computing the hilbert transform on the real line we introduce a new method for computing the hilbert transform on the real line it is a collocation method based on an expansion in rational eigenfunctions of the hilbert transform operator and implemented through the fast fourier transform an error analysis is given and convergence rates for some simple classes of functions are established numerical tests indicate that the method compares favorably with existing methods

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

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

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

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

vers un algorithme pour la reduction stable des revetements p cycliques de la droite projective sur un corps p adique in his ph d thesis c lehr offers an algorithm which gives the stable model for p cyclic covers of the projective line over a p adic field under the conditions that the branch locus whose cardinal is m 1 has the so called equidistant geometry and m p in this note we give an algorithm also in the equidistant geometry case but without condition on m in particular we are able to study the reduction at 2 of hyperelliptic curves with equidistant branch locus

finding the zeros of a univariate equation proxy rootfinders chebyshev interpolation and the companion matrix when a function f x is holomorphic on an interval x in a b its roots on the interval can be computed by the following three step procedure first approximate f x on a b by a poly

a complete orthonormal system of functions in l 2 infty infty space a new complete orthonormal system of functions in the l 2 infty infty space is introduced the system consists of two sequences composed of odd and even functions respectively unlike hermite and laguerre sets of functions which behave exponentially at infinity the new system exhibits asymptoticbehavior x 1 for the odd sequence and x 2 for the even one formulae representing products derivatives etc in series in the system are developed a nonlinear differential equation with a requirement for summability of the square of its solutions instead of boundary conditions is solved the example displays most of the features of the new method important for essentially nonlinear problems

a stable toolkit method in quantum control recently the toolkit discretization introduced to accelerate the numerical resolution of the time dependent schrodinger equation arising in quantum optimal control problems demonstrated good results on a large range of models however when coupling this class of methods with the so called monotonically convergent algorithms numerical instabilities affect the convergence of the discretized scheme we present an adaptation of the toolkit method which preserves the monotonicity of the procedure the theoretical properties of the new algorithm are illustrated by numerical simulations

a chebyshev rational chebyshev spectral method for the helmholtz equation in a sector on the surface of a sphere defeating corner singularities when the boundaries of a domain meet at an angle the solutions to an elliptic partial differential equation will usually be singular at the corner using the example of the helmholtz equation on the surface of a sphere in a domain bounded by meridians we show how corner singularities can be defeated by mapping the corner to infinity by applying a chebyshev series in longitude and a rational chebyshev series in the mercator coordinate y arctanh cos colatitude we obtain an exponential rate of convergence despite the corner singularities

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

the crane equation uuxx 2 the boundary value problem uuxx 2 appears in cranes theory of laminar convection from a point source we show that the solution is real only when x 2 on this interval denoting the constants of integration by a and s the general solution is av xs a where the crane function v is the parameter free function v exp erfinv 2 x 2 and erfinv z is the inverse of the error function v x is weakly singular at both endpoints its chebyshev polynomial coefficients an decrease proportionally to 1 n3 exponential convergence can be restored by writing v x n 0a2nt2n z x where the mapping is z arctanh x l2 arctanh x 2 2 another option is singular basis functions v 1x2 2 10 216log 1x2 2 has a maximum pointwise error that is less 1 2000 of the maximum of the crane function

rational chebyshev series for the thomas fermi function endpoint singularities and spectral methods we solve the thomas fermi problem for neutral atoms u y y 1 y u 3 2 0 on y 0 with u 0 1 and u 0 using rational chebyshev functions tl n y l to illustrate some themes in solving differential equations on a semi infinite interval l is a user choosable numerical parameter the thomas fermi equation is singular at the origin giving a tl convergence rate of only fourth order but this can be removed by the change of variables z y with v z u y z the function v z decays as z with a term in z 3 which is consistent with a geometric rate of convergence however the asymptotic series has additional terms with irrational fractional powers beginning with z 4 5 4 4 in spite of the faster spatial decay the irrational powers degrade the convergence rate to slightly larger than tenth order this vividly illustrates the subtle connection between the spatial decay of u x and the decay with degree of its rational chebyshev series the tl coefficients a n l are hostages to a tug of war between a singularity on the negative real axis which gives a geometric rate of convergence that slows with increasingl and the slow inverse power decay for large z which gives quasi tenth order convergence with a proportionality constant that decreasesinversely as a power of l for l 2 we can approximate u y 0 v z z 0 to 1 part in a million with a truncation n of only 20 l 64 and n 600 gives u y 0 1 5880710226113753127186845 correct to 25 decimal places

weakly non local solutions for capillary gravity waves fifth degree korteweg de vries equation hunter and scheurle have shown that capillary gravity water waves in the vicinity of bond number bo l 3 are consistently modelled by the korteweg de vries equation with the addition of a fifth derivative term this wave equation does not have strict soliton solutions for bo l 3 because the near solitons have oscillatory wings that extend indefinitely from the core of the wave however these solutions are arbitrarily small perturbations of solitary waves because the amplitude of the wings is exponentially small in the amplitude 6 of the core pomeau ramani and grammaticos have calculated the amplitude of the wings by applying matched asymptotics in the complex plane in the limit e 0 in this article we describe a mixed chebyshev radiation function pseudospectral method which is able to calculate the weakly non local solitons for all e we show that for fixed phase speed the solitons form a three parameter family because the linearized wave equation has three eigensolutions we also show that one may repeat the soliton with even spacing to create a three parameter of periodic solutions which we also compute because the amplitude of the wings is exponentially small these non local capillary gravity solitons are as interesting as the classical localized solitons that solve the korteweg de vries equation

traps and snares in eigenvalue calculations with application to pseudospectral computations of ocean tides in a basin bounded by meridians we make several observations about eigenvalue problems using as examples laplace s tidal equations and the differential equation satisfied by the associated legendre functions whatever the discretization only some of the eigenvalues of then dimensional matrix eigenvalue problem will be good approximations to those of the differential equation usuallythen 2 eigenvalues of smallest magnitude for the tidal problem however the good eigenvalues are scattered so our first point is it is important to plot the drift of eigenvalues with changes in resolution we suggest plotting the difference between a low resolution eigenvalue and the nearest high resolution eigenvalue divided by the magnitude of the eigenvalue or the intermodal separation whichever is smaller second as a final safeguard it is important to look at the chebyshev coefficients of the mode we show a numerically computed anti kelvin wave which has little eigenvalue drift but is completely spurious as is obvious from its spectral series third inverting the roles of parameters can drastically modify the spectrum legendre s equation may have either an infinite number of discrete modes or only a handful depending on which parameter is the eigenvalue fourth when the modes are singular but decay to zero at the endpoints as is true of tides a tanh mapping can retrieve the usual exponential accuracy of spectral methods fifth the pseudospectral method is more reliable than deriving a banded galerkin matrix by means of recurrence relations the pseudospectral code is simple to check whereas it is easy to make an untestable mistake with the intricate algebra required for the galerkin method sixth we offer a brief cautionary tale about overlooked modes all these cautions are applicable toallforms of spatial discretization including finite difference and finite element methods however we limit our illustrations to spectral schemes where these difficulties are most easily resolved with a bit of care the pseudospectral method is a very robust and efficient method for solving differential eigenproblems even with singular eigenmodes

a fourier hermite pseudospectral method for penetrative convection a fourier hermite pseudospectral method is developed to study numerically the three dimensional penetrative convection problem under the boussinesq approximation an s shaped temperature profile in the absence of motion is prescribed in the vertical direction all variables are expanded in terms of fourier hermite basis functions the hermite functions are scaled to adjust the length of the computational domain in the vertical a semi implicit scheme is used for time marching with the third order adam bashforth and crank nicholson scheme for the nonlinear and linear terms respectively an implementation of the numerical method on a parallel computer is also described numerical simulation results of resolution 643are presented for low to moderate rayleigh numbers with a prandtl number of unity the highly stable outer regions are seen to act as effective lids and all penetrative flow are contained within the computational box variances heat fluxes and their budgets are reported for several rayleigh numbers to demonstrate the efficacy of the numerical method

on colmez s product formula for periods of cm abelian varieties colmez conjectured a product formula for periods of abelian varieties with complex multiplication by a field k analogous to the standard product formula in algebraic number theory he proved this conjecture up to a rational power of 2 for k mathbb q abelian in this paper we complete the proof of colmez for k mathbb q abelian by eliminating this power of 2 our proof relies on analyzing the galois action on the de rham cohomology of fermat curves in mixed characteristic 0 2 which in turn relies on understanding the stable reduction of mathbb z 2 n covers of the projective line branched at three points

complex coordinate methods for hydrodynamic instabilities and sturm liouville eigenproblems with an interior singularity makes a wide detour around the singularity specific guidelines for choosing f x for different problems are given in the text results are impressive for an eigenvalue problem with a pole in the middle of the original real interval a sturm liouville problem of the fourth kind just six basis functions suffice to calculate the real and imaginary parts of the lowest eigenvalue to within 1 4 for strong instability i e modes whose phase speeds have large imaginary parts the complex mapping is unnecessary because the critical latitudes are complex and distant from the real axis even so the mapping is useful for instability problems because it can be used to make calculations for very slowly growing modes to follow the changes in c right up to the neutral curve where the imaginary part of c 0 although especially valuable for spectral algorithms the same trick can be applied with finite difference methods also the main disadvantage of the algorithm is that the eigenfunction must be calculated in a second separate step but this is usually a minor flaw in comparison to the complex mapping s virtues for coping with singular eigenvalue problems

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

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

wild monodromy and automorphisms of curves let r be a complete discrete valuation ring dvr of mixed characteristic 0 p with field of fractions k containing the pth roots of unity this article is concerned with semistable models of p cyclic covers of the projective line c pk1 we start by providing a new construction of a semistable model of c in the case of an equidistant branch locus if the cover is given by the kummer equation zp f x0 we define what we call the monodromy polynomiall y of f x0 a polynomial with coefficients in k its zeros are key to obtaining a semistable model of c as a corollary we obtain an upper bound for the minimal extension k k over which a stable model of the curve c exists consider the polynomial l y yp f yi where the yi range over the zeros of l y we show that the splitting field of this polynomial always contains k and that in some instances the two fields are equal

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

periodic excitations of bilinear quantum systems a well known method of transferring the population of a quantum system from an eigenspace of the free hamiltonian to another is to use a periodic control law with an angular frequency equal to the difference of the eigenvalues for finite dimensional quantum systems the classical theory of averaging provides a rigorous explanation of this experimentally validated result this paper extends this finite dimensional result known as the rotating wave approximation to infinite dimensional systems and provides explicit convergence estimates

multi hump stationary waves for a korteweg de vries equation with nonlocal perturbations abstract periodic and solitary wave solutions are investigated numerically for a perturbed korteweg de vries equation with unstable and dissipation terms in hilbert transform u t uu x u xxx h u x h u xxx 0 a family of solitary wave solutions s 1 s 2 s n whose members are distinguished by the number of humps is numerically identified the tails of these waves decay as o 1 x 2 when x irrespective of the magnitude of it is also found that for a given there exist families of periodic wave solutions p 1 p 2 p n which originate from one near sinusoidal wave and end up in the infinite periodicity to the corresponding solitary waves the numerical results are consistent with the theoretical estimates based on the conservation properties

the eigenvalues of hermite and rational spectral differentiation matrices we derive expressions for the eigenvalues of spectral differentiation matrices for unbounded domains in particular we consider galerkin and collocation methods based on hermite functions as well as rational functions a fourier series combined with a cotangent mapping we show that i first derivative matrices have purely imaginary eigenvalues and second derivative matrices have real and negative eigenvalues ii for the hermite method the eigenvalues are determined by the roots of the hermite polynomials and for the rational method they are determined by the laguerre polynomials and iii the hermite method has attractive stability properties in the sense of small condition numbers and spectral radii

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

discrete orthogonal function expansions for non uniform grids using the fast fourier transform abstract discrete orthogonal function expansions are obtained for non uniformly spaced data by combined use of mapping functions and the fast fourier transform the result yields alias free differentiation and integration operators the method is applicable to periodic zero or zero derivative boundary conditions or combinations thereof truncation is required to avoid aliasing and or singularity but the truncation error is explicit quantitatively expressible and generally small this approach is demonstrated in an infinite physical domain with application to linear convection and diffusion the resulting errors are orders of magnitude smaller than those generated by standard finite difference methods difficulties arising from image fields are an important phenomena the method of differentiation as well as the boundary conditions imply artificial image fields a vortex pairing problem is presented which shows that image flows can totally alter the solution the new scheme avoids this undesirable effect by keeping image flows infinitely far away

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

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

subgroups of inertia groups arising from abelian varieties abstract given an abelian variety over a field with a discrete valuation grothendieck defined a certain open normal subgroup of the absolute inertia group this subgroup encodes information on the extensions over which the abelian variety acquires semistable reduction we study this subgroup and use it to obtain information on the extensions over which the abelian variety acquires semistable reduction

the rate of convergence of fourier coefficients for entire functions of infinite order with application to the weideman cloot sinh mapping for pseudospectral computations on an infinite interval we analytically compute the asymptotic fourier coefficients for several classes of functions to answer two questions the numerical question is to explain the success of the weideman cloot algorithm for solving differential equations on an infinite interval their method combines fourier expansion with a change of coordinate using the hyperbolic sine function the sinh mapping transforms a simple function like exp z2 into an entire function of infinite order this raises the second analytical question what is the fourier rate of convergence for entire functions of an infinite order the answer is sometimes even slower than a geometric series in this case the fourier series converge only on the real axis even when the function u z being expanded is free of singularities except at infinity earlier analysis ignored stationary point contributions to the asymptotic fourier coefficients when u z had singularities off the real z axis but we show that sometimes these stationary point terms are more important than residues at the poles of u z

a generalization of the wiener rational basis functions on infinite intervals part i derivation and properties we formulate and derive a generalization of an orthogonal rational function basis for spectral expansions over the infinite or semi infinite interval the original functions first presented by wiener are a mapping and weighting of the fourier basis to the infinite interval by identifying the fourier series as a biorthogonal composition of jacobi polynomials functions we are able to define generalized fourier series which when appropriately mapped to the whole real line and weighted generalize wiener s basis functions it is known that the original wiener rational functions inherit sparse galerkin matrices for differentiation and can utilize the fast fourier transform fft for computation of the expansion coefficients we show that the generalized basis sets also have a sparse differentiation matrix and we discuss connection problems which are necessary theoretical developments for application of the fft

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

a rational approximation and its applications to nonlinear partial differential equations on the whole line an orthogonal system of rational functions is discussed some inverse inequalities imbedding inequalities and approximation results are obtained two model problems are considered the stabilities and convergences of proposed rational spectral schemes and rational pseudospectral schemes are proved the techniques used in this paper are also applicable to other problems on the whole line numerical results show the efficiency of this approach

newton methods for the optimal control of closed quantum spin systems an efficient and robust computational framework for solving closed quantum spin optimal control and exact controllability problems with control constraints is presented closed spin systems are of

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 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

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

stationary travelling wave solutions of an unstable kdv burgers equation abstract both solitary and periodic stationary travelling wave solutions are investigated numerically for an unstable korteweg de vries burgers equation u t uu x u xxx u u xx 0 0 a family of stationary solitary wave solutions whose members are distinguished by the number of humps is found for a given corresponding to each solitary wave thus found a family of stationary periodic waves with the same number of humps exists under periodic condition and ends up in the infinite periodicity to the corresponding solitary wave the numerical results are consistent with the theoretical estimates based on the conservation properties

numerical investigation of the stability of the rational solutions of the nonlinear schrdinger equation hightlightsa broad numerical investigation involving large ensembles of perturbed initial data indicates the peregrine and second order rational solutions of the nls equation are linearly unstable a highly accurate chebyshev pseudo spectral method is developed for solving the nls equation that uses the map x cot in combination with the fast fourier transform to approximate uxx on an infinite domain a modified fourier spectral method that can treat initial data with discontinuous derivatives over periodic domains is developed for solving the nls equation of which the resolution of the peregrine solution is a particular example the rational solutions of the nonlinear schrdinger nls equation have been proposed as models for rogue waves in this article we develop a highly accurate chebyshev pseudo spectral method cps4 to numerically study the stability of the rational solutions of the nls equation the scheme cps4 using the map x cot and the fft to approximate uxx correctly handles the infinite line problem a broad numerical investigation using cps4 and involving large ensembles of perturbed initial data indicates the peregrine and second order rational solutions are linearly unstable although standard fourier integrators are often used in current studies of the nls rational solutions they do not handle solutions with discontinuous derivatives correctly using standard fourier pseudo spectral method fps4 for peregrine initial data yields tiny gibbs oscillations in the first steps of the numerical solution these oscillations grow to o 1 providing further evidence of the instability of the peregrine solution to resolve the gibbs oscillations we modify fps4 using a spectral splitting technique which significantly improves the numerical solution

a pseudo spectral method for a non local kdv burgers equation posed on r abstract in this paper we present a new pseudo spectral method to solve the initial value problem associated to a non local kdv burgers equation involving a caputo type fractional derivative the basic idea is using an algebraic map to transform the whole real line into a bounded interval where we can apply a fourier expansion special attention is given to the correct computation of the fractional derivative in this setting

galerkin schemes and the sinc galerkin method for singular sturm liouville problems abstract galerkin schemes for the computation of the eigenvalues of both regular and singular sturm liouville problems are compared to the sine galerkin method contrasted are numerical results as well as qualitative features the schemes discussed include finite element spectral and collocation the equivalence of the spectral method with sine basis functions to the sinc collocation scheme provides this technique with many distinctive features highlighted among these features through the course of the computational comparisons is its ease of implementation its exponential accuracy in the presence of singularities and its all around versatility

rational chebyshev spectral methods for unbounded solutions on an infinite interval using polynomial growth special basis functions abstract in the method of matched asymptotic expansions one is often forced to compute solutions which grow as a polynomial in y as y similarly the integral or repeated integral of a bounded function f y is generally unbounded also the k th integral of a function f y solves we describe a two part algorithm for solving linear differential equations on y where u y grows as a polynomial as y first perform an explicit analytic transformation to a new unknown v so that v is bounded second expand v as a rational chebyshev series and apply a pseudospectral or galerkin discretization for our examples it is convenient to perform a preliminary step of splitting the problem into uncoupled equations for the parts of u which are symmetric and antisymmetric with respect to y 0 but although this is very helpful when applicable it is not necessary for the integral and interated integrals and for constant coefficient differential equations in general the galerkin matrices are banded with very low bandwidth we derive an improvement on the last coefficient error estimate of the author s book which applies to series with a subgeometric rate of convergence as is normally true of rational chebyshev expansions

numerical methods based on whittaker cardinal or sinc functions this paper summarizes the results known to date for using sine functions composed with other functions as bases for approximations in numerical analysis described in this paper are methods of interpolation and approximation of functions and their derivatives quadrature the approximate evaluation of transforms hilbert fourier laplace hankel and mellin and the approximate solution of differential and integral equations the methods have many advantages over classical methods which use polynomials as bases in addition all of the methods converge at an optimal rate if singularities on the boundary of approximation are ignored

a spectral method for unbounded domains a spectral method for an unbounded domain is presented rational basis functions which are algebraically mapped legendre functions are used for expansion in the radial direction of polar coordinates r or r z they satisfy the pole condition exactly at the coordinate singularity and their behavior asr is suitable for expanding smooth functions which decay algebraically or exponentially asr the method is not stiff when it is applied to initial value problems despite the presence of the coordinate singularity solenoidal vector fields are treated efficiently by the toroidal and poloidal decomposition which reduces the number of dependent variables from 3 to 2 examples include the computation of vortex dynamics in two and three dimensions

prolate spheroidal wavefunctions as an alternative to chebyshev and legendre polynomials for spectral element and pseudospectral algorithms prolate spheroidal functions of order zero are generalizations of legendre polynomials which when the bandwidth parameter c 0 oscillate more uniformly on x 1 1 than either chebyshev or legendre polynomials this suggests that compared to these polynomials prolate functions give more uniform spatial resolution further when used as the spatial discretization for time dependent partial differential equations in combination with explicit time marching prolate functions allow a longer stable timestep than legendre polynomials we show that these advantages are real and further that it is almost trivial to modify existing pseudospectral and spectral element codes to use the prolate basis however improvements in spatial resolution are at most a factor of 2 approached slowly as n the timestep can be lengthened by several times but not by a factor that grows rapidly with n the prolate basis is not likely to radically expand the range of problems that can be done on a workstation however for production runs on the bleeding edge edge of arithmurgy such as numerical weather prediction the rewards for switching to a prolate basis may be considerable

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

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

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

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

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

jacobi rational approximation and spectral method for differential equations of degenerate type we introduce an orthogonal system on the half line induced by jacobi polynomials some results on the jacobi rational approximation are established which play important roles in designing and analyzing the jacobi rational spectral method for various differential equations with the coefficients degenerating at certain points and growing up at infinity the jacobi rational spectral method is proposed for a model problem appearing frequently in finance its convergence is proved numerical results demonstrate the efficiency of this new approach

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

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