on using a modified legendre spectral method for solving singular ivps of lane emden type in this paper approximate solutions of singular initial value problems ivps of the lane emden type in second order ordinary differential equations odes are obtained by an improved legendre spectral method the legendre gauss points are used as collocation nodes and lagrange interpolation is employed in the volterra term the results reveal that the method is effective simple and accurate

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

detecting algebraic in dependence of explicitly presented functions some applications of nevanlinna theory to mathematical logic we consider algebraic relations between explicitly presented analytic functions with particular emphasis on tarski s high school algebra problem the part not related directly to tarski s high school algebra problem let u be a connected complex analytic manifold denote by f u the minimal field containing all functions meromorphic on u and closed under exponentiation f e f let f j f u p j m u 0 for 1 j m and g k f u q k m u 0 for 1 k n where m u is the field of functions meromorphic on u let f i f j h u for i j and g k g l h u for k l where h u is the ring of functions holomorphic on u if all zeros and singularities of formule are contained in an analytic subset of u then m n and there exists a permutation of 1 m such that h p j q j e fj g j for 1 j m when h m u additionally f j g j h u for all j

accuracy analysis of explicit runge kutta methods applied to the incompressible navier stokes equations this paper investigates the temporal accuracy of the velocity and pressure when explicit runge kutta methods are applied to the incompressible navier stokes equations it is shown that at least up to and including fourth order the velocity attains the classical order of accuracy without further constraints however in case of a time dependent gradient operator which can appear in case of time varying meshes additional order conditions need to be satisfied to ensure the correct order of accuracy furthermore the pressure is only first order accurate unless additional order conditions are satisfied two new methods that lead to a second order accurate pressure are proposed which are applicable to a certain class of three and four stage methods a special case appears when the boundary conditions for the continuity equation are independent of time since in that case the pressure can be computed to the same accuracy as the velocity field without additional cost relevant computations of decaying vortices and of an actuator disk in a time dependent inflow support the analysis and the proposed methods

influence of soil structure and root water uptake strategy on unsaturated flow in heterogeneous media 1 we analyze the combined effects of the spatial variability of soil hydraulic properties and the water uptake by plant roots on unsaturated water flow for this analysis we use a simplified macroscopic root water uptake model which is usually applied only for homogeneous or layered soil and therefore we also determine whether it is applicable for multidimensional heterogeneous media analytical solutions for mean and variance of pressure head first order second moment approximations in layered media and numerical solutions of two dimensional 2 d autocorrelated multi gaussian and non multi gaussian parameter fields are analyzed for steady state and transient flow conditions for non gaussian topological features that have little influence on the mean and the variance of the pressure field if root water uptake is ignored we test whether the influence is significant if root water uptake is accounted for the results reveal that in field structures with large patches of coarse material local regions with pressure head values at the wilting point develop these are surrounded by wet material without a compensation mechanism for local stress the global transpiration demand is not met if local wilting occurs various compensation mechanisms are tested that depend respectively on the saturation the relative conductivity or a strategy where the deficit in the global uptake rate is equally distributed to unstressed locations the strategies lead to a global actual transpiration rate at the potential value and attenuate the formation of locally wilted areas wilted regions can however still occur and may be an artifact of the simplified model concept as root soil interactions are neglected therefore simplified macroscopic models for root water uptake should be used with caution in heterogeneous media

on the effect of numerical errors in large eddy simulations of turbulent flows aliased and dealiased numerical simulations of a turbulent channel flow are performed using spectral and finite difference methods analytical and numerical studies show that aliasing errors are more destructive for spectral and high order finite difference calculations than for low order finite difference simulations numerical errors have different effects for different forms of the nonlinear terms in the navier stokes equations for divergence and convective forms spectral methods are energy conserving only if dealiasing is performed for skew symmetric and rotational forms both spectral and finite difference methods are energy conserving even in the presence of aliasing errors it is shown that discrepancies between the results of dealiased spectral and standard nondialiased finite difference methods are due to both aliasing and truncation errors with the latter being the leading source of differences the relative importance of aliasing and truncation errors as compared to subgrid scale model terms in large eddy simulations is analyzed and discussed for low order finite difference simulations truncation errors can exceed the magnitude of the subgrid scale term

energy conserving runge kutta methods for the incompressible navier stokes equations energy conserving methods have recently gained popularity for the spatial discretization of the incompressible navier stokes equations in this paper implicit runge kutta methods are investigated which keep this property when integrating in time firstly a number of energy conserving runge kutta methods based on gauss radau and lobatto quadrature are constructed these methods are suitable for convection dominated problems such as turbulent flows because they do not introduce artificial diffusion and are stable for any time step secondly to obtain robust time integration methods that work also for stiff problems the energy conserving methods are extended to a new class of additive runge kutta methods which combine energy conservation with l stability in this class the radau iia b method has the best properties results for a number of test cases on two stage methods indicate that for pure convection problems the additive radau iia b method is competitive with the gauss methods however for stiff problems such as convection dominated flows with thin boundary layers both the higher order gauss and radau iia b method suffer from order reduction overall the gauss methods are the preferred method for energy conserving time integration of the incompressible navier stokes equations

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

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

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

energy preserving turbulent simulations at a reduced computational cost energy conserving discretizations are widely regarded as a fundamental requirement for high fidelity simulations of turbulent flows the skew symmetric splitting of the nonlinear term is a well known approach to obtain semi discrete conservation of energy in the inviscid limit however its computation is roughly twice as expensive as that of the divergence or advective forms alone a novel time advancement strategy that retains the conservation properties of skew symmetric based schemes at a reduced computational cost has been developed this method is based on properly constructed runge kutta schemes in which a different form advective or divergence for the convective term is adopted at each stage a general framework is presented to derive schemes with prescribed accuracy on both solution and energy conservation simulations of homogeneous isotropic turbulence show that the new procedure is effective and can be considerably faster than skew symmetric based techniques

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

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

definable sets in ordered structures i this paper introduces and begins the study of a well behaved class of linearly ordered structures the minimal structures the definition of this class and the corresponding class of theories the strongly dlminlmal theories is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories theorems 2 1 and 2 3 respectively provide characterizations of 41 minimal ordered groups and rings several other simple results are collected in 3 the primary tool in the analysis of minimal structures is a strong analogue of ii forking symmetry given by theorem 4 2 this result states that any parametri cally definable unary function in an yminimal structure is piecewise either constant or an ore er preserving or reversing bijection of intervals the results that follow include the existence and uniqueness of prime models over sets theorem 5 1 and a characterization of all bto categorical aminimal structures theorem 6 1 1 introduction the class of linearly ordered structures has long been an im portant subject of concern to model theorists impressive results have been obtained in the study of models of several particular theories that extend the theory of linear order among those theories that have been approached successfully are peano arithmetic the theory of ordered abelian groups that of real closed fields and that of linear order itself yet very little has been done in the way of developing a general model theory for ordered structures in this paper we develop the model theory for a class of linearly ordered structures that we isolate by demanding that a structure in this class satisfy a condition whose effect is that the linear ordering and the algebraic part of the structure behave quite well with respect to one another let l be a finitary first order language and w an l structure a set of n tuples a c x is said to be parametrically definable if there is some l formula

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

explicit runge kutta schemes for incompressible flow with improved energy conservation properties the application of pseudo symplectic runge kutta methods to the incompressible navier stokes equations is discussed in this work in contrast to fully energy conserving implicit methods these are explicit schemes of order p that preserve kinetic energy to order q with q p use of explicit methods with improved energy conservation properties is appealing for convection dominated problems especially in case of direct and large eddy simulation of turbulent flows a number of pseudo symplectic methods are constructed for application to the incompressible navier stokes equations and compared in terms of accuracy and efficiency by means of numerical simulations

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

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

solutions of singular ivps of lane emden type by the variational iteration method abstract in this paper approximate exact solutions of a class of lane emden type singular ivps problems by the variational iteration method are presented the variational iteration method yields solutions in the forms of convergent series with easily calculable terms the scheme is shown to be highly accurate and in some cases yields exact solutions in few iterations

definable sets in ordered structures on introduit la notion de theorie o minimale des structures ordonnees une telle theorie etant telle que les sous ensembles definissables de ses modeles soient particulierement simples

fully conservative higher order finite difference schemes for incompressible flow conservation properties of the mass momentum and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations existing finite difference schemes in regular and staggered grid systems are checked for violations of the conservation requirements and a few important discrepancies are pointed out in particular it is found that none of the existing higher order schemes for a staggered mesh system simultaneously conserve mass momentum and kinetic energy this deficiency is corrected through the derivation of a general family of fully conservative higher order accurate finite difference schemes for staggered grid systems finite difference schemes in a collocated grid system are also analyzed and a violation of kinetic energy conservation is revealed the predicted conservation properties are demonstrated numerically in simulations of inviscid white noise performed in a two dimensional periodic domain the proposed fourth order schemes in a staggered grid system are generalized for the case of a non uniform mesh and the resulting scheme is used to perform large eddy simulations of turbulent channel flow

smooth functions in o minimal structures abstract fix an o minimal expansion of the real exponential field that admits smooth cell decomposition we study the density of definable smooth functions in the definable continuously differentiable functions with respect to the definable version of the whitney topology this implies that abstract definable smooth manifolds are affine moreover abstract definable smooth manifolds are definably c diffeomorphic if and only if they are definably c 1 diffeomorphic

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

kinetic energy conservation issues associated with the collocated mesh scheme for incompressible flow the collocated mesh scheme is often favored over the staggered mesh scheme for turbulence simulation in complex geometries due to its simpler form in curvilinear coordinates the collocated mesh scheme does not conserve kinetic energy however and few careful checks of the impact of these errors have been made in this work analysis is used to identify two sources of kinetic energy conservation error in the collocated mesh scheme 1 errors arising from the interpolations used to estimate the velocity on the cell faces and 2 errors associated with the slightly inconsistent pressure field used to ensure mass conservation for the cell face volume fluxes it is shown that the interpolation error can be eliminated through the use of first order accurate centered interpolation operators with mesh independent weights the pressure error appears to be intrinsic to the scheme and it is shown to scale as o t2 x2 the effects of the conservation errors are investigated numerically through simulations of inviscid flow over an airfoil and in large eddy simulations of turbulent channel flow neither the interpolation error nor the pressure error appear to lead to significant problems in the channel flow simulations where viscous dissipation is present and where the cartesian mesh is stretched in only one direction the inviscid airfoil simulations performed on a curvilinear mesh show a much greater sensitivity to the interpolation error the standard second order centered interpolation is shown to lead to severe numerical oscillations while the kinetic energy conserving first order centered interpolation produces solutions that are almost as smooth as those obtained with a third order upwind interpolation when compared in channel flow simulations however the first order centered interpolation is shown to be far superior to the third order upwind interpolation the latter being adversely affected by numerical dissipation only slight differences were noted when comparing channel flow simulations using first order and second order centered interpolations these results suggest that numerical oscillations can be controlled in curvilinear coordinates through the use of properly constructed non dissipative centered interpolations

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

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

the six grothendieck operations on o minimal sheaves in this paper we develop the formalism of the grothendieck six operations on o minimal sheaves the grothendieck formalism allows us to obtain o minimal versions of i derived projection formula ii universal coefficient formula iii derived base change formula iv kunneth formula v local and global verdier duality

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 homotopy perturbation method for nonlinear oscillators with discontinuities in the paper the homotopy perturbation method is applied to the nonlinear oscillators with discontinuities only one iteration leads to high accuracy of the solutions

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

symmetry preserving discretization of turbulent flow we propose to perform turbulent flow simulations in such manner that the difference operators do have the same symmetry properties as the underlying differential operators i e the convective operator is represented by a skew symmetric coefficient matrix and the diffusive operator is approximated by a symmetric positive definite matrix mimicking crucial properties of differential operators forms in itself a motivation for discretizing them in a certain manner we give it a concrete form by noting that a symmetry preserving discretization of the navier stokes equations is stable on any grid and conserves the total mass momentum and kinetic energy for the latter the physical dissipation is to be turned off of coarse being stable on any grid the choice of the grid may be based on the required accuracy solely and the main question becomes how accurate is a symmetry preserving discretization its accuracy is tested for a turbulent flow in a channel by comparing the results to those of physical experiments and previous numerical studies the comparison is carried out for a reynolds number of 5600 which is based on the channel width and the mean bulk velocity based on the channel half width and wall shear velocity the reynolds number becomes 180 the comparison shows that with a fourth order symmetry preserving method a 64 64 32 grid suffices to perform an accurate numerical simulation

an efficient time advancing strategy for energy preserving simulations energy conserving numerical methods are widely employed within the broad area of convection dominated systems semi discrete conservation of energy is usually obtained by adopting the so called skew symmetric splitting of the non linear convective term defined as a suitable average of the divergence and advective forms although generally allowing global conservation of kinetic energy it has the drawback of being roughly twice as expensive as standard divergence or advective forms alone in this paper a general theoretical framework has been developed to derive an efficient time advancement strategy in the context of explicit runge kutta schemes the novel technique retains the conservation properties of skew symmetric based discretizations at a reduced computational cost it is found that optimal energy conservation can be achieved by properly constructed runge kutta methods in which only divergence and advective forms for the convective term are used as a consequence a considerable improvement in computational efficiency over existing practices is achieved the overall procedure has proved to be able to produce new schemes with a specified order of accuracy on both solution and energy the effectiveness of the method as well as the asymptotic behavior of the schemes is demonstrated by numerical simulation of burgers equation

o minimality and the andre oort conjecture for mathbb c n we give an unconditional proof of the andre oort conjecture for arbitrary products of modular curves we establish two generalizations the first includes the manin mumford conjecture for arbitrary products of elliptic curves defined over q as well as lang s conjecture for torsion points in powers of the multiplicative group the second includes the manin mumford conjecture for abelian varieties defined over q our approach uses the theory of o minimal structures a part of model theory and follows a strategy proposed by zannier and implemented in three recent papers a new proof of the manin mumford conjecture by pila zannier a proof of a special but new case of pink s relative manin mumford conjecture by masser zannier and new proofs of certain known results of andre oort manin mumford type by pila 2010 mathematics subject classification 11g18 03c64

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

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

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

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

explicit analytical solutions for one dimensional steady state flow in layered heterogeneous unsaturated soils under random boundary conditions 1 in this study we directly derive first order analytical solutions to the pressure head moments mean and variance for one dimensional steady state unsaturated flow in randomly heterogeneous layered soil columns under various random boundary conditions we assume that the constitutive relation between the unsaturated hydraulic conductivity and the pressure head follows an exponential model and treat the saturated hydraulic conductivity ks as a random function and the pore size distribution parameter as a random constant unlike the solution given in lu and zhang 2004 in which kirchhoff transformation was used and the solution to pressure head variance was presented as a function of cross covariances related to the intermediate kirchhoff transformed variable the solution to the pressure head variance presented in this paper is an explicit function of the input variabilities in addition we also give analytical solutions to the statistics of the unsaturated hydraulic conductivity and the effective water content these first order analytical solutions are compared with those from monte carlo simulations we also investigated the effect of uncertain boundary conditions the relative contribution of input variabilities to the head variance and the possible errors introduced by treating the correlated field as a random constant in the analytical solutions the results indicate that the uncertain constant head at the bottom of a deep soil column may not have a significant effect on predicting flow statistics in the upper portion of the column furthermore it is found that treating as a random constant is justified when the correlation length of is relatively large as compared to the layer thickness

the elementary theory of restricted analytic fields with exponentiation numbers with exponentiation is model complete when we combine this with hovanskii s finiteness theorem 9 it follows that the real exponential field is o minimal in o minimal expansions of the real field the definable subsets of r share many of the nice structural properties of semialgebraic sets for example definable subsets have only finitely many connected components definable sets can be stratified and triangulated and continuous definable maps are piecewise trivial see 5 in this paper we will prove a quantifier elimination result for the real field augmented by exponentiation and all restricted analytic functions and use this result to obtain o minimality we were led to this while studying work of ressayre 13 and several of his ideas emerge here in simplified form however our treatment is formally independent of the results of 16 17 9 and 13

on the variational iteration method and other iterative techniques for nonlinear differential equations abstract a variety of iterative methods for the solution of initial and or boundary value problems in ordinary and partial differential equations is presented these iterative procedures provide the solution or an approximation to it as a sequence of iterates for initial value problems it is shown that these iterative procedures can be written in either an integral or differential form the integral form is governed by a volterra integral equation whereas the differential one can be obtained from the volterra representation by simply differentiation it is also shown that integration by parts variation of parameters adjoint operators green s functions and the method of weighted residuals provide the same volterra integral equation and that this equation in turn can be written as that of the variational iteration method it is therefore shown that the variational iteration method is nothing else by the picard lindelof theory for initial value problems in ordinary differential equations and banach s fixed point theory for initial value problems in partial differential equations and the convergence of these iterative procedures is ensured provided that the resulting mapping is lipschitz continuous and contractive it is also shown that some of the iterative methods for initial value problems presented here are special cases of the bellman kalaba quasilinearization technique provided that the nonlinearities are differentiable with respect to the dependent variable and its derivatives but such a condition is not required by the techniques presented in this paper for boundary value problems it is shown that one may use the iterative procedures developed for initial value problems but the resulting iterates may not satisfy the boundary conditions and two new iterative methods governed by fredholm integral equations are proposed it is shown that the resulting iterates satisfy the boundary conditions if the first one does so the iterative integral formulation presented here is applied to ten nonlinear oscillators with odd nonlinearities and it is shown that its results coincide with those of differential two and three level iterative techniques harmonic balance procedures and standard and modified linstedt poincare techniques the method is also applied to two boundary value problems

numerically stable formulations of convective terms for turbulent compressible flows abstract a systematic analysis of the discrete conservation properties of non dissipative central difference approximations of the compressible navier stokes equations is reported a generalized splitting of the nonlinear convective terms is considered and energy preserving formulations are fully characterized by deriving a two parameter family of split forms previously developed formulations reported in literature are shown to be particular members of this family novel splittings are introduced and discussed as well furthermore the conservation properties yielded by different choices for the energy equation i e total and internal energy entropy are analyzed thoroughly it is shown that additional preserved quantities can be obtained through a suitable adaptive selection of the split form within the derived family local conservation of primary invariants which is a fundamental property to build high fidelity shock capturing methods is also discussed in the paper numerical tests performed for the taylor green vortex at zero viscosity fully confirm the theoretical findings and show that a careful choice of both the splitting and the energy formulation can provide remarkably robust and accurate results

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

on pillay s conjecture in the general case abstract let m be an arbitrary o minimal structure let g be a definably compact definably connected abelian definable group of dimension n here we compute i the new intrinsic o minimal fundamental group of g ii for each k 0 the k torsion subgroups of g iii the o minimal cohomology algebra over q of g as a corollary we obtain a new uniform proof of pillay s conjecture an o minimal analogue of hilbert s fifth problem relating definably compact groups to compact real lie groups extending the proof already known in o minimal expansions of ordered fields

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

analytical solution to transient richards equation with realistic water profiles for vertical infiltration and parameter estimation a general analytical model for one dimensional transient vertical infiltration is presented the model is based on a combination of the brooks and corey soil water retention function and a generalized hydraulic conductivity function this leads to power law diffusivity and convective term for which the exponents are functions of the inverse of the pore size distribution index accordingly the proposed analytical solution covers many existing realistic models in the literature the general form of the analytical solution is simple and it expresses implicitly the depth as function of water content and time it can be used to model infiltration through semi infinite dry soils with prescribed water content or flux boundary conditions some mathematical expressions of practical importance are also derived the general form solution is useful for comparison between models validation of numerical solutions and for better understanding the effect of some hydraulic parameters based on the analytical expression a complete inverse procedure which allows the estimation of the hydraulic parameters from water content measurements is presented

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

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

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

inclusion dynamics hybrid automata hybrid systems are dynamical systems with the ability to describe mixed discrete continuous evolution of a wide range of systems consequently at first glance hybrid systems appear powerful but recalcitrant neither yielding to analysis and reasoning through a purely continuous time modeling as with systems of differential equations nor open to inferential processes commonly used for discrete state transition systems such as finite state automata a convenient and popular model called hybrid automata was introduced to model them and has spurred much interest on its tractability as a tool for inference and model checking in a general setting intuitively a hybrid automaton is simply a finite state automaton with each state augmented by continuous variables which evolve according to a set of well defined continuous laws each specified separately for each state this article investigates both the notion of hybrid automaton and the model checking problem over such a structure in particular it relates first order theories and analysis results on multivalued maps and reduces the bounded reachability problem for hybrid automata whose continuous laws are expressed by inclusions x f x t to a decidability problem for first order formulaeover the reals furthermore the paper introduces a class of hybrid automata for which the reachability problem can be decided and shows that the problem of deciding whether a hybrid automaton belongs to this class can be again decided using first order formulaeover the reals despite the fact that the bisimulation quotient for this class of hybrid automata can be infinite we show that our techniques permit effective model checking for a nontrivial fragment of ctl

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

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