adaptive numerical simulation of traffic flow density in this paper we describe a one dimensional adaptive moving mesh method and apply it to the traffic flow model introduced in 2006 by sopasakis and katsoulakis and its modified model proposed in 2009 by kurganov and polizzi these models can be written as a scalar conservation law with a global flux the proposed scheme is an extension of the moving non oscillatory central scheme which belongs to a class of moving finite volume methods we also modify the model given by kurganov and polizzi to account for the driver s reaction i e delayed response finally the moving finite volume method is extended accordingly

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

the approximation for the boundary optimal control problem of burgers fisher equation with constraints this paper deals with the numerical approximation with meshless method for the boundary optimal control problem with some control and state constraints governed by the burgers fisher equation which is a nonlinear evolution equation and is the prototype model for the reaction convection and diffusion phenomena arising in many spatial temporal processes by making use of the element free galerkin efg method the original optimal control problem is discretized spatially to a semi discrete optimal control problem governed by a system of nonlinear ordinary differential equations then by using the control parameterization method the original problem can be reduced to an optimal parameter selection problem governed by a lumped parameter system which can be solved as a nonlinear optimization problems by using the sequential quadratic programming sqp algorithm the numerical simulations are given to illustrate the effectiveness of the proposed numerical approximation method

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

sturm liouville eigenproblems with an interior pole the eigenvalues and eigenfunctions of self adjoint sturm liouville problems with a simple pole on the interior of the interval a b are investigated three general theorems are proved and it is shown that as n the eigenfunctions more and more closely resemble those of an ordinary sturm liouville problem and n m2 2 b a 2 just as if there were no singularity the low order modes however differ drastically from those of a nonsingular eigenproblem in that i both eigenvalues and eigenfunctions are complex despite the fact the problem is self adjoint ii the real and imaginary parts of the nth eigenfunction may both have ever increasing numbers of interior zeros as b instead of just n 1 zeros and iii as b the eigenvalues for all small n may cluster about a common value in contrast to the widely separated eigenvalues of the corresponding nonsingular problem these results are general but in order to present quantitative solutions for the low order modes too special attention is gi

efficient high order method for differential equations in unbounded domains using generalized coordinate transformation abstract many physical phenomena are localized in space where physical quantities of interest extend to large distances effectively to infinity where they vanish smoothly thus its modeling necessitates the imposition of vanishing boundary conditions bcs to the formulated differential equations des numerical solutions for such des can be complicated as they involve operations on physical quantities expanding infinitely in space currently many approaches have been implemented in this context where they mostly use techniques like domain truncation using functions intrinsic to unbounded domains and domain transformation however since the actual solutions have a decaying nature solving such problems with high accuracy and efficiency while taking into consideration a wide range of decaying rates and oscillation frequencies is a major challenge in this work we present an efficient high order method based on galerkin method to solve des in unbounded domains using properly designed domain transformations where the physical unbounded domain is mapped into a computational bounded domain 0 1 the mapped sine series is then used to solve the differential equation in the computational domain 0 1 where the vanishing quantities at infinities are mapped to 0 corresponding to physical and 1 corresponding to physical the designed transformations maintain the orthogonality in both physical and computational spaces moreover using sine series basis allows in most cases analytical integrations for verification the proposed method is applied to solve 1 helmholtz equation in one dimension where the exact solutions for various cases decaying exponential or algebraic function are known and 2 three dimensional schrodinger wave equation with harmonic oscillator for the helmholtz problem the error in the approximate solution was less than 10 40 using 100 basis functions while for the second problem the error is less than 10 25 using the same number of bases all results and the analyses demonstrate how the properly designed mapped basis sets can be used to develop efficient high order methods to solve des with vanishing boundary conditions and with wide range of decaying rates and oscillation frequencies simultaneously

moving mesh methods for problems with blow up in this paper we consider the numerical solution of pdes with blow up for which scaling invariance plays a natural role in describing the underlying solution structures it is a challenging numerical problem to capture the qualitative behaviour in the blow up region and the use of nonuniform meshes is essential we consider moving mesh methods for which the mesh is determined using so called moving mesh partial differential equations mmpdes specifically the underlying pde and the mmpde are solved for the blow up solution and the computational mesh simultaneously motivated by the desire for the mmpde to preserve the scaling invariance of the underlying problem we study the effect of different choices of mmpdes and monitor functions it is shown that for suitable ones the mmpde solution evolves towards a moving mesh which close to the blow up point automatically places the mesh points in such a manner that the ignition kernel which is well known to be a natural coordinate in describing the behaviour of blow up approaches a constant as t to t the blow up time several numerical examples are given to verify the theory for these mmpde methods and to illustrate their efficacy

the variational iteration method for nonlinear oscillators with discontinuities in this paper he s variational iteration method vim is applied to nonlinear oscillators with discontinuities we illustrate that the vim is very effective and convenient and does not require linearization or small perturbation contrary to the conventional methods in vim only one iteration leads to high accuracy of the solutions moreover we show that the obtained approximate solutions are valid for the whole solution domain and the approximations are uniformly valid not only for small parameters but also for very large parameters

an analytic solution to the thomas fermi equation a perturbative procedure due to bender et al here referred to as the bmps procedure j math phys 30 1447 1989 and useful in solving difficult nonlinear problems has been used here to solve the thomas fermi t f equation the present work attempts to balance the ease of the ensuing analysis with the use of an analytic zero order function that already contains a good deal of the nonlinearity of the t f equation the initial slope of the t f potential is computed with 0 35 error in a second order application of the theory

two dimensional blasius viscous flow of a power law fluid over a semi infinite flat plane analytic results are obtained for the similarity equation governing the two dimensional blasius viscous flow of a power law fluid over a semi infinite flat plane via taylor series for small values of the independent similarity variable then an analytic perturbative procedure is used to determine an approximate solution that exhibits the correct asymptotic behavior this perturbation method allows for the computation of the shear stress at the wall something which is impossible with a taylor series approach it is found that the perturbation solutions converge sufficiently rapidly indeed a first order approximation gives qualitatively accurate results furthermore we employ the perturbation method to deduce the influence of the power law index n on the obtained similarity solutions

analytical and numerical investigations of spontaneous imbibition in porous media we present semi analytical solutions for co current displacements with some degree of counter current flow the solution assumes a one dimensional horizontal displacement of two immiscible incompressible fluids with arbitrary viscosities and saturation dependent relative permeability and capillary pressures we address the impact of the system length on the degree of counter current flow when there is no pressure drop in the non wetting phase across the system assuming negligible capillary back pressure at the inlet boundary of the system it is shown that in such displacements the fractional flow can be used to determine a critical water saturation from which regions of both co current and counter current flow are identified this critical saturation changes with time as the saturation front moves into the porous medium furthermore the saturation profile in the approach presented here is not necessarily a function of distance divided by the square root of time we also present approximate solutions using a perturbative approach which is valid for a wide range of flow conditions this approach requires less computational power and is much easier to implement than the implicit integral solutions used in previous work finally a comprehensive comparison between analytical and numerical solutions is presented numerical computations are performed using traditional finite difference formulations and convergence analysis shows a generally slow convergence rate for water imbibition rates and saturation profiles this suggests that most coarsely gridded simulations give a poor estimate of imbibition rates while demonstrating the value of these analytical solutions as benchmarks for numerical studies complementing buckley leverett analysis this article is protected by copyright all rights reserved

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

numerical simulation of the generalized huxley equation by he s variational iteration method abstract by means of variational iteration method the solution of generalized huxley equation are obtained comparison with the adomian decomposition method is made showing that the former is more effective than the later in this paper he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomials

linearization techniques for singular initial value problems of ordinary differential equations linearization methods for singular initial value problems in second order ordinary differential equations are presented these methods result in linear constant coefficients ordinary differential equations which can be integrated analytically thus yielding piecewise analytical solutions and globally smooth solutions the accuracy of these methods is assessed by comparisons with exact and asymptotic solutions of homogeneous and non homogeneous linear and nonlinear lane emden equations it is shown that linearization methods provide accurate solutions even near the singularity or the zeros of the solution in fact it is shown that linearization methods provide more accurate solutions than methods based on perturbation methods it is also shown that the accuracy of these techniques depends on the nonlinearity of the ordinary differential equations and may not be a monotonic function of the step size

efficient modal analysis of systems with local stiffness uncertainties the characterization of the uncertainty in modal quantities of an uncertain linear structural system is essential to the rapid determination of its response to arbitrary loadings although the size of many computational structural models used is extremely large i e thousands of equations the uncertainty to be analyzed is oftentimes localized to very small regions of the model this paper addresses the development of an efficient computational methodology for the modal analysis of linear structural systems with local stiffness uncertainties the newly developed methodology utilizes an enriched basis that consists of the sub spectrum of a nominal structural system augmented with additional basis vectors generated from a knowledge of the structure of the stiffness uncertainty in addition methods for determining bounds on the approximate modal frequencies and mode shapes are discussed numerical results demonstrate that the algorithm produces highly accurate results with greatly reduced computational effort copyright 2009 john wiley sons ltd

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

an analytic study of fisher s equation by using adomian decomposition method in this paper we develop a framework to obtain exact solutions to fisher s equation and to a nonlinear diffusion equation of the fisher type by employing adomian decomposition method the proposed scheme is supported by examining nonlinear diffusion equations of the fisher type

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

computations of the complex error function rational expansions for computing the complex error function w z e z 2 text erfc iz are presented these expansions have the following attractive properties 1 they can be evaluated using a polynomial evaluation routine such as homer s method 2 the polynomial coefficients can be computed once and for all by a single fast fourier transform fft and 3 high accuracy is achieved uniformly in the complex plane with only a small number of terms comparisons reveal that in some parts of the complex plane certain competitors may be more efficient however the difference in efficiency is never great and the new algorithms are simpler than existing ones a complete program takes eight lines of matlab code

moving mesh discontinuous galerkin methods for pdes with traveling waves abstract in this paper a moving mesh discontinuous galerkin dg method is developed for nonlinear partial differential equations pdes with traveling wave solutions the moving mesh strategy for one dimensional pdes is based on the rezoning approach which decouples the solution of the pde from the moving mesh equation we show that the dg moving mesh method is able to resolve sharp wave fronts and wave speeds accurately for the optimal arc length and curvature monitor functions numerical results reveal the efficiency of the proposed moving mesh dg method for solving burgers burgers fisher and schlogl nagumo equations

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

taylor series and δ perturbation expansions for a nonlinear semiconductor transport equation a one dimensional nonlinear equation for steady state carrier transport in semiconductors is solved by two approaches a taylor series first integral method and the perturbation method the square of the carrier speed is found as a function of the distance where collisions are treated by an energy independent mean free time the coefficients of the taylor series at the first maximum are exact expressions in the function determined from the first integral in the perturbation expansion the function is evaluated to first order in for both methods expressions for the distance from the first maximum to the emitting electrode are found and the distance is evaluated numerically for the taylor series first integral method

analysis of moving mesh partial differential equations with spatial smoothing two moving mesh partial differential equations mmpdes with spatial smoothing are derived based upon the equidistribution principle this smoothing technique is motivated by the robust moving mesh method of dorfi and drury j comput phys 69 1987 pp 175 195 it is shown that under weak conditions the basic property of no node crossing is preserved by the spatial smoothing and a local quasi uniformity property of the coordinate transformations determined by these mmpdes is proven it is also shown that discretizing the mmpdes using centered finite differences these basic properties are preserved

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

chebyshev spectral methods and the lane emden problem the three dimensional spherical polytropic lane emden problem is yrr 2 r yr y m 0 y 0 1 yr 0 0 where m 0 5 is a constant parameter the domain is r 0 where is the first root of y r we recast this as a non linear eigenproblem with three boundary conditions and as the eigenvalue allow ing imposition of the extra boundary condition by making the change of coordinate x r yxx 2 x yx 2 y m 0 y 0 1 yx 0 0 y 1 0 we find that a newton kantorovich iteration always converges from an m independent starting point y 0 x cos 2 x 0 3 we apply a chebyshev pseudospectral method to discretize x the lane emden equation has branch point singularities at the endpoint x 1 whenever m is not an integer we show that the chebyshev coefficients are a n constant n 2m 5 as n however a chebyshev truncation of n 100 always gives at least ten decimal places of accuracy much more accuracy when m is an inte ger the numerical algorithm is so simple that the complete code in maple is given as a one page table ams subject classifications 65l10 65d05 85a15

adomian decomposition method for burger s huxley and burger s fisher equations the approximate solutions for the burger s huxley and burger s fisher equations are obtained by using the adomian decomposition method solving frontier problems of physics the decomposition method kluwer boston 1994 the algorithm is illustrated by studying an initial value problem the obtained results are presented and only few terms of the expansion are required to obtain the approximate solution which is found to be accurate and efficient

a modified tau spectral method that eliminated spurious eigenvalues abstract a modified tau spectral method is presented which eliminates the spurious eigenvalues produced by the usual tau method the modified tau method essentially involves an appropriate factorization of the differential operators in the eigenvalue problem it is developed for eigenvalue problems posed as single differential equations or as systems of such equations several eigenvalue problems are solved using both the usual and modified chebyshev tau methods including the orr sommerfeld stability equation for plane poiseuille flow the convergence of the modified tau method is shown to be at least as rapid as that of the usual tau method the use of the tau coefficients in indicating convergence is also discussed

laguerre pseudospectral approximation to the thomas fermi equation we propose an iterative method to solve the nonlinear thomas fermi equation based on laguerre pseudospectral approximation different from the direct use of pseudospectral methods for solving the problem in the literature we represent the solution of the thomas fermi equation as sum of two parts due to its singularity at the origin both parts regard the initial slope as a parameter unknown to be determined one singular part is a power series expansion the other smooth part satisfies a nonlinear two point boundary value problem we use a laguerre pseudospectral method to discretize and solve it numerically by the newton iteration moreover an introduced new boundary condition is employed innovatively to construct an outer iteration to determine the initial slope we show numerically this separation of solution as two parts is necessary for improvement of accuracy by direct laguerre approximations of the thomas fermi equation comparisons of our numerical results with those obtained by other methods show effectiveness and accuracy of our approach

linearized galerkin and artificial parameter techniques for the determination of periodic solutions of nonlinear oscillators abstract linearized harmonic balance galerkin methods for the determination of the periodic solutions of nonlinear oscillators are presented and the results are compared with those of an artificial parameter method based on the introduction of a linear stiffness term and the linstedt poincare technique the linearized harmonic balance methods presented in this paper are not iterative and are based on taylor series expansions the neglect of nonlinear terms in the resulting series a fourier approximation to the solution and orthogonality conditions it is shown that the first approximation provided by the linearized harmonic balance technique is identical to that of the artificial parameter method whereas the second one depends on how the equation is linearized and written and the discrepancies have been found to be caused by the neglect of the nonlinear terms in the taylor series expansion because a fourier series expansion of these terms introduces harmonics which are not accounted for in the linearization procedure it is also shown that the first approximation of the artificial parameter method and the linearized harmonic balance procedure are asymptotically equivalent to that derived from a modified linstedt poincare method when the nonlinearities are small and that these techniques may not be applicable to some oscillators

an explicit analytic solution to the thomas fermi equation a new kind of analytic technique namely the homotopy analysis method is employed to give an explicit analytic solution of the thomas fermi equation and the related recurrence formulae of constant coefficients this solution can be regarded as the definition of the exact solution of the thomas fermi equation

kawahara solitons in boussinesq equations using a robust christov galerkin spectral method we develop a robust christov galerkin spectral technique for computing interacting localized wave solutions of and fourth and sixth order generalized wave equations to this end a special complete orthonormal system of functions in l 2 is used whose rate of convergence is shown to be exponential for the cases under consideration for the time stepping an implicit algorithm is chosen which makes use of the banded structure of the matrices representing the different spatial derivatives as featuring examples the head on collision of solitary waves is investigated for a sixth order generalized boussinesq equation and a fourth order boussinesq type equation with a linear term its solutions comprise monotone shapes sech es and damped oscillatory shapes kawahara solitons the numerical results are validated against published data in the literature using the method of variational imbedding

a rescaling algorithm for the numerical calculation of blowing up solutions on presente un algorithme a invariance d echelle pour le calcul numerique des solutions a singularites explosives des equations d evolution non lineaires

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

one dimensional nonlinear steady infiltration an approximate solution of the one dimensional nonlinear steady infiltration with arbitrary plant withdrawal functions is presented the solution is derived using perturbation expansion methods and is applicable for homogeneous soils for which the hydraulic conductivity function is of the rational power form the soil domain is either a semi infinite flow medium or a finite depth medium overlying a shallow water table the boundary condition at the surface of the soil is taken as the steady constant flux and the bottom boundary condition is a prescribed head at a depth the water extraction functions used herein include the uniform the piecewise constant and the exponential uptake the perturbation solution showed very good agreement with the exact implicit solutions for various soil types and surface flux conditions the approximate solution provided a basis on which explicit algebraic equations for parameter estimation can be derived numerical results show the effects of the soil type surface flux and rooting depths on the pressure distribution in the soil

on the determination of the limit cycles using the harmonic balance method several methods have been developed to search for limit cycles or bifurcation curves of nonlinear differential equations recently some non perturbative methods have been proposed however the majority of these methods are based on the so called harmonic balance method in this paper we will see how to use the harmonic balance method to obtain information of the number period and form of the limit cycles of certain nonlinear equations the method also gives a sequence of algebraic approximations of the limit cycles we will also compare our results with other non perturbative methods the results can be generalized to other nonlinear differential equations

analytic study on burgers fisher huxley equations and combined forms of these equations abstract in this work we formally derive travelling wave solutions for the burgers fisher huxley equations and combined forms of these equations the tanh coth method is used to determine these sets of travelling wave solutions kinks and periodic wave solutions are constructed the proposed scheme can be used to a wide class of nonlinear evolution equations

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

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

an exact result for the thomas fermi equation a priori bounds for the potential slope at the origin we examine the nonlinear boundary value problem formed by the thomas fermi equation 3 2x 1 2 complemented with the boundary conditions 0 1 and 0 we show that the value of at the origin which plays a crucial role in this problem can be accurately bounded a priori by exploiting integral properties of the thomas fermi equation and without any assumption on the functional dependence of x extension of the approach to more general equations of the emden fowler type is also briefly considered

solitary wave solutions of the generalised burgers huxley equation exact solitary wave solutions of the generalised burgers huxley equation delta u delta t alpha udelta delta u delta chi delta 2u delta chi 2 beta u 1 udelta udelta gamma are obtained by using the relevant nonlinear transformations the results obtained are the generalisation of former work the method in this paper can also be applied to the burgers fisher equation

topological kinks in p 2n field theories a variational approach the expansion and the linear expansion are analytical perturbation techniques that enable one to find approximate analytic solutions to nonlinear problems these expansions augmented by new variational strategies often yield excellent results already in first order we study in this paper the static kinks in scalar field theories with v 12m2 2 g 2n using these techniques we find excellent agreement between the lowest order variational approximation in both methods and the exact answer we also estimate the energy of the first excited quantum state by considering small oscillations about the kink motion and using our variational wave functions and a shape parameter ansatz for the first excited state wave function

the origin and nature of spurious eigenvalues in the spectral tau method the chebyshev tau spectral method for approximating eigenvalues of boundary value problems may produce spurious eigenvalues with large positive real parts even when all true eigenvalues of the problem are known to have negative real parts we explain the origin and nature of the spurious eigenvalues in an example problem the explanation will demonstrate that the large positive eigenvalues are an approximation of infinite eigenvalues in a nearby generalized eigenvalue problem

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

asymptotology by homotopy perturbation method in the paper an heuristical example is given to illustrate the basic idea of the homotopy perturbation method so that homotopy perturbation method has made all that is necessary simple and all that is complex unnecessary

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

dynamic analysis using a reduced basis of exact modes and ritz vectors the problem of obtaining an approximate solution for the response of a structure under dynamic forces is addressed the structure is modeled using mass and stiffness matrices and has n degrees of freedom an dimensional approximation is sought in a subspace defined by an arbitrary m orthogonal array a residual vector which is smaller in magnitude the more accurate the approximation is derived the residual contains two terms one is the error involved representing the force vector in the subspace the other is smaller the better the array leads to m approximate solutions of the exact eigenproblem numerical experiments are performed using arrays that are combinations of exact eigenvectors and of ritz vectors in one case when 11 different 10 dimensional bases are used to approximate the solution of a 60 degree of freedom problem it is found that a root mean square measure of relative error is smallest for the basis formed using the 6 lowest exact eigenvectors plus 4 ritz vectors generated from the spatial distribution of the loading

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

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

near optimality of the sinc approximation near optimality of the sinc approximation is established in a variety of spaces of functions analytic in a strip region about the real axis each space being characterized by the decay rate of their elements functions in the neighborhood of the infinity

a numerical simulation and explicit solutions of the generalized burgers fisher equation we consider solitary wave solutions of the generalized burgers fisher bf equation u t pu ru x u x x qu 1 u r 0 in this paper by considering the decomposition scheme we first obtain the exact solutions of the generalized bf equation for the initial condition without using any classical transformations and then its numerical solutions are constructed without using any discretization technique the numerical solutions are compared with the known analytical solutions its remarkable accuracy is finally demonstrated in the study of some values r 2 of the generalized bf equation

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

deleted residuals the qr factored newton iteration and other methods for formally overdetermined determinate discretizations of nonlinear eigenproblems for solitary cnoidal and shock waves abstract solitary waves cnoidal waves and shock waves can be computed by solving a nonlinear eigenvalue problem which in discretized form is a system of nonlinear algebraic equations unfortunately many such systems are singular because the solution is not unique until one or more additional constraints are imposed for example if the waves are translationally invariant and u x is a solution then so also is u x for arbitrary to obtain a unique solution one must impose an additional condition to reduce the one parameter family of solutions by constraining we describe five methods for coping with such singular systems i reformulation of the problem ii deleting residuals iii keller s bordered matrix scheme iv qr factored overdetermined newton iteration and v pseudoinverse newton iteration we illustrate these ideas using the cnoidal waves of the korteweg de vries equation the traveling shocks of the korteweg de vries burgers equation and the weakly nonlocal solitary waves of the nonlinear equatorial beta plane equations finite difference fourier and rational chebyshev pseudospectral methods and spectrally upgraded finite differences are applied reformulation and deleting residuals are the cheapest strategies but the qr factored newton iteration is needed for the shock waves which lack the symmetry of the other two wave species

adaptive numerical method for burgers type nonlinear equations abstract in this paper a moving mesh partial differential equation mmpde approach will be applied to some types of nonlinear partial differential equations the method is described in detail and a number of computational examples are presented using different monitor functions applied to the burgers type equations

travelling wave solutions of generalized forms of burgers burgers kdv and burgers huxley equations in this work exact travelling wave solutions of generalized forms of burgers burgers kdv and burgers huxley equations are obtained the analysis rests mainly on the standard tanh method the work emphasizes the need for a transformation formula for the case where the parameter m is non integer the approach can be used in a variety of many types of nonlinearity

five themes in chebyshev spectral methods applied to the regularized charney eigenproblem the charney problem a second order ordinary differential equation eigenproblem on z 0 with complex eigenvalues is of great historical importance in meteorology and oceanography here it is used as a testbed for several extensions of spectral methods the first is to parameterize a plane curve which is singular at an endpoint as very common in applications the second stretch is to extend the chebyshev tau method to compute eigenfunctions of the form m z log z v z where m z and v z are entire functions and where the approximation interval is a line segment in the complex plane third we offer a special procedure for finding the roots of a function which is not a polynomial but rather the combination of a polynomial plus a logarithm multiplied by a second polynomial lastly to resolve the very thin boundary layer of the regularized charney problem we combine a rational chebyshev t l n pseudospectral method with a change of coordinate which is quadratic at the ground remarkably best results are obtained by applying four boundary conditions even though the charney problem is a differential equation of only second order

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

a non iterative derivative free method for nonlinear ordinary differential equations most methods for the determination of approximate solutions of nonlinear ordinary differential equations require that the nonlinearities be sufficiently differentiable with respect to the dependent variable and its derivatives in this paper a non iterative method for obtaining approximate solutions of nonlinear ordinary differential equations which does not require the derivatives of the nonlinearities is presented and its convergence is proved the method is also generalized for the determination of the periodic solutions of autonomous nonlinear ordinary differential equations by introducing as dependent variable the unknown frequency of oscillation the method does not require the presence of small parameters and can be used in a piecewise fashion

the delta expansion in zero dimensions the recently introduced expansion or logarithmic expansion technique for obtaining nonperturbative information about quantum field theories is reviewed in the zero dimensional context there it is easy to study questions of analytic continuation that arise in the construction of the feynman rules that generate the series it is found that for six and higher point green s functions a cancellation occurs among the most divergent terms and that divergences that arise from summing over an infinite number of internal lines are illusory the numerical accuracy is studied in some detail the series converges inside a circle of radius one for positive bare mass squared and diverges if the bare mass squared is negative but in all cases low order pade approximants are extremely accurate these general features are expected to hold in higher dimensions such as four

laguerre galerkin method for nonlinear partial differential equations on a semi infinite interval summary a laguerre galerkin method is proposed and analyzed for the burgers equation and benjamin bona mahony bbm equation on a semi infinite interval by reformulating these equations with suitable functional transforms it is shown that the laguerre galerkin approximations are convergent on a semi infinite interval with spectral accuracy an efficient and accurate algorithm based on the laguerre galerkin approximations to the transformed equations is developed and implemented numerical results indicating the high accuracy and effectiveness of this algorithm are presented

numerical algorithm for computing acoustic and vortical spatial instability waves local linear stability is often invoked in computational aeroacoustics to predict mach wave radiation or to prescribe inflow conditions in order to drive turbulent transition in large eddy simulations in this work the governing equations are reformulated for the nonoscillatory part of eigenfunctions boundary conditions can thus be explicitly enforced and moreover the numerical cost is drastically reduced regardless of the method chosen to solve this problem an efficient method based on a matrix formulation is proposed in this study one single small collocation domain is used even for computing the stability of supersonic flows vortical and acoustic instability waves of supersonic plane jets are briefly revisited to demonstrate the efficiency of this new approach

an efficient computational approach to solving singular initial value problems for lane emden type equations in the paper we present the new computational approach for solving singular initial problems for the lane emden type equations with nonlinear terms based on the differential transformation method dtm and the modified general formula for the adomian polynomials with differential transformation components to illustrate the capability and efficiency of the method three examples are given the results obtained in this work are also compared with numerical results of heydari et al 2012 wazwaz 2005 singh et al 2009 and the standard dtm as well these results show that the proposed technique without linearization or small perturbation is very effective and convenient for solving of linear and nonlinear differential equations the general differential transformation formula is proved the general formula is presented for any analytic nonlinear terms no discretization linearization or perturbation is required nonlinear differential equations are solved using the proposed formula

an adaptive moving mesh method with static rezoning for partial differential equations abstract adaptive mesh methods are valuable tools in improving the accuracy and efficiency of the numerical solution of evolutionary systems of partial differential equations if the mesh moves to track fronts and large gradients in the solution then larger time steps can be taken than if it were to remain stationary we derive explicit differential equations for moving the mesh so that the time variation of the solution at the mesh points is minimized moving the mesh based on this approach allows for larger time steps but does not guarantee that the solution is well resolved in space we maintain spatial accuracy when there are new emerging layers or wave fronts by adaptively rezoning the mesh points to equidistribute an error estimate when using a multistep integration method the past solution values are also interpolated so that the same multistep method can be used after rezoning the resulting algorithm has very few problem dependent numerical parameters and is appropriate for a large class of one dimensional partial differential equations we illustrate the performance of the algorithm by examples and demonstrate that the proposed algorithm is efficient and accurate when compared with other adaptive mesh strategies

asymptotic approach for non linear periodical vibrations of continuous structures abstract an asymptotic approach for determining periodic solutions of non linear vibration problems of continuous structures such as rods beams plates etc is proposed starting with the well known perturbation technique the independent displacement and frequency is expanded in a power series of a natural small parameter it leads to infinite systems of interconnected non linear algebraic equations governing the relationships between modes amplitudes and frequencies a non trivial asymptotic technique based on the introduction of an artificial small parameter is used to solve the equations an advantage of the procedure is the possibility to take into account a number of vibration modes as examples free longitudinal vibrations of a rod and lateral vibrations of a beam under cubically non linear restoring force are considered resonance interactions between different modes are investigated and asymptotic formulae for corresponding backbone curves are derived

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

application of fixed point method to obtain semi analytical solution to blasius flow and its variation the classical blasius flow and its variation sakiadis flow are investigated by the fixed point method fpm and the semi analytical solutions are obtained different from early perturbation series solutions the present semi analytical solutions are valid in the whole region without using matching procedure moreover these solutions can correctly provide the asymptotic property of the governing equations the comparison of the present results with numerical results is carried out which reveals that fpm is a convenient and effective tool for nonlinear problems

a new perturbative approach to nonlinear problems a recently proposed perturbative technique for quantum field theory consists of replacing nonlinear terms in the lagrangian such as 4 by 2 1 and then treating as a small parameter it is shown here that the same approach gives excellent results when applied to difficult nonlinear differential equations such as the lane emden thomas fermi blasius and duffing equations

on the zero dispersion limit of the benjamin ono cauchy problem for positive initial data we study the cauchy initial value problem for the benjamin ono equation in the zero dispersion limit and we establish the existence of this limit in a certain weak sense by developing an appropriate analogue of the method invented by lax and levermore to analyze the corresponding limit for the korteweg de vries equation 2010 wiley periodicals inc

comparison of homotopy perturbation method and homotopy analysis method comparison of homotopy perturbation method hpm and homotopy analysis method is made revealing that the former is more powerful than the later furthermore the hpm is further developed in this paper by applying the modern perturbation methods

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

evaluation of feynman diagrams in the logarithmic approach to quantum field theory the techniques necessary to compute the o 2 contributions to the green s functions of a scalar field theory with self interaction 2 1 in d dimensional space time are developed the resulting expressions are evaluated explicitly for d 1 0 and some negative even dimensions for d 3 and 4 we calculate their leading behavior in terms of a spatial cutoff a

rapid reanalysis of the generalized eigenvalue problem of locally modified linear dynamical systems abstract a computationally efficient method for the reanalysis of the damped generalized eigenvalue problems associated with linear dynamical systems with local damping and stiffness modifications is developed no restrictions are imposed on the damping of the system and the modifications in the damping and the stiffness can be independent of each other the application of the developed method to the undamped generalized eigenvalue problem with mass and stiffness modifications is also discussed the proposed method achieves up to three orders of magnitude gains in computational efficiency compared to the conventional solution procedures

eigenvalue error analysis of viscously damped structures using a ritz reduction method the efficient solution of the eigenvalue problem that results from inserting passive dampers with variable stiffness and damping coefficients into a structure is addressed eigenanalysis of reduced models obtained by retaining a number of normal modes augmented with ritz vectors corresponding to the static solutions resulting from the load patterns introduced by the dampers has been empirically shown to yield excellent approximations to the full eigenvalue problem an analysis of this technique in the case of a single damper is presented a priori and a posteriori error estimates are generated and tested on numerical examples comparison theorems with modally truncated models and a markov parameter matching reduced order model are derived these theorems corroborate the heuristic that residual flexibility methods improve low frequency approximation of the system the analysis leads to other techniques for eigenvalue approximation approximate closed form solutions are derived that include a refinement to eigenvalue derivative methods for approximation an efficient newton scheme is also developed a numerical example is presented demonstrating the effectiveness of each of these methods

on the rational second kind chebyshev pseudospectral method for the solution of the thomas fermi equation over an infinite interval in this paper we propose a pseudospectral method for solving the thomas fermi equation which is a nonlinear singular ordinary differential equation on a semi infinite interval this approach is based on the rational second kind chebyshev pseudospectral method that is indeed a combination of tau and collocation methods this method reduces the solution of this problem to the solution of a system of algebraic equations the slope at origin is provided with high accuracy comparison with some numerical solutions shows that the present solution is effective and highly accurate