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

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

behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations we establish nonlinear stability and asymptotic behavior of traveling periodic waves of viscous conservation laws under localized perturbations or nonlocalized perturbations asymptotic to constant shifts in phase showing that long time behavior is governed by an associated second order formal whitham modulation system a key point is to identify the way in which initial perturbations translate to initial data for this formal system a task accomplished by detailed estimates on the linearized solution operator about the background wave notably our approach gives both a common theoretical treatment and a complete classification in terms of phase coupling or decoupling of general systems of conservation or balance laws encompassing cases that had previously been studied separately or not at all at the same time our refined description of solutions gives the new result of nonlinear asymptotic stability with respect to localized perturbations in the phase decoupled case further distinguishing behavior in the different cases an interesting technical aspect of our analysis is that for systems of conservation laws the whitham modulation description is of system rather than scalar form as a consequence of which renormalization methods such as have been used to treat the reaction diffusion case in general do not seem to apply

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

a non linear stochastic differential equation exact critical statics and dynamics abstract the nonlinear stochastic differential equation we present is a generalization of a class of equations describing various physical and chemical systems coupled to external sources of noise the static behaviour of this system exhibits first and second order nonequilibrium transitions which are purely induced by the external noise exact analytical expressions for the time dependent solutions are found

stability of cellular states of the kuramoto sivashinsky equation this paper is concerned with the instability property of a particular type of solution to the kuramoto sivashinsky equation namely the symmetric time independent cellular states attention is focused on the dimension of their unstable manifolds we show that as a control parameter varies the dimension changes in an ordered way governed by certain properties of an associated ordinary differential equation further change in dimension is shown to be related to hopf bifurcations from the cellular states an asymptotic analysis explains the onset of these bifurcations and predicts approximate parameter values at which they occur

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

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

numerical proof of stability of roll waves in the small amplitude limit for inclined thin film flow abstract we present a rigorous numerical proof based on interval arithmetic computations categorizing the linearized and nonlinear stability of periodic viscous roll waves of the kdv ks equation modeling weakly unstable flow of a thin fluid film on an incline in the small amplitude kdv limit the argument proceeds by verification of a stability condition derived by bar nepomnyashchy and johnson noble rodrigues zumbrun involving inner products of various elliptic functions arising through the kdv equation one key point in the analysis is a bootstrap argument balancing the extremely poor sup norm bounds for these functions against the extremely good convergence properties for analytic interpolation in order to obtain a feasible computation time another is the way of handling analytic interpolation in several variables by a two step process carving up the parameter space into manageable pieces for rigorous evaluation these and other general aspects of the analysis should serve as blueprints for more general analyses of spectral stability

analytical approach to initial value problems in nonlinear systems a method based on the equivalence between finite dimensional nonlinear and infinite dimensional linear systems of ordinary differential equations is presented in order to calculate the time dependent solutions of nonlinear physico chemical systems the solution is cast in the form of power series whose general term is known analytically

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

nonlinear fourier analysis for the infinite interval korteweg de vries equation i an algorithm for the direct scattering transform the nonlinear fourier analysis of wave motion governed approximately by the korteweg de vries kdv equation on the infinite line is the central point of discussion we assume that the wave amplitude is recorded in the form of a discrete space or time series which is determined either by experimental measurement or by computer simulation of the physical system of interest we develop numerical data analysis procedures based upon the scattering transform solution to the kdv equation as given by gardner et al 1 we are motivated by the observation that historically the fourier transform has been ubiquitously used to spectrally analyze linear wave data here we develop methods for employing the scattering transform as a tool to similarly analyze nonlinear wave data specifically we develop numerical methods to evaluate the direct scattering transform dst of a space or time series the approach thus provides a basis for analyzing and interpreting nonlinear wave behavior in the wavenumber or frequency domain the dst spectrum separates naturally into soliton and radiation components and may be simply interpreted in terms of the large time asymptotic state of the infinite line kdv equation

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

order parameter equations for long wavelength instabilities abstract we consider order parameter equations describing the nonlinear evolution of finite amplitude disturbances near the threshold of a large scale instability it is shown that the ginzburg landau equation is not valid in many cases the structure of typical order parameter equations is discussed

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

modulated amplitude waves and defect formation in the one dimensional complex ginzburg landau equation the transition from phase chaos to defect chaos in the complex ginzburg landau equation cgle is related to saddle node bifurcations of modulated amplitude waves maws first the spatial period p of maws is shown to be limited by a maximum psn which depends on the cgle coefficients maw like structures with period larger than psn evolve to defects second slowly evolving near maws with average phase gradients 0 and various periods occur naturally in phase chaotic states of the cgle as a measure for these periods we study the distributions of spacings p between neighbouring peaks of the phase gradient a systematic comparison of p and psn as a function of coefficients of the cgle shows that defects are generated at locations where p becomes larger than psn in other words maws with period psn represent critical nuclei for the formation of defects in phase chaos and may trigger the transition to defect chaos since rare events where p becomes sufficiently large to lead to defect formation may only occur after a long transient the coefficients where the transition to defect chaos seems to occur depend on system size and integration time we conjecture that in the regime where the maximum period psn has diverged phase chaos persists in the thermodynamic limit 2001 published by elsevier science b v

numerical bifurcation and stability analysis of solitary pulses in an excitable reaction diffusion medium abstract we present a systematic computer assisted study of the bifurcations and instabilities of solitary pulses in an excitable medium capable of displaying both stable pulse propagation and spatiotemporally chaotic dynamics over intervals of parameter space the reaction diffusion model used is of the activator inhibitor type only the activator diffuses in this medium the control parameters are the ratio of time scales of the activator and inhibitor dynamics and the excitation threshold this study focuses on travelling pulses their domain of existence and the bifurcations that render them unstable these pulses are approximated as a homoclinic orbits in a travelling wave ode frame and b as solutions of the full partial differential equation pde with periodic boundary conditions in large domains a variety of bifurcations in the travelling wave ode frame are observed including heteroclinic loops so called t points a r champneys and y a kuznetsov numerical detection and continuation of codimension 2 homoclinic bifurcations int j bif chaos 4 1994 785 h kokobu homoclinic and heteroclinic bifurcations of vectorfields japan j appl math 5 1988 455 instabilities in the full pde frame include both hopf bifurcations to modulated travelling waves involving the discrete pulse spectrum as well as transitions involving the continuous spectrum such as the so called backfiring transition m bar m hildebrand m eiswirth m falcke h engel and m neufeld chemical turbulence and standing waves in a surface reaction model the influence of global coupling and wave instabilities chaos 4 1994 499 the stability of modulated pulses is computed through numerical floquet analysis and a cascade of period doubling bifurcations is observed as well as certain global bifurcations these results corroborated by observations from direct numerical integration provide a skeleton around which many features of the overall complex spatiotemporal dynamics of the pde are organized

on symmetric long wave patterns in two layer flows abstract the doubly periodic pattern formation of nonlinear partial differential equations that describe the small amplitude long wave interfacial dynamics of a two layer fluid flow in a channel for small volumetric flow rates is investigated this system consists of a kuramoto sivashinsky type equation governing the evolution of the interfacial deflection coupled to a poisson equation for the disturbance pressure field the system has reflectional and translational symmetries in the direction of the flow and transverse to the flow a local analysis near the mutual bifurcation of a two dimensional wave and oblique waves yields criteria for different bifurcation scenarios near criticality a numerical study away from the initial bifurcation reveals that for linearly coupled systems the patterns appear to be steady for moderate values of the bifurcation parameter when the equations are coupled nonlinearly instabilities to the local solutions lead to a low dimensional chaotic attractor through a period doubling scenario oscillatory solutions such as an exchange of energy from the two dimensional wave to oblique modes are also seen along with a travelling square pattern that is quasi periodic

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

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

local stability theory of solitary pulses in an active medium abstract if the spatio temporal chaos of an extended active medium is driven by repulsive or attractive interaction among localized coherent structures such as kinks and pulses its dynamics can often be described by a low dimensional dynamical system this welcomed reduction in the degrees of freedom only occurs when the localized structures remain stable during their mutual interaction it is shown that the structures can be destroyed by two types of local disturbance an expanding radiation packet and a localized disturbance which travels with the structure these two instabilities and the bifurcations they induce select the proper coherent structure and specify the dynamics of the generalized kuramoto sivashinsky gks equation in an extended domain the possibility of describing pulse radiation interaction by resonance poles also suggests that a more general coherent structure theory can be developed

on steady state travelling solutions of an evolution equation describing the behaviour of disturbances in active dissipative media abstract a nonlinear evolution equation is considered which is often encountered in modelling the behaviour of perturbations in various active dissipative media e g in problems of fluid film flow hydrodynamics periodic steady state travelling solutions have been found numerically for it stability of these solutions has been investigated and bifurcation analysis has been carried out the analysis has demonstrated that decrease of the wave number causes more and more new families of steady state travelling solutions a countable set of such solutions is formed in the limit when the wave number tends to zero it is also shown that time oscillating solutions can be generated from steady state ones due to bifurcation of the landau hopf type

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

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

on traveling wave solutions of the kuramoto sivashinsky equation abstract previous numerical results indicate that the kuramoto sivashinsky equation admits three classes of non periodic traveling wave solutions namely regular shocks oscillatory shocks and solitary waves however it has been shown that regular monotonic shocks cease to exist in the weak shock limit due to the radiation of oscillatory waves of exponentially small with respect to the shock strength but growing in space amplitude here oscillatory shocks and solitary waves are constructed by asymptotic analysis it thus transpires that in the weak shock limit oscillatory shocks can only be antisymmetric otherwise oscillatory and monotonic waves of exponentially small but growing in space amplitude would inevitably be excited under certain conditions however the growing waves can link a nearly antisymmetric oscillatory shock with a radiating regular shock to form a solitary wave the predictions of the asymptotic theory are supported by numerical results

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

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

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

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

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

hierarchy of bifurcations of space periodic structures in a nonlinear model of active dissipative media abstract this work is a detailed research of spatial periodic structures and their bifurcations in a nonlinear model of media with an active dissipation presented here is an obvious bifurcation diagram of stationary solutions attractors on a wide scale of wave numbers are studied and classified significant progress in understanding of behavior of nonstationary solutions is attained

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 new approximate stationary probability distribution for processes driven by ornstein uhlenbeck noise a novel approximate probability distribution for a class of nonlinear processes driven by multiplicative linear ornstein uhlenbeck noise is presented a comparison is made with other approximate distributions reported in the literature

falling films on flexible inclines the nonlinear stability and dynamic behavior of falling fluid films is studied for flow over a flexible substrate we use asymptotic methods to deduce governing equations valid in various limits long wave theory is used to derive benney like coupled equations for the film thickness and substrate deflection weakly nonlinear equations are then derived from these equations that in the limit of large wall damping and or large wall tension reduce to the kuramoto sivashinsky equation these models break down when inertia becomes more significant so we also use a long wave approximation in conjunction with integral theory to derive three strongly coupled nonlinear evolution equations for the film thickness substrate deflection and film volumetric flow rate valid at higher reynolds numbers these equations accounting for inertia capillary viscous wall tension and damping effects are solved over a wide range of parameters our results suggest that decreasing wall damping and or wall tension can promote the development of chaos in the weakly nonlinear regime and lead to severe substrate deformations in the strongly nonlinear regime these can give rise to situations in which the free surface and underlying substrate come into contact in finite time

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

the kuramoto sivashinsky equation a bridge between pde s and dynamical systems abstract through extensive numerical simulation we have characterized the transition to chaos of the solutions to the kuramoto sivashinsky equation the attracting solution manifolds undergo a complex bifurcation sequence including multimodal fixed points invariant tori traveling wave trains and homoclinic orbits we relate this behavior to earlier work where the kuramoto sivashinsky equation was shown to behave as a finite dimensional dynamical system of ordinary differential equations

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

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 decay properties and asymptotic behavior of solutions to a non local perturbed kdv equation abstract we consider the kdv equation with an additional non local perturbation term defined through the hilbert transform also known as the ost equation we prove that the solutions u t x have a pointwise decay in spatial variable u t x 1 1 x 2 provided that the initial data has the same decaying and moreover we find the asymptotic profile of u t x when x next we show that decay rate given above is optimal when the initial data is not a zero mean function and in this case we derive an estimate from below 1 x 2 u t x for x large enough in the case when the initial datum is a zero mean function we prove that the decay rate above is improved to 1 1 x 2 e for 0 e 1 finally we study the local well posedness of the ost equation in the framework of lebesgue spaces

the strong coupling expansion for anharmonic oscillators we accurately calculate the strong coupling expansion for anharmonic oscillators by means of a robust and stable numerical algorithm the method applies to any state and to any anharmonicity degree by means of the perturbation coefficients we estimate the location of the branch points that determine the convergence radius of the strong coupling expansion

integral equations a tool to solve the schrodinger equation from the integral equation equivalent to the one dimensional schrodinger equation with the boundary conditions appropriate for a bound state problem an exact three point integration rule is derived the approximate evaluation of this rule by means of the euler mclaurin sum rule gives rise to various o h2 and o h4 integration methods associated with the different ways of splitting the starting equation into a free part and an interaction term the method is particularly useful to deal with the singularities of the potential

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

temporal evolutions and stationary waves for dissipative benjamin ono equation abstract initial value problems as well as stationary solitary and periodic waves are investigated for dissipative benjamin ono dbo equation multi hump stationary waves and their structures are identified numerically and the stability regions of stationary periodic waves are also examined numerically these results elucidate a close relation between irregular behaviours in the initial value problem and the multiplicity of stationary waves

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

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

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

nonlinear modulational stability of periodic traveling wave solutions of the generalized kuramoto sivashinsky equation abstract in this paper we consider the spectral and nonlinear stabilities of periodic traveling wave solutions of a generalized kuramoto sivashinsky equation in particular we resolve the long standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized integrable perturbations our analysis is based upon detailed estimates of the linearized solution operator which are complicated by the fact that the necessarily essential spectrum of the associated linearization intersects the imaginary axis at the origin we carry out a numerical evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves in close agreement with previous numerical studies of frisch she thual bar nepomnyashchy chang demekhin kopelevich and others carried out by other techniques we also compare predictions of the associated whitham modulation equations which formally describe the dynamics of weak large scale perturbations of a periodic wave train with numerical time evolution studies demonstrating their effectiveness at a practical level for the reader s convenience we include in an appendix the corresponding treatment of the swift hohenberg equation a nonconservative counterpart of the generalized kuramoto sivashinsky equation for which the nonlinear stability analysis is considerably simpler together with numerical evans function analyses extending spectral stability analyses of mielke and schneider

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

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

steady solutions of the kuramoto sivashinsky equation abstract steady solutions of the kuramoto sivashinsky equation are studied these solutions are defined on the whole x line and propagate with a constant speed c 2 in time for large c 2 it is shown that the solution is unique and has a conical form for small c 2 there is a periodic solution and an infinite set of quasi periodic solutions as asserted by moser s twist map theorem numerical computations for intermediate values of c 2 suggest that below c 1 6 of every speed there is a continuum of odd quasi periodic solutions or a cantor set of chaotic solutions wrapped by infinite sequences of conic solutions

on the numerical integration of the schrodinger equation numerical tests abstract the various numerical methods obtained in a preceding work in the finite difference scheme are compared among themselves and against the exact solutions of some simple exactly soluble potentials the tests strongly favour the limiting method the discretized plane wave method a discussion of the results in the continuous spectrum is also presented

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

nonlinear analysis of the eckhaus instability modulated amplitude waves and phase chaos with nonzero average phase gradient abstract we analyze the eckhaus instability of plane waves in the one dimensional complex ginzburg landau equation cgle and describe the nonlinear effects arising in the eckhaus unstable regime modulated amplitude waves maws are quasi periodic solutions of the cgle that emerge near the eckhaus instability of plane waves and cease to exist due to saddle node sn bifurcations these maws can be characterized by their average phase gradient and by the spatial period p of the periodic amplitude modulation a numerical bifurcation analysis reveals the existence and stability properties of maws with arbitrary and p maws are found to be stable for large enough and intermediate values of p for different parameter values they are unstable to splitting and attractive interaction between subsequent extrema of the amplitude defects form from perturbed plane waves for parameter values above the sn of the corresponding maws the break down of phase chaos with average phase gradient 0 wound up phase chaos is thus related to these sns a lower bound for the break down of wound up phase chaos is given by the necessary presence of sns and an upper bound by the absence of the splitting instability of maws

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

order and complexity in the kuramoto sivashinsky model of weakly turbulent interfaces abstract we present a large number of new geometric ergodic and statistical properties of the kuramoto sivashinsky equation modeling interfacial turbulence in various physical contexts in addition this equation has the remarkable property of inertial manifolds where some finite dimensional dynamical system is rigorously equivalent to this infinite dimensional partial differential equation in moderate size domains up to ten periods in length a low dimensional vector field skeleton underpins even strongly chaotic regimes and controls the bifurcations of the inertial manifold the extreme numerical sensitivity of chaos in this dissipative pde requires very high precision methods despite the geometrical complexities of the bifurcation structure some statistical properties remain remarkably simple there is overwhelming evidence that for some parameter values a permanent unsteady state exists an unexpectedly simple diffusive relaxation of the large scale fluctuations is extracted from extensive numerical simulations in these calculations we observe long time tails for the correlation functions of relevant quantities we propose an explanation in terms of an effective viscosity and compare the transport in the weakly turbulent interface with related theories for random interfaces and developed turbulence

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

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

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

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

laminarizing effects of dispersion in an active dissipative nonlinear medium abstract the kuramoto sivashinsky equation has become a popular prototype for systems which exhibit spatial temporal chaos we show here that a linear dispersion term h xxx tends to arrest such irregular behavior in favor of spatially periodic cellular structures as is consistent with prior numerical and experimental observations the study includes a normal form analysis and a numerical investigation of periodic and solitary stationary wave solutions of the equation it is shown that by 1 1 the infinite families of stationary periodic waves of the kuramoto sivashinsky equation each ending in a solitary wave have been annihilated successively such that only a lone family of periodic waves consisting of one hump kdv pulses for 3 7 remains as the only periodic stationary wave attractors of the system these periodic waves have much larger domains of attraction than the strange attractors and hence tend to dominate spatial temporal chaos in an extended domain with significant dispersion

dynamics of a nonlocal kuramoto sivashinsky equation abstract in this paper we study the effects of a nonlocal term on the global dynamics of the kuramoto sivashinsky equation we show that the equation possesses a family of maximal attractors parameterized by the mean value of the initial data the dimension of the attractor is estimated as a function of the coefficient of the nonlocal term and the width of the periodic domain

an approximate master equation for systems driven by linear ornstein uhlenbeck noise abstract by use of an operator method we construct a novel approximate evolution equation for a one dimensional probability distribution of a single degree of freedom system driven by ornstein uhlenbeck noise this equation is an integro differential equation of the time convolutionless type and its steady state solution is presented a class of non linear equations with multiplicative noise is considered