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

growth and integrability in the dynamics of mappings the growth of some numerical characteristics of the mappings under their iterations in the context of the general problem of integrability is discussed in the general case such characteristics as complexity by arnold or the number of the different images for the multiple valued mappings are growing exponentially it is shown that the integrability is closely related with thepolynomial growth the analogies with quantum integrable systems are discussed

diophantine non integrability of a third order recurrence with the laurent property we consider a one parameter family of third order nonlinear recurrence relations each member of this family satisfies the singularity confinement test has a conserved quantity and moreover has the laurent property all of the iterates are laurent polynomials in the initial data however we show that these recurrences are not diophantine integrable according to the definition proposed by halburd 2005 j phys a math gen 38 l1 explicit bounds on the asymptotic growth of the heights of iterates are obtained for a special choice of initial data as a by product of our analysis infinitely many solutions are found for a certain family of diophantine equations studied by mordell that includes markoff s equation

recovery of a source term or a speed with one measurement and applications we study the problem of recovery the source a t x f x in the wave equation in anisotropic medium with a known so that a 0 x 6d 0 with a single measurement we use carleman estimates combined with geometric arguments and give sharp conditions for uniqueness we also study the non linear problem of recovery the sound speed in the equation utt c x u d 0with one measurement we give sharp conditions for stability as well an application to thermoacoustic tomography is also presented

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

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

bilinear discrete painleve equations based on the results of singularity confinement we derive bilinear expressions for the discrete painleve equations in these cases where a bilinear expression is not sufficient we obtain trilinear or higher multilinear expressions we show that the bilinear approach provides a natural framework for the derivation of backlund and schlesinger transforms for the discrete painleve equations

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

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

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

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

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

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

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

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

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure