rational approximation to the thomas fermi equations abstract we show that a simple and straightforward rational approximation to the thomas fermi equation provides the slope at origin with unprecedented accuracy and that pade approximants of relatively low order are far more accurate than more elaborate approaches proposed recently by other authors we consider both the thomas fermi equation for isolated atoms and for atoms in strong magnetic fields

accurate calculation of the solutions to the thomas fermi equations abstract we obtain highly accurate solutions to the thomas fermi equations for atoms and atoms in very strong magnetic fields we apply the pade hankel method numerical integration power series with pade and hermite pade approximants and chebyshev polynomials both the slope at origin and the location of the right boundary in the magnetic field case are given with unprecedented accuracy

a new analytical and numerical treatment for singular two point boundary value problems via the adomian decomposition method based on the adomian decomposition method a new analytical and numerical treatment is introduced in this research to investigate linear and non linear singular two point bvps the effectiveness of the proposed approach is verified by several linear and non linear examples

a general method for recovering equilibrating element tractions abstract a new geometrical presentation of a general method for recovering equilibrating element tractions is developed for 2d finite element models of problems in structural mechanics the data for the method consists of the prescribed loading and the stress fields resulting from a conventional linear elastic finite element analysis the geometrical presentation utilises the physical concepts of nodal forces and maxwell force diagrams based on nodal equilibrium only small problems have to be solved for each node this presentation complements the original algebraic approach of ladeveze and extends the method to cover the use of hierarchical p elements in regular meshes and general higher order elements in 1 irregular meshes as encountered in h p mesh refinement the same presentation is considered for the recovery of residual tractions as required in the solution of the local residual problem comparison is made with related methods for obtaining residual tractions in particular those proposed by bank and weiser and ainsworth et al detailed examples are included to illustrate the presentation this paper considers only the fundamental aspects of recovering equilibrating tractions the knowledge gained may be used in various applications such as error analysis limit analysis and structural design

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

quasilinearization method and its verification on exactly solvable models in quantum mechanics the proof of the convergence of the quasilinearization method of bellman and kalaba whose origin lies in the theory of linear programming is extended to large and infinite domains and to singular functionals in order to enable the application of the method to physical problems this powerful method approximates solution of nonlinear differential equations by treating the nonlinear terms as a perturbation about the linear ones and is not based unlike perturbation theories on existence of some kind of small parameter the general properties of the method particularly its uniform and quadratic convergence which often also is monotonic are analyzed and verified on exactly solvable models in quantum mechanics namely application of the method to scattering length calculations in the variable phase method shows that each approximation of the method sums many orders of the perturbation theory and that the method reproduces properly the singular structure of the exact solutions the method provides final and reasonable answers for infinite values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist

jacobi approximations in certain hilbert spaces and their applications to singular differential equations jacobi approximations in certain hilbert spaces are investigated several weighted inverse inequalities and poincare inequalities are obtained some approximation results are given singular differential equations are approximated by using jacobi polynomials this method keeps the spectral accuracy some linear problems and a nonlinear logistic equation are considered the stabilities and the convergences of proposed schemes are proved strictly the main idea and techniques used in this paper are also applicable to other singular problems in multiple dimensional spaces

necessary and sufficient conditions for oscillations of higher order delay differential equations we obtain necessary and sufficient conditions under which ail solutions of certain n th order delay differential equations oscillate when n is odd and necessary and sufficient conditions under which all bounded solutions oscillate when n is even the conditions derived involve the characteristic equation of the delay differential equation

the improved homotopy analysis method for the thomas fermi equation abstract the homotopy analysis method ham is sharpened to solve the thomas fermi equation some techniques are employed including the use of asymptotic analysis to introduce proper transformation and the use of optimal initial guess and optimal auxiliary linear operator to accelerate the convergence of homotopy approximations the optimal convergence control parameters are determined by the minimum of the squared residual error as a result the initial slop is provided with more than 10 digit accuracy which is far more accurate than the results obtained by other authors using the same method it demonstrates the flexibility and power of the ham equipped with these techniques

an asymptotic analysis of positive solutions of generalized thomas fermi differential equations the sub half linear case abstract the existence and the asymptotics behavior for the large value of the variable of the positive solutions of generalized thomas fermi equation presented in this article are proved it is assumed that coefficient q t belongs to the class of regularly varying functions in the sense of karamata properties of these functions and the schauder tychonoff fixed point theorem are the main tools for the proofs

guaranteed computable error bounds for conforming and nonconforming finite element analyses in planar elasticity we obtain fully computable a posteriori error estimators for the energy norm of the error in second order conforming and nonconforming finite element approximations in planar elasticity these estimators are completely free of unknown constants and give a guaranteed numerical upper bound on the norm of the error the estimators are shown to also provide local lower bounds up to a constant and higher order data oscillation terms numerical examples are presented illustrating the theory and confirming the effectiveness of the estimator

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

variation and series approach to the thomas fermi equation abstract the thomas fermi equation describing the screening of the coulomb potential inside heavy neutral atoms is reconsidered an accurate representation for its numerical solution was obtained by means of the variational principle the proposed new solution has more precise asymptotic behaviour at large distances from the origin and allows us to obtain the exact value of the initial slope the obtained new variational solution can also be developed in power series similar to the baker s ones but more precise even than some series solutions that have been recently obtained within the homotopy analysis method and a modified variational method

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

finite difference finite element method for two dimensional space and time fractional bloch torrey equations in this paper a class of two dimensional space and time fractional bloch torrey equations 2d stfbtes are considered some definitions and properties of fractional derivative spaces are presented by finite difference method and galerkin finite element method a semi discrete variational formulation for 2d stfbtes is obtained the stability and convergence of the semi discrete form are discussed then a fully discrete scheme of 2d stfbtes is derived and the convergence is investigated finally some numerical examples based on linear piecewise polynomials and quadratic piecewise polynomials are given to prove the correctness of our theoretical analysis

homotopy perturbation method a new nonlinear analytical technique in this paper a new perturbation method is proposed in contrast to the traditional perturbation methods this technique does not require a small parameter in an equation in this method according to the homotopy technique a homotopy with an imbedding parameter email protected 0 1 is constructed and the imbedding parameter is considered as a small parameter so the method is called the homotopy perturbation method which can take the full advantages of the traditional perturbation methods and homotopy techniques to illustrate its effectiveness and its convenience a duffing equation with high order of nonlinearity is used the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter and is more accurate than the perturbation solutions

asymptotic and oscillatory behavior of solutions of first order nonlinear neutral delay differential equations the authors consider the first order nonlinear neutral delay differential equation e y t p t y t q t y t 0 where p q and are continuous q t 0 0 0 and u u 0 if u 0 they give sufficient conditions for all nonoscillatory solutions of e to converge to zero as t two oscillation theorems for equation e are also proved

oscillatory and asymptotic behavior of delay differential equations with a nonlinear damping term in this paper we study the oscillatory and asymptotic behavior of the solutions of delay differential equations of the form x n t f t x t x t 0 n even by comparing with certain differential equations of the same or lower order whose behavior is known the obtained criteria can be applied to the delay differential equation x n t q t x t x t sign x t 0 n even where and are positive constants

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

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

a posteriori estimates of the solution error caused by discretization in the finite element finite difference and boundary element methods a theory is described which guarantees an upper and lower bound estimate of the discretization error in numerical solutions of elliptic boundary value problems this method gives bounded global estimates of the error in the energy norm pointwise estimates of the error in the solution variable or its derivatives can then be obtained if the numerical solution is exhibiting pointwise monotonic convergence the versatility of this method is illustrated by its application to numerical solutions from finite element finite difference and boundary element methods

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

oscillations of advanced differential inequalities on considere le comportement oscillatoire d inegalites differentielles avancees de la forme 1 2 x n t sign x t p t i 1 m x g i t i et 1 z x n t sign x t i 1 m p i t x g i t ou n 2 z 1 2 n n 1 i i 1 m sont des nombres non negatifs avec 1 m 1 les fonctions p p i r r 0 sont integrables sur chaque segment fini et ne sont pas identiquement nulles dans tout voisinage de l infini

element by element a posteriori error estimation and improvement of stress solutions for two dimensional elastic problems a quantitative method of a posteriori error estimation based on finite element solutions is developed the proposed method can estimate the error of solutions obtained by the finite element method for two dimensional elastic problems with 4 node elements since the error is estimated element by element the proposed method does not require a large memory or long computing time not only rectangular elements but also arbitrarily shaped 4 node elements can be used in this method for estimating the error in the finite element computation with high accuracy the finite element solutions are improved by adding the estimated errors onto the original solutions the proposed method can be utilized for any type of linear problem if an isoparametric finite element method is used

homotopy perturbation technique abstract the homotopy perturbation technique does not depend upon a small parameter in the equation by the homotopy technique in topology a homotopy is constructed with an imbedding parameter p 0 1 which is considered as a small parameter some examples are given the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters

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

a finite element method for singular solutions of the navier stokes equations on a non convex polygon it is shown in choi and kweon 2013 that a solution of the navier stokes equations with no slip boundary condition on a non convex polygon can be written as u p c 1 1 1 c 2 2 2 u r p r near each non convex vertex where u r p r h 2 h 1 i i are corner singularity functions for the stokes problem with no slip condition and c i r are coefficients which are called the stress intensity factors we design a finite element method to approximate the coefficients c i and the regular part u r p r show the unique existence of the approximations and derive their error estimates some numerical examples are given confirming convergence rates for the approximations

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

oscillation theorems for nth order differential equations with deviating arguments abstract this paper is concerned with the study of the oscillatory behavior of solutions of the n th order nonlinear differential equation a t x n v t v q t f x g t 0 where n is even and 1 v n 1 a systematic study is attempted which extends and correlates a number of existing results

on weak residual error estimation a general framework for weak residual error estimators applying to various types of boundary value problems in connection with finite element and finite volume approximations is developed basic ideas commonly shared by various applications in error estimation and adaptive computation are presented and illustrated some numerical results are given to show the effectiveness and efficiency of the estimators

a general formulation of equilibrium macro elements with control of spurious kinematic modes the exorcism of an old curse this paper illustrates a method whereby a family of robust equilibrium elements can be formulated in a general manner the effects of spurious kinematic modes present to some extent in all primitive equilibrium elements are eliminated by judicious assembly into macro equilibrium elements these macroelements are formulated with sufficient generality so as to retain the polynomial degree of the stress field as a variable such a family of macro elements is a new development and results for polynomials of degree greater than two have not been seen before the quality of results for macro equilibrium elements with varying degrees of polynomial is demonstrated by numerical examples

realistic computable error bounds for three dimensional finite element analyses in linear elasticity we obtain a computable estimator for the energy norm of the error in piecewise affine and piecewise quadratic finite element approximations of linear elasticity in three dimensions we show that the estimator provides guaranteed upper bounds on the energy norm of the error as well as up to a constant and data oscillation terms local lower bounds

analytic continuation of taylor series and the boundary value problems of some nonlinear ordinary differential equations abstract we compare and discuss the respective efficiency of three methods with two variants for each of them based respectively on taylor maclaurin series pade approximants and conformal mappings for solving quasi analytically a two point boundary value problem of a nonlinear ordinary differential equation ode six configurations of ode and boundary conditions are successively considered according to the increasing difficulties that they present after having indicated that the taylor series method almost always requires the recourse to analytical continuation procedures to be efficient we use the complementarity of the two remaining methods pade and conformal mapping to illustrate their respective advantages and limitations we emphasize the importance of the existence of solutions with movable singularities for the efficiency of the methods particularly for the so called pade hankel method we show that this latter method is equivalent to pushing a movable pole to infinity for each configuration we determine the singularity distribution in the complex plane of the independent variable of the solution sought and show how this distribution controls the efficiency of the two methods in general the method based on pade approximants is easy to use and robust but may be awkward in some circumstances whereas the conformal mapping method is a very fine method which should be used when high accuracy is required

oscillatory properties of functional differential equations abstract in the present paper we establish some new criteria for the oscillatory and asymptotic behavior of the functional differential equations with deviating arguments of the form d dt 1 a n 1 t d dt 1 a n 2 t d dt 1 a 1 t d dt x t t x g 1 t x g m t 0 where 1 the obtained results improve unify and correlate a number of existing criteria

approximation of some diffusion evolution equations in unbounded domains by hermite functions spectral and pseudospectral approximations of the heat equation are analyzed the solution is represented in a suitable basis constructed with hermite polynomials stability and convergence estimates are given and numerical tests are discussed

a maximum entropy method for solving the boundary value problem of second order ordinary differential equations we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent

validation of a posteriori error estimators by numerical approach abstract a numerical methodology which determines the quality or robustness of a posteriori error estimators is described the methodology accounts precisely for the factors which affect the quality of error estimators for finite element solutions of linear elliptic problems namely the local geometry of the grid and the structure of the solution the methodology can be employed to check the robustness of any estimator for the complex grids which are used in engineering computations

explicitly symmetrical treatment of three body phase space we derive expressions for three body phase space that are explicitly symmetrical in the masses of the three particles we study geometrical properties of the variables involved in elliptic integrals and demonstrate that it is convenient to use the jacobian zeta function to express the results in four and six dimensions

orthogonal rational functions on a semi infinite interval abstract by applying a mapping to the chebyshev polynomials we define a new spectral basis the rational chebyshev functions on the semi infinite interval denoted by tl n y continuing earlier work by the author and by grosch and orszag we show that these rational functions inherit most of the good numerical characteristics of the chebyshev polynomials orthogonality completeness exponential or infinite order convergence matrix sparsity for equations with polynomial coefficients and simplicity seven numerical examples illustrate their versatility the charney stability problem of meteorology for example is solved to show the feasibility of applying spectral methods to a semi infinite atmosphere for functions that are singular at both endpoints such as k 1 y one may combine rational chebyshev functions with a preliminary mapping to obtain a single exponentially convergent expansion on y 0 finally we successfully generalize the wkb method to obtain for the j 0 bessel function an amplitude phase approximation which is convergent rather than asymptotic and is accurate not merely for large y but for all y even the origin

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

iterative method for constrained and mixed approximation an inexpensive improvement of f e m performance a general iterative procedure for solving mixed lagrangian finite element equations is introduced the method allows a dramatic improvement in accuracy of simple irreducible formulations to be achieved in a few resolutions applications to incompressible flow stress analysis and metal forming problems are given the procedures can be easily implemented into standard programs

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

oscillation theorems for nth order delay differential equations abstract oscillation criteria are obtained for a class of functional differential equations including the generalized emden fowler equation x n q t x g t sgn x g t 0 0 as a special case the purpose of this paper is to extend and improve known fundamental oscillation criteria for superlinear sublinear and linear differential equations with retarded arguments

variational approach to the thomas fermi equation by the semi inverse method a variational principle is obtained for the thomas fermi equation then the ritz method is applied to solve an analytical solution which is a much simpler and more efficient method

a variational method for finite element stress recovery and error estimation abstract a variational method for obtaining smoothed stresses from a finite element derived non smooth stress field is presented the method is based on minimizing a functional involving discrete least squares error plus a penalty constraint that ensures smoothness of the stress field an equivalent accuracy criterion is developed for the smoothing analysis which results in a c1 continuous smoothed stress field possessing the same order of accuracy as that found at the superconvergent optimal stress points of the original finite element analysis application of the smoothing analysis to residual error estimation is also demonstrated

some oscillation criteria for delay differential equations of even order abstract some new oscillation criteria for the delay differential equations of the form a t x n v v q t x g 1 t h x g 2 t 0 d dt where n is even and 1 v n 1 are established

pricing european and american options by radial basis point interpolation a rbpi method for pricing european and american options is proposed rbpi has not yet been used for option pricing the overall efficiency is improved by coupling rbpi with other approaches three different algorithms for american options are tested and compared numerical results demonstrate the effectiveness of the techniques employed we propose the use of the meshfree radial basis point interpolation rbpi to solve the black scholes model for european and american options the rbpi meshfree method offers several advantages over the more conventional radial basis function approximation nevertheless it has never been applied to option pricing at least to the very best of our knowledge in this paper the rbpi is combined with several numerical techniques namely an exponential change of variables which allows us to approximate the option prices on their whole spatial domain a mesh refinement algorithm which turns out to be very suitable for dealing with the non smooth options payoff and an implicit euler richardson extrapolated scheme which provides a satisfactory level of time accuracy finally in order to solve the free boundary problem that arises in the case of american options three different approaches are used and compared the projected successive overrelaxation method psor the bermudan approximation and the penalty approach numerical experiments are presented which demonstrate the computational efficiency of the rbpi and the effectiveness of the various techniques employed

structure conserving parametrization of feynman diagrams using x space parameters instead of feynman parameters dimensionally regularized feynman diagrams are expanded with respect to the external momenta the internal masses the logarithms of these masses and the regularization parameter 12n a general formula for an arbitrary feynman diagram is obtained for any dimension n all ultraviolet divergences appear in a direct and transparent way as poles their residues in the limit 12n 0 are subseries that are recognized as subdiagrams relations between diagrams are transparent in all orders of perturbation theory

the asymptotic and oscillatory behavior of the solutions of higher order neutral equations abstract consider the higher order neutral differential equation y t i 1 n1 p i y t i j 1 n2 r j y t j n s 1 n3 q s y t 0 we study the asymptotic behavior of solutions of eq 1 and we obtain several sufficient conditions for the oscillation of solutions of eq 1 this paper generalizes all results of ladas and sficas 1

asymptotic and oscillatory behavior of solutions of nonlinear neutral delay equations of arbitrary order abstract the authors obtain results on the oscillatory and asymptotic behavior of solutions of the nonlinear neutral delay differential equation y t p t y g t n f t y h t 0 where n 1 1 and uf t u 0 for u 0 several known results of this type are included as special cases examples illustrating their results are included

a model study of the quality of a posteriori error estimators for linear elliptic problems error estimation in the interior of patchwise uniform grids of triangles abstract this paper is the first in a series in which we discuss computational methodologies for checking the quality of a posteriori error estimators for finite element approximations of linear elliptic problems in this first part we study the asymptotic properties of error estimators in the interior of patchwise uniform grids of triangles a completely numerical methodology for the analysis of the quality of estimators is presented results from the application of the methodology to the study of the quality of several well known error estimators are reported in subsequent papers we shall discuss methods to study the properties of estimators for meshes of quadrilaterals non uniform grids at boundaries grid interfaces and near singular points

a series solution to the thomas fermi equation abstract nonlinear thomas fermi equation is solved by an analytic technique named homotopy analysis method ham in this paper for a further improvement of the convergence and precision of the solution to thomas fermi equation by ham different from previous work however a more generalized set of basis function and consequential auxiliary linear operator are introduced to provide a series solution the comparisons are also made among the results of the present work some well known numerical solution and previous work with the same technique which shows the present work has provided a better series solution by far

the existence of solutions to second order singular boundary value problems abstract this article analyzes qualitative properties of solutions to two point boundary value problems for singular ordinary differential equations in particular we form new approaches that ensure that all possible solutions satisfy certain a priori bounds the methods involve differential inequalities the ideas are then applied to generate new existence results for solutions to the boundary value problems under consideration many of the results are novel for both the singular and the nonsingular cases the main concepts are further illustrated via applications to various equations arising in modeling such as the poisson boltzmann equation and the thomas fermi equation with an ionized atom

oscillatory and asymptotic behavior of solutions of differential equations with deviating arguments the oscillatory and asymptotic behavior of solutions of a class of nth order nonlinear differential equations with deviating arguments of the form e lnx t q t f x g1 t x gm t 0 where 1 and l0x t x t lkx t ak t lk 1x t k 1 2 n ddt is examined a classification of solutions of e with respect to their behavior as t and their oscillatory character is obtained the comparisons of e 1 and e 1 with first and second order equations of the form y t c1 t f y g1 t y gm t 0 and an 1 t z t c2 t f z g1 t z gm t 0 respectively are presented the obtained results unify extend and improve some of the results by graef grammatikopoulos and spikes philos and staikos

multidimensional phase space and sunset diagrams we derive expressions for the phase space of a particle of momentum p decaying into n particles that are valid for any number of dimensions these are the imaginary parts of so called sunset diagrams which we also obtain the results are given as a series of hypergeometric functions which terminate for odd dimensions and are also well suited for deriving the threshold behavior

the self equilibration of residuals and complementary a posteriori error estimates in the finite element method it is demonstrated that the residual in a compatible displacement finite element solution can be partitioned into local self equilibrating systems on each element an a posteriori error analysis is then based on a complementary approach and examples indicate that the guaranteed upper bound on the energy of the error is preserved

on the oscillatory and asymptotic behavior of even order nonlinear differential equations with retarded arguments abstract some new oscillation criteria for the retarded differential equations of the form d dt a n 1 t d dt a 2 t d dt a 1 t dx t dt q t x g 1 t h dx g 2 t dt 0 n is even are established

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

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

the modified decomposition method and pade approximants for solving the thomas fermi equation in this paper we propose a new approach to develop a nonperturbative approximate solution for the thomas fermi equation this approach rests on the recently developed modification of the adomian decomposition method the initial slope of the thomas fermi potential y 0 is computed by converting the obtained series solution into several diagonal pade approximants the proposed scheme is presented in a general way so that it can be used in applied sciences

increasing solutions of thomas fermi type differential equations the superlinear case abstract an asymptotic analysis of increasing solutions x t of equations of super linear thomas fermi type is performed in the framework of regular variation leading to the accurate behavior of all such x t for t

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