision assumed as sixteen decimal digits by computing the roots of a polynomial of degree thirteen alternatively newton s iteration converges in three iterations with an initialization accurate to four decimal places that is the chosen root of a cubic equation

chebyshev fourier spectral methods in bipolar coordinates bipolar coordinates provide an efficient cartography for a variety of geometries the exterior of two disks or cylinders a half plane containing a disk an eccentric annulus with a small disk offset from the center of an outer boundary that is a large circle and so on a pseudospectral method that employs a tensor product basis of fourier functions in the cyclic coordinate and chebyshev polynomials in the quasi radial coordinate gives easy to program spectral accuracy we show however that as the inner disk becomes more and more offset from the center of the outer boundary circle the grid is increasingly non uniform and the rate of exponential convergence increasingly slow one dimensional coordinate mappings significantly reduce the non uniformity in spite of this non uniformity the chebyshev fourier method is quite effective in an idealized model of the wind driven ocean circulation resolving both internal and boundary layers bipolar coordinates are also a good starting point for solving problems in a domain which is not one of the bipolar compatible domains listed above but is a sufficiently small perturbation of such this is illustrated by applying boundary collocation with bipolar harmonics to solve laplace s equation in a perturbed eccentric annulus in which the disk shaped island has been replaced by an island bounded by an ellipse similarly a perturbed bipolar domain can be mapped to an eccentric annulus by a smooth change of coordinates

a two dimensional adaptive pseudo spectral method we develop a two dimensional adaptive pseudo spectral procedure which is capable of improving the approximation of functions which are rapidly varying in two dimensions the method is based on introducing two dimensional coordinate transformations chosen to minimize certain functionals of the solution to be approximated the method is illustrated by numerical computation of the solutions to a system of reaction diffusion equations modeling the gasless combustion of a solid fuel spatio temporal patterns are computed as a parameter related to the activation energy is increased above a critical value c the spatial patterns are characterized by a very rapid variation in the direction of the axis of the cylinder together with a standing wave pattern in the direction of the azimuthal angle for small values of c the solutions exhibit a nearly sinusoidal dependence in both time and as is increased further relaxation oscillations in both time and occur beyond a critical value of stable time periodic solutions are no longer found and the solution exhibits a quasi periodic time dependence

adaptive spectral methods with application to mixing layer computations this paper reports some experiments on the use of adaptive chebyshev pseudospectral methods for compressible mixing layer computations different functionals measuring the optimality of the polynomial approximation are discussed and compared in particular we address the problem of the practical computation of the various functionals the utility of the self adaptive method is then demonstrated by some examples from compressible mixing layer calculations

three dimensional fully spectral numerical method for mantle convection with depth dependent properties a semi implicit fully spectral collocation method for the simulation of three dimensional mantle convection with depth dependent thermodynamic and transport properties is presented the variable property navier stokes equation expressed in terms of the primitive variable velocity and pressure is solved with the mass continuity and temperature equations the periodic horizontal boundary conditions allow a fourier expansion for the two horizontal directions the stress free impermeable isothermal boundary conditions along with the depth dependent coefficients are handled with a chebyshev expansion in the vertical direction in the limit of an infinite prandtl number appropriate to mantle convection the inertial terms in the momentum equation are unimportant in this case an explicit solution of a poisson equation for pressure can be avoided instead a fourth order equation for vertical velocity can be solved simultaneous imposition of both impermeable and continuity boundary conditions during the vertical velocity evaluation is discussed the pressure distributions on the top and bottom bounding planes were determined by means of an influence matrix technique the numerical method employed here avoids time splitting errors and enforces velocity boundary conditions and continuity over the entire domain including the boundaries to machine accuracy strongy time dependent three dimensional solutions up to a surface rayleigh number of 1 107 have been obtained strong upwellings pulsating chaotically are formed by the collective merging of cylindrical plumes

sharp interface limits of the cahn hilliard equation with degenerate mobility in this work sharp interface limits for the degenerate cahn hilliard equation with a polynomial double well free energy and a mobility that vanishes at the minima of the double well are derived for the choice of a quadratic mobility the leading order sharp interface motion is not governed by pure surface diffusion as has been previously claimed in the literature but contains a contribution from nonlinear porous medium type bulk diffusion at the same order our analysis reveals that there are two subcases one where the solution for the order parameter is bounded between the minima proven to exist for the first mobility by elliott and garcke siam j math anal 27 1996 pp 404 423 and one where it converges to the classical stationary solution of the cahn hilliard equation consistent treatment of the bulk diffusion requires the matching of exponentially large and small terms in combination with multiple inner layers moreover the leading order sharp interface motion depends sensitively on the choice of mobility the asymptotic analysis shows that for example with a biquadratic mobility the leading order sharp interface motion is driven only by surface diffusion the sharp interface models are corroborated by comparing relaxation rates of perturbations to a radially symmetric stationary state with those obtained by the phase field model

theta functions gaussian series and spatially periodic solutions of the korteweg de vries equation it has been shown by novikov funct anal appl 8 236 1974 dubrovin et al russian math surveys 31 59 1976 lax commun pure appl math 28 141 1975 mckean and van moerbeke inv math 30 217 1975 and others that the nonlinear evolution equations which admit solitary waves also have spatially periodic exact solutions polycnoidal waves which can be expressed in terms of multidimensional riemann theta functions here it is shown that via poisson summation the fourier series that define the theta functions can be transformed into an infinite series of gaussian functions because the lowest terms of the gaussian series generate the usual solitary waves it is possible to intimately explore the relationship between solitary waves and these spatially periodic polycnoidal waves also by using the gaussian series one can perturbatively calculate phase velocities and wave structure for the polycnoidal wave even in the strongly nonlinear regime for which the soliton or mult

an extended procedure for finding exact solutions of partial differential equations arising from potential symmetries applications to gas dynamics lie point symmetries and nonlocal symmetries of partial differential equation pde systems are widely used for construction of exact invariant solutions in this paper we describe an extended algorithmic procedure that for a given nonlocal potential symmetry can yield additional exact solutions which cannot be found using the usual algorithm in particular such additional solutions are exact solutions of the given pde system but are not invariant solutions of the corresponding potential system as an example we consider a tree of nonlocally related pde systems for lagrange planar gas dynamics equations and classify its nonlocal symmetries for an ideal polytropic gas for two different nonlocal symmetries of the lagrange system we demonstrate that the extended method yields wider classes of exact solutions than the usual method

new conservation laws obtained directly from symmetry action on a known conservation law two formulas are introduced to directly obtain new conservation laws for any system of partial differential equations from a known conservation law and admitted symmetries the first formula maps any conservation law of a given system to the corresponding conservation law of the system obtained through a contact transformation when the contact transformation is a symmetry of the given system then the corresponding conservation law is a conservation law of the given system the second formula checks a priori whether or not the action of a symmetry continuous or discrete on a conservation law can yield one or more new conservation laws of the given system several examples are considered including the use of a discrete symmetry to obtain a new conservation law and the use of a continuous symmetry to generate two new conservation laws

parallel algorithms with local fourier basis abstract we present a nonoverlapping domain decomposition method with local fourier basis applied to a model problem in liquid flames the introduction of domain decomposition techniques in this paper is for numerical and parallel efficiency purposes when one requires a large number of grid points to catch complex structures we obtain then a high order accurate domain decomposition method that allows us to generalize our previous work on the use of local fourier basis to solve combustion problems with nonperiodic boundary conditions m garbey and d tromeur dervout j comput phys 145 316 1998 local fourier basis methodology fully uses the superposition principle to split the searched solution in a numerically computed part and an analytically computed part our present methodology generalizes the israeli et al 1993 j sci comput 8 135 method which applies domain decomposition with local fourier basis to the helmholtz s problem in the present work several new difficulties occur first the problem is unsteady and nonlinear which makes the periodic extension delicate to construct in terms of stability and accuracy second we use a streamfunction biharmonic formulation of the incompressible navier stokes equation in two space dimensions the application of domain decomposition with local fourier basis to a fourth order operator is more difficult to achieve than for a second order operator a systematic investigation of the influence of the method s parameters on the accuracy is done a detail parallel mimd implementation is given we give an a priori estimate that allows the relaxation of the communication between processors for the interface problem treatment results on nonquasi planar complex frontal polymerization illustrate the capability of the method

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

loop soliton interaction in an integrable nonlinear telegraphy model reciprocal and backlund transformations an integrable telegraphy equation linked in turn to the geometry of pseudospherical surfaces and a class of model nonlinear elastic stress strain laws descriptive of ideally hard material response is investigated a constant coefficient ideal formalism is introduced which leads to a reciprocal transformation between this nonlinear telegraphy equation and a system compatible with the classical sine gordon equation a backlund transformation is then constructed which is used to analyse two loop soliton interaction the latter exhibits a novel exchange particle phenomenon

asymptotic numerical study of supersensitivity for generalized burgers equations this article addresses some asymptotic and numerical issues related to the solution of burgers equation varepsilon u xx u t u u x 0 on 1 1 subject to the boundary conditions u 1 1 delta u 1 1 and its generalization to two dimensions varepsilon delta u u t u u x u u y 0 on 1 1 times pi pi subject to the boundary conditions u x 1 1 delta u x 1 1 with 2 pi periodicity in y the perturbation parameters delta and varepsilon are arbitrarily small positive and independent when they approach 0 they satisfy the asymptotic order relation delta o s rm e a varepsilon for some constant a in 0 1 the solutions of these convection dominated viscous conservation laws exhibit a transition layer in the interior of the domain whose position as t to infty is supersensitive to the boundary perturbation algorithms are presented for the computation of the position of the transition layer at steady state the algorithms generalize to viscous conservation laws with a convex nonlinearity and are scalable in a parallel computing environment

conservation laws for nonlinear telegraph equations a complete conservation law classification is given for nonlinear telegraph nlt systems with respect to multipliers that are functions of independent and dependent variables it turns out that a very large class of nlt systems admits four nontrivial local conservation laws the results of this work are summarized in tables which display all multipliers fluxes and densities for the corresponding conservation laws a physical example is considered for possible applications

the exponential accuracy of fourier and chebyshev differencing methods it is shown that when differencing analytic functions using the pseudospectral fourier or chebyshev methods the error committed decays to zero at an exponential rate

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

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

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

rectangular vandermonde matrices on chebyshev nodes abstract we consider a rectangular vandermonde matrix v on chebyshev nodes using combinatorial identities we give an explicit factorization of the matrix v a factorization of thepseudo inverse of v and prove that the condition number 2 v does not depend onthe dimension of nodes set we also discuss the numerical properties of the proposed formulas

spectral implementation of an adaptive moving mesh method for phase field equations phase field simulations have been extensively applied to modeling microstructure evolution during various materials processes however large scale simulations of three dimensional 3d microstructures are still computationally expensive among recent efforts to develop advanced numerical algorithms the semi implicit fourier spectral method is found to be particularly efficient for systems involving long range interactions as it is able to utilize the fast fourier transforms fft on uniform grids in this paper we report our recent progress in making grid points spatially adaptive in the physical domain via a moving mesh strategy while maintaining a uniform grid in the computational domain for the spectral implementation this approach not only provides more accurate treatment at the interfaces requiring higher resolution but also retains the numerical efficiency of the semi implicit fourier spectral method numerical examples using the new adaptive moving mesh semi implicit fourier spectral method are presented for both two and three space dimensional microstructure simulations and they are compared with those obtained by other methods by maintaining a similar accuracy the proposed method is shown to be far more efficient than the existing methods for microstructures with small ratios of interfacial widths to the domain size

an adaptive pseudo spectral method for reaction diffusion problems the spectral interpolation error was considered for both the chebyshev pseudo spectral and galerkin approximations a family of functionals i sub r u with the property that the maximum norm of the error is bounded by i sub r u j sub r where r is an integer and j is the degree of the polynomial approximation was developed these functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize i sub r u the number of collocation points is then chosen to maintain a prescribed error bound the method is illustrated by various examples from combustion problems in one and two dimensions

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

lower bounds for the condition number of vandermonde matrices we derive lower bounds for the condition number of then n vandermonde matrixvn x in the cases where the node vectorxt x1 x2 xn has positive elements or real elements located symmetrically with respect to the origin the bounds obtained grow exponentially inn witho 2n ando 2n 2 respectively we also compute the optimal spectral condition numbers ofvn x for the two node configurations including the optimal nodes and compare them with the bounds obtained

conservation laws for some systems of nonlinear pdes via the symmetry adjoint symmetry pair method abstract this paper applies the symmetry adjoint symmetry pair method to find conservation laws for nonlinear partial differential equations as examples conservation laws are obtained for a combined potential system with nonlinear telegraph and nonlinear diffusion convection equations as special cases as well as for an extended nonlinear wave equation system in addition for both examples conservation laws are obtained for and nonlocally related equivalent systems the paper concludes with a posed open problem

periodic solutions of the intermediate long wave equation a nonlinear superposition principle periodic stationary wave solutions of the intermediate long wave ilw equation are derived using the bilinear transformation method and a new expression for the dispersion relation is obtained the class of physically important real valued solutions is identified these solutions may be represented as an infinite superposition of solitary wave profiles a property shared by the related korteweg de vries kdv and benjamin ono bo equation this nonlinear superposition principle which has been the subject of various interpretations in the literature is discussed the ilw periodic solution approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes respectively for intermediate amplitudes the solution can be well approximated by either a sine wave or solitary wave in the shallow water kdv limit the ilw periodic solution leads to the familiar cnoidal wave whereas the deep water bo limit yields benjamin s periodic wave a previously unknown expression for the cnoidal wave dispersion relation in terms of theta functions is obtained

an efficient spectral method for ordinary differential equations with rational function coefficients the authors present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients they employ expansions in terms of a large class of orthogonal polynomial families including all the classical orthogonal polynomials these families obey a simple 3 term recurrence relation for differentiation which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse the inverse is an integration operator for the family and it is simply the tridiagonal coefficient matrix for the recurrence since in these families convolution operators i e matrix representations of multiplication by a function are banded for polynomials the authors are able to obtain a banded representation for linear differential operators with rational coefficients this leads to a method of solution of initial or boundary value problems that besides having an operation count that scales linearly with the order of truncation n is computationally well conditioned among the applications considered is the use of rational maps for the resolution of sharp interior layers 19 refs 2 figs 3 tabs

lattice soliton solutions to the kadomtsev petviashvili equation with positive dispersion lattice soliton solutions that have doubly periodic array of the localized structures in the plane are presented to the kadomtsev petviashvili equation with the positive dispersion by using the superposition of the rational functions each structure is similar to the rational soliton the existence condition and some properties are also presented

group properties of generalized quasi linear wave equations abstract in this paper complete group classification of a class of 1 1 dimensional generalized quasi linear wave equations is performed by using the lie ovsiannikov method additional equivalent transformation and furcate split method lie reductions of some truly variable coefficient wave equations which are singled out from the classification results are investigated some classes of exact solutions of these variable coefficient wave equations are constructed by means of both the reductions and the additional equivalent transformations the nonclassical symmetries to the generalized quasi linear wave equation are also studied this enabled to obtain some exact solutions of the wave equations which are invariant under certain conditional symmetries

comparing symmetries and conservation laws of nonlinear telegraph equations a comparison is made between the symmetries and conservation laws admitted by nonlinear telegraph nlt systems such systems are not variational unlike the situation for variational systems where all conservation laws arise from symmetries there are many nlt systems that admit more conservation laws than symmetries the results are summarized in a table which includes the numbers of symmetries and conservation laws for each nlt system it is also indicated when symmetries map conservation laws to other conservation laws

on exact solutions of the regularized long wave equation a direct approach to partially integrable equations ii periodic solutions in this second of two articles designated i and ii the bilinear transformation method is used to obtain stationary periodic solutions of the partially integrable regularized long wave rlw equation these solutions are expressed in terms of riemann theta functions and this approach leads to a new and compact expression for the important dispersion relation the periodic solution or cnoidal wave can be represented as an infinite sum of sech2 solitary waves this remarkable property may be interpreted in the context of a nonlinear superposition principle the rlw cnoidal wave approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes respectively analytic approximations and error estimates are given which shed light on the character of the cnoidal wave in the different parameter regimes similar results are presented in brief for the related rlw boussinesq rlwb equation

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

vandermonde systems on equidistant nodes in 0 1 accurate computation this paper deals with vandermonde matrices v whose nodes are the equidistant points in 0 1 we give an analytic factorization and explicit formula for the entries of their inverse and explore its computational issues we also give asymptotic estimates of the frobenius norm of both v and its inverse and show that a new representation of the floating point number system allows one to build an accurate algorithm for the interpolation problem on equidistant nodes in 0 1

global properties of pseudospectral methods abstract we apply polynomial interpolation methods both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations the derivative matrix for a general sequence of collocation points is explicitly constructed we explore the effect of several factors on the performance of these methods we show that global methods cannot be interpreted in terms of local methods for example the accuracy of the approximation differs when large gradients of the function occur near the center of the region or in the vicinity of the boundary this difference does not depend on the density of the collocation points near the boundaries hence intuition based on finite difference methods can lead to false results

peakons and coshoidal waves traveling wave solutions of the camassa holm equation abstract camassa and holm 1 have recently derived a new integrable wave equation for the special case 0 they showed it has solitary waves of the form c exp x ct which they named peakons in this work we derive a perturbation series for general which converges even at the peakon limit we also give three analytical representations for the spatially periodic generalization of the peakon the coshoidal wave the three representations are i a closed form analytical solution ii a fourier series with coefficients that are explicit rational functions and iii an imbricate fourier series which is the superposition of an infinite number of peakons each separated from its neighbors by distance p where p is the spatial period lastly we have numerically tested the soliton superposition principle although the camassa holm equation is integrable for general it appears that imbricating solitary waves generates an exact spatially periodic solution only for the special cases 0 c 1 2 however the imbricate soliton series is a very good approximate solution for general even when the spatial period is small and the solution resembles a sine wave more than a solitary wave

some novel nonlinear coherent excitations of the davey stewartson system exact solutions of many integrable 2 1 2 spatial and 1 temporal dimensional systems of nonlinear evolution equations e g the davey stewartson model can be obtained by a special separation of variables procedure by choosing the jacobi elliptic functions as the building blocks exact doubly periodic solutions are obtained analytically here two sets of elliptic functions with two different independent moduli are employed and the resulting wave packets are expressed as rational functions of elliptic functions by taking the long wave limit in one spatial variable peculiar wave patterns localized in one direction but periodic in the other direction will arise by taking the long wave limit in both spatial variables exponentially localized wave patterns which differ from the known dromions will result the boundary conditions relating to these localized structures are studied

framework for nonlocally related partial differential equation systems and nonlocal symmetries extension simplification and examples any partial differential equation pde system can be effectively analyzed through consideration of its tree of nonlocally related systems if a given pde system has n local conservation laws then each conservation law yields potential equations and a corresponding nonlocally related potential system moreover from these n conservation laws one can directly construct 2n 1 independent nonlocally related systems by considering these potential systems individually n singlets in pairs n n 1 2couplets taken all together one n plet in turn any one of these 2n 1 systems could lead to the discovery of new nonlocal symmetries and or nonlocal conservation laws of the given pde system moreover such nonlocal conservation laws could yield further nonlocally related systems a theorem is proved that simplifies this framework to find such extended trees by eliminating redundant systems the planar gas dynamics equations and nonlinear telegraph equations are used as illustrative examples many new local

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

the 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

framework for potential systems and nonlocal symmetries algorithmic approach an algorithmic framework is presented to find an extended tree of nonlocally related systems for a given system of differential equations des each system in an extended tree is equivalent in the sense that the solution set for any system in a tree can be found from the solution set for any other system in the tree useful conservation laws play an essential role in the construction of an extended tree a useful conservation law yields potential variables and equivalent nonlocally related potential systems and subsystems for any given system nonlocal symmetries for a given system of des can arise from any system in its extended tree we construct extended trees for the systems of planar gas dynamics and nonlinear telegraph equations and in both cases obtain new nonlocal symmetries more importantly due to the equivalence of solution sets any coordinate independent method of analysis qualitative numerical perturbation etc can be applied to any system within the tree and may yield simpler computations and or results that cannot be obtained when the method is directly applied to the given system

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

matrix approach to polynomials 2 abstract we continue our study of the structure initiated in t arponen a matrix approach to polynomials linear algebra appl 359 2003 181 196 our main emphasis is exploring further structure into the formalism introduced in t arponen a matrix approach to polynomials linear algebra appl 359 2003 181 196 this formalism reveals beautiful interplay between certain elementary operators and provides tools for example for checking handling and generalizing combinatorial identities as we show in examples in addition to that we discover a group structure among vandermonde matrices a fascinating diophantine equation and new proofs and generalizations of recently found related results

cnoidal waves as exact sums of repeated solitary waves new series for elliptic functions via poisson summation of the corresponding fourier series it is shown that the cnoidal wave of the korteweg de vries equation can be written as an infinite sum of periodically repeated solitary waves a result first proved by toda through a different method similar series of hyperbolic secant or tangent functions are also derived for the elliptic dn sn and cn functions which make it possible to express the cnoidal wave of almost any evolution equation in terms of a series of the corresponding solitary waves as a by product new series are also obtained for the complete elliptic integral k and modulus k

group theoretical framework for potential symmetries of evolution equations we develop algebraic approach to the problem of classification of potential symmetries of nonlinear evolution equations it is essentially based on the recently discovered fact r zhdanov j math phys 50 053522 2009 that any such symmetry is mapped into a contact symmetry the approach enables using the classical results on classification of contact symmetries of nonlinear evolution equations by sokolov and magadeev to classify evolution equations admitting potential symmetries we construct several examples of new nonlinear fourth order evolution equations admitting potential symmetries since the symmetries obtained depend on nonlocal variables they cannot be derived by the infinitesimal lie approach

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

local and nonlocal symmetries for nonlinear telegraph equation in this paper local and nonlocal symmetry classifications are considered for four equivalent nonlinear telegraph equations a complete potential symmetry classification of a scalar nonlinear telegraph equation is given through the point symmetry classification of a related potential system six new classes of equations are shown to admit potential symmetries the relationships between local including contact and nonlocal potential symmetries of these equations are explored a physical example is considered for possible applications of the obtained potential symmetries

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

a consistent equilibrium in a cross section of an elastic plastic beam abstract a phenomenon of inequality of equilibrium and constitutive internal forces in a cross section of elastic plastic beams is common to many finite element formulations it is here discussed in a rate independent elastic plastic beam context and a possible treatment is presented the starting point of our discussion is reissners finite strain beam theory and its finite element implementation the questions of the consistency of interpolations for displacements and rotations and the related locking phenomena are fully avoided by considering the rotation function of the centroid axis of a beam as the only unknown function of the problem approximate equilibrium equations are derived by the use of the distribution theory in conjunction with the collocation method the novelty of our formulation is an inclusion of a balance function that measures the error between the equilibrium and constitutive bending moments in a cross section an advantage of the present approach is that the locations where the balance of equilibrium and constitutive moments should be satisfied can be prescribed in advance in order to minimize the error explicit analytical expressions are used for the constitutive forces for a rectangular cross section and bilinear constitutive law they are given in appendix a the comparison between the results of the two finite element formulations the one using consistent and the other inconsistent equilibrium in a cross section is shown for a cantilever beam subjected to a point load the problem of high curvature gradients in a plastified region is also discussed and solved by using an adapted collocation method in which the coordinate system is transformed such to follow high gradients of curvature

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