new dissipated energy and partial regularity for thin film equations abstract we study the thin film equation h t h n h x x x x and prove that the energy h p h x 2 d x dissipates for suitable weak solutions when n p lies in the whole domain e defined by laugesen which extends partially the corresponding results of laugesen chugunova and taranets and then improve some previous results of partial regularity by chou and the first author

global bifurcation and nodal solutions for fourth order problems with sign changing weight in this paper we shall establish unilateral global bifurcation result for a class of fourth order eigenvalue problems with sign changing weight under some natural hypotheses on perturbation function we show that m k n 0 is a bifurcation point of the above problems and there are two distinct unbounded continua c k n and c k n consisting of the bifurcation branch c k n from m k n 0 where m k n is the kth positive or negative eigenvalue of the linear problem corresponding to the above problems email protected as the applications of the above result we study the existence of nodal solutions for a class of fourth order eigenvalue problems with sign changing weight moreover we also establish the sturm type comparison theorem for fourth order problems with sign changing weight

higher order nonlinear degenerate parabolic equations this paper is concerned with nonlinear degenerate parabolic equations of the form ut 1 m 1 d f u d2m 1u 0 with f u u n n 1 near u 0 and d x under appropriate boundary conditions it is shown that there exists a weak solution u some of the main results of the paper are that u 0 if u0 0 and that the support of u t when u0 0 increases with t for the last property we require that n 2 and m 1

phase plane analysis and traveling wave solution of third order nonlinear singular problems arising in thin film evolution a nonlinear third order singular ordinary differential equation is a model for a viscous fluid which drains over a wet surface as a thin film down a vertical wall here we discuss the physical processes of a thin liquid film evolution the classical thin film equation is being derived and the series solution is obtained by employing the adomian decomposition method adm in this paper a new way for the phase planes analysis and traveling wave solution by adm is presented the proposed algorithm is implemented using mathematical software mathematica 6 0

a generalized thin film equation in multidimensional space abstract this paper deals with the existence and nonnegativity of the weak periodic domain solution for a degenerate fourth order parabolic equation modeling the evolution of thin films this study in the multidimensional domain follows the recent results related to the resolution in one dimensional space j e rakotoson j m rakotoson c verbeke generalized lubrification models blow up and global existence racsam 99 2 2005 235 241 c verbeke quelques modeles d equations d evolution de surfaces explosion en temps fini et diverses proprietes qualitatives these de doctorat de l universite de poitiers 15 decembre 2005 j e rakotoson j m rakotoson c verbeke higher order equations related to thin film blow up and global existence the influence of the initial data submitted for publication

on the determination of the steady film profile for a non newtonian thin droplet abstract a shooting method is used to determine a solution to a third order ode modeling the steady profile of a non newtonian thin droplet we compare a direct approach to an iterative approach using a secant method we obtain a nonlinear relationship between the contact angle and the position of the contact line r from this nonlinear relationship we use curve fitting to obtain an empirical law of the form tan r f k where k is the power law coefficient and f is a nonlinear function of k

energy levels of steady states for thin film type equations abstract we study the phase space of the evolution equationht f h hxxx x g h hx x by means of a dissipated energy here h x t 0 and at h 0 the coefficient functions f 0 and g can either degenerate to 0 or blow up to or tend to a nonzero constant we first show that all positive periodic steady states are saddles in the energy landscape with respect to zero mean perturbations if g f 0 or if the perturbations are allowed to have period longer than that of the steady state for power law coefficients f y yn and g y b ym for some b 0 we analytically determine the relative energy levels of distinct steady states for example with m n 1 2 and for suitable choices of the period and mean value we find three fundamentally different steady states the first is a constant steady state that is stable and is a local minimum of the energy the second is a positive periodic steady state that is linearly unstable and has higher energy than the constant steady state it is a saddle point for the energy the third is a periodic collection of droplet compactly supported steady states having lower energy than either the positive steady state or the constant one since the energy must decrease along every orbit these results significantly constrain the dynamics of the evolution equation

maximum entropy formalism for a dense gas abstract the maximum entropy formalism is developed to implement a complete description of a dense gas in non equilibrium situations two variational schemes are discussed in this context the kinetic equations describing the system are consistently closed with the n particle distribution function obtained via this formalism the equilibrium state obtained on this basis is consistent with the description of the theory of liquids in the equilibrium state

traveling wave solutions to thin film equations thin films can be effectively described by the lubrication approximation in which the equation of motion is mathit h mathit t mathit h mathit n mathit h mathit x mathit x mathit x mathit x 0 here h is a necessarily positive quantity which represents the height or thickness of the film different values of n especially 1 2 and 3 correspond to different physical situations this equation permits solutions in the form of traveling disturbances with a fixed form if u is the propagation velocity the resulting equation for the disturbance is mathit uh mathit x mathit h mathit n mathit h mathit x mathit x mathit x mathit x here quantitative and qualitative solutions to the equation are presented the study has been limited to the intervals in x where the solutions are positive it is found that transitions between different qualitative behaviors occur at n 3 2 3 2 and 1 2 for example if u is not zero solitonlike solutions defined on a finite interval are only possible for n3 more specific results can be obtained in the case in which the velocity is zero solitons occur for n2 for n 1 the region 3 2n is characterized by the presence of advancing front solutions with support on ensuremath infty t for ng1 2 single minimum solutions diverging at ifmmode pm else textpm fi ensuremath infty are possible the generic solution present for all positive values of n is a receding front which diverges at finite x for n0

existence and positivity of solutions of a fourth order nonlinear pde describing interface fluctuations we study the partial differential equation which arose originally as a scaling limit in the study of interface fluctuations in a certain spin system in that application x lies in r but here we study primarily the periodic case x e s we establish existence uniqueness and regularity of solutions locally in time for positive initial data in h s and prove the existence of several families of lyapunov functions for the evolution from the latter we establish a sharp connection between existence globally in time and positivity preservation if o t is a maximal half open interval of existence for a positive solution of the equation with t 1 16r2 then t 03 and lim m w r exists and is constant we discuss also some explicit solutions and propose a generalization to higher dimensions 0 1994 john wiley

the lubrication approximation for thin viscous films regularity and long time behavior of weak solutions we consider the fourth order degenerate diffusion equation in one space dimension this equation derived from a lubrication approximation models the surface tension dominated motion of thin viscous films and spreading droplets 15 the equation with f h h also models a thin neck of fluid in the hele shaw cell 10 11 23 in such problems h x t is the local thickness of the the film or neck this paper considers the properties of weak solutions that are more relevant to the droplet problem than to hele shaw for simplicity we consider periodic boundary conditions with the interpretation of modeling a periodic array of droplets we consider two problems the first has initial data h0 0 and f h h n 0 n 3 we show that there exists a weak nonnegative solution for all time also we show that this solution becomes a strong positive solution after some finite time t and asymptotically approaches its means as t the weak solution is in the classical sense of distributions for 3 8 n 3 and in a weaker sense introduced in 1 for the remaining 0 n 3 8 furthermore the solutions have high enough regularity to just include the unique source type solutions 2 with zero slope at the edge of the support they do not include any of the less regular solutions with positive slope at the edge of the support second we consider strictly positive initial data h0 m 0 and f h h n 0 n for this problem we show that even if a finite time singularity of the form h 0 does occur there exists a weak nonnegative solution for all time t this weak solution becomes strong and positive again after some critical time t as in the first problem we show that the solution approaches its mean as t the main technical idea is to introduce new classes of dissipative entropies to prove existence and higher regularity we show that these entropies are related to norms of the difference between the solution and its mean to prove the relaxation result 1996 john wiley sons inc

the scaled boundary finite element method in structural dynamics the scaled boundary finite element method is extended to solve problems of structural dynamics the dynamic stiffness matrix of a bounded finite domain is obtained as a continued fraction solution for the scaled boundary finite element equation the inertial effect at high frequencies is modeled by high order terms of the continued fraction without introducing an internal mesh by using this solution and introducing auxiliary variables the equation of motion of the bounded domain is expressed in high order static stiffness and mass matrices standard procedures in structural dynamics can be applied to perform modal analyses and transient response analyses directly in the time domain numerical examples for modal and direct time domain analyses are presented rapid convergence is observed as the order of continued fraction increases a guideline for selecting the order of continued fraction is proposed and validated high computational efficiency is demonstrated for problems with stress singularity

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

local variational description for a dense gas abstract we present the study of a dense gas at a hydrodynamical level of description we start from the entropy functional obtained at a mesoscopic regime l romero salazae et al physica a 237 1997 150 mef1 and we apply the maximum entropy formalism where the restrictions are now the hydrodynamic variables and the density fluctuations an entropy functional is obtained both in the local and the equilibrium inhomogeneous situation

existence and uniqueness of weak solutions for a generalized thin film equation abstract in this paper we establish the existence and uniqueness of weak solutions for the initial boundary value problem of a generalized thin film equation in addition we present some interesting results on the regularity of weak solutions

stability and regularity of weak solutions for a generalized thin film equation in this paper we establish a stability result and an error estimate of weak solutions for the initial boundary value problem of a generalized thin film equation and also obtain some higher regularity results for weak solutions

analysis of adomian decomposition applied to a third order ordinary differential equation from thin film flow abstract the adomian decomposition method is applied to the third order ordinary differential equation y y k obtained by considering a travelling wave solution admitted by a generalized thin film equation or for thin film flow down a vertical wall the adomian decomposition method leads to a power series approximation to the solution of y y k we show that the domain of convergence of the adomian decomposition solution is dependent on k we truncate the adomian decomposition solution at a suitable order to ensure that the truncated solution satisfies the contact line condition y 0 at some point x x we then determine the contact angle where tan d y d x at x x and plot the variation of contact angle with increasing k

a point interpolation method for two dimensional solids a point interpolation method pim is presented for stress analysis for two dimensional solids in the pim the problem domain is represented by properly scattered points a technique is proposed to construct polynomial interpolants with delta function property based only on a group of arbitrarily distributed points the pim equations are then derived using variational principles in the pim the essential boundary conditions can be implemented with ease as in the conventional finite element methods the present pim has been coded in fortran the validity and efficiency of the present pim formulation are demonstrated through example problems it is found that the present pim is very easy to implement and very flexible for obtained displacements and stresses of desired accuracy in solids as the elements are not used for meshing the problem domain the present pim opens new avenues to develop adaptive analysis codes for stress analysis in solids and structures copyright 2001 john wiley sons ltd

a thin film equation with a singular diffusion in this paper a thin film equation with a singular diffusion term u t u n u a u u 0 is studied with periodic boundary conditions in multidimensional space it has a lot of applications in fluids theory such as draining of foams and the movement of contact lenses in order to overcome the difficulty of singular diffusion term the approximation problems need to be constructed and the entropy functional method is applied for the limit process moreover a compactness argument shows the existence of nonnegative weak solutions

the closure hypothesis in the maximum entropy formalism abstract closure hypotheses in kinetic theory constitute a point of comparison between different approaches to describe the behavior of a macroscopic system the maximum entropy formalism mef i gives us a way to close the bbgky hierarchy equations and its discussion is the main objective of this paper

symmetric singularity formation in lubrication type equations for interface motion fourth order degenerate diffusion equations arise in a lubrication approximation of a thin film or neck driven by surface tension numerical studies of the lubrication equation le h t h n h xxx x 0 with various boundary conditions indicate that singularity formation in which h x t t to 0 occurs for small enough n with anomalous or second type scaling inconsistent with usual dimensional analysis this paper considers locally symmetric or even singularities in the le and in the modified lubrication equation mle h t h n h xxxx 0 both equations have the property that entropy bounds forbid finite time singularities when n is sufficiently large power series expansions for local symmetric similarity solutions are proposed for equation le with n 1 and mle for all n in mathbb r in the latter case special boundary conditions that force singularity formation are required to produce singularities when n is large matching conditions at higher order terms in the

nonnegative solutions of a fourth order nonlinear degenerate parabolic equation elena beretta michiel bertsch roberta dal passo communicated by m guatin o introduction in this paper we study nonnegative solutions of the fourth order equation 0 1 ut u uxxx x o where n is a real positive constant equation 0 1 is a nonlinear parabolic partial differential equation which is degenerate at points at which u vanishes it arises in several applications see the references in 2 4 5 6 in which u x t stands for a nonnegative quantity for example in the study of the dynamics of thin liquid films u denotes the thickness of the film or another relevant nonnegative quantity the case n 1 models the flow in a hele shaw cell while n 3 corresponds to viscous flow on a solid surface without slip driven by surface tension see also 6 7 8 for a related equation it is well known that fourth order equations do not satisfy a maximum principle in particular classical solutions of the linear equation ut u 0 may become negative in finite time bernis 8 friedman 2 have observed that if n is positive equation 0 1 has generalized solutions which remain nonnegative for all times in addition they show that the qualitative positivity of these solutions strongly depends on the value of n in more recent papers by kadanoff et al a combination of numerical calculations and matched asymptotics 4 and the analysis of travelling wave solutions 5 leads to results which suggest a very rich structure of the solutions of 0 1 it is the purpose of the present paper to clarify several aspects of the nonlinear equation 0 1 and its dependence on the parameter n our results lead to several conjectures and open problems as was pointed out in 4 for sufficiently small values of n solutions of 0 1 which are initially strictly positive may vanish at some point x 0 after a finite time to but to the best of our knowledge there is no rigorous proof of this phenomenon in the literature on the other hand it is of particular interest for applications for example if u denotes the thickness of a liquid film it implies that at time to the film breaks at xo

the linear limit of thin film flows as an obstacle type free boundary problem we study the limit as n to 0 of the nonnegative self similar source type solutions of the thin film equationut unuxxx x 0 we obtain a unique limiting function u which is a solution of an obstacle type free boundary problem with the constraint u geq 0 associated with the linear equationut uxxxx 0 the function u is c1 for t 0 and has finite speed of propagation the positivity set um 0 being bounded by two contact lines x pm a t 1 4 a constant the function u has a dirac mass as initial condition and satisfies the linear equation in the positivity set but not across the free boundaries or contact lines also known as moving boundaries we give an integral representation of u in the positivity set we set up a precise definition of the general non self similar obstacle type free boundary problem which is different from a standard parabolic variational inequality and compare it with the cauchy problem we also consider source type solutions for negative values of n which are so

spreading of a thin film with suction or blowing including surface tension effects in this paper we derive a fourth order nonlinear partial differential equation modelling the effects of suction blowing on the evolution of the free surface of a thin viscous film on a porous base gravity and surface tension effects are included the suction blowing is modelled as an arbitrary function v n t r the lie group classification method is used to determine a first order quasi linear partial differential equation satisfied by v n t r this equation is coupled with the model equation and solved numerically new results are obtained which show how the free surface and the apparent contact angle vary with suction blowing

numerical investigation of the generalized lubrication equation abstract explicit implicit explicit and crank nicolson implicit explicit numerical schemes for solving the generalized lubrication equation are derived we prove that the implicit explicit and crank nicolson implicit explicit numerical schemes are unconditionally stable numerical solutions obtained from both schemes are compared initial curves with both zero and finite contact angles are considered

blow up and extinction for a thin film equation with initial boundary value conditions abstract in this paper we consider an initial boundary value problem for a thin film equation with the nonlocal source u q u q d x in a bounded domain due to the comparison principle being invalid we use the potential well method to give a threshold result of global existence and non existence for the sign changing weak solutions further we obtain the conditions under which the global solutions extinct in finite time and study the asymptotic behavior of the non extinct solutions

thin films with high surface tension this paper is a review of work on thin fluid films where surface tension is a driving mechanism its aim is to highlight the substantial amount of literature dealing with relevant physical models and also analytic work on the resultant equations in general the introduction of surface tension into standard lubrication theory leads to a fourth order nonlinear parabolic equation pad h t pad x left c frac h 3 3 frac partial 3 h partial x 3 f h h x h xx right 0 label abeq1 where h h x t is the fluid film height for steady situations this equation may be integrated once and a third order ordinary differential equation is obtained appropriate forms of this equation have been used to model fluid flows in physical situations such as coating draining of foams and the movement of contact lenses in the introduction a form of the above equation is derived for flow driven by surface tension surface tension gradients gravity and long range molecular forces modifications to the equation due to slip the effect of two free surfaces two phase fluids and higher dimensional forms are also discussed the second section of this paper describes physical situations where surface tension driven lubrication models apply and the governing equations are given the third section reviews analytical work on the model equations as well as the generalized lubrication equation pad h t pad x left h n h xxx right 0 in particular the discussion focusses on asymptotic results travelling waves stability and similarity solutions numerical work is also discussed while for analytical results the reader is directed to existing literature

an adaptive moving mesh method for thin film flow equations with surface tension we present an adaptive moving mesh method for the numerical solution of thin liquid film spreading flows with surface tension we follow the r adaptive moving mesh technique which utilises a mesh density function and moving mesh partial differential equations mmpdes to adapt and move the mesh coupled to the pde s describing the thin film flow problem numerical experiments are performed on two one dimensional thin film flow equations to test the accuracy and efficiency of the method this technique accurately resolves the multiple one dimensional structures observed in these test problems moreover it reduces the computational effort in comparison to the numerical solution using the finite difference scheme on a fixed uniform mesh we are the first to implement r adaptive schemes to higher order parabolic pdes this method accurately resolves the solution and reduces the computational effort we introduce a new mesh density function that resolves multiple solution structures this mesh density function is also adapted to include multiple solution components

entropy balance equation for a dense gas abstract this work is concerned with the description of the time evolution for the entropy functional in a dense fluid out of equilibrium we consider the fluid as a collection of n particles interacting via the superposition of pairwise additive short range intermolecular potentials such an interaction is assumed to be central and the particles do not have internal structure to reach this goal we use the maximum entropy formalism in which the gibbs entropy is maximized under the restrictions imposed by the normalization of the n particle distribution function and the knowledge of the one and two particle distribution functions this procedure leads to an expression for the entropy functional in terms of such distribution functions on the other hand the evolution equations allow for the construction of a balance equation for the entropy functional in the system an analysis of the entropy flux and entropy source is given for a particular expression in the lagrange multipliers coming from the maximization procedure such an analysis shows the connection of the entropy production with the conversion rate of kinetic and potential energy between particles interesting conclusions about the temperature are reached in this case

group theoretical framework for potential symmetries of evolution equations we develop algebraic approach to the problem of classification of potential symmetries of nonlinear evolution equations it is essentially based on the recently discovered fact r zhdanov j math phys 50 053522 2009 that any such symmetry is mapped into a contact symmetry the approach enables using the classical results on classification of contact symmetries of nonlinear evolution equations by sokolov and magadeev to classify evolution equations admitting potential symmetries we construct several examples of new nonlinear fourth order evolution equations admitting potential symmetries since the symmetries obtained depend on nonlocal variables they cannot be derived by the infinitesimal lie approach

a family of fourth order q logarithmic equations we prove the existence of global in time weak nonnegative solutions to a family of nonlinear fourth order evolution equations parametrized by a real parameter q 0 1 which includes the well known thin film q 1 2 and the derrida lebowitz speer spohn dlss equation q 1 subject to periodic boundary conditions in one spatial dimension in contrast to the gradient flow approach in 25 our method relies on dissipation property of the corresponding entropy functionals tsallis entropies resulting in required a priori estimates and extends the existence result from 25 to a wider range of the family members namely to 0 q 1 2 generalized beckner type functional inequalities yield an exponential decay rate of relative entropies which in further implies the exponential stability in the l1 norm of the constant steady state finally we provide illustrative numerical examples supporting the analytical results

mathematical analysis to a nonlinear fourth order partial differential equation abstract the paper first study the steady state thin film type equation u n u q 2 u u m u f x u with navier boundary conditions in multidimensional space by the truncation method a fixed point argument and some energy estimates the existence and asymptotic limit 0 for the positive weak solutions are given second the parabolic equation u t u n u x x x q 2 u x x x x u m u x x 0 with a navier boundary in one dimensional space is researched the existence is obtained by applying a semi discrete method for the time variable and solving the corresponding elliptic problem the uniqueness is shown for q 2 depending on an energy estimate in addition the iteration relation of the semi discrete problem gives an exponential decay result for the time t the thin film equation which is usually used to describe the motion of a very thin layer of viscous in compressible fluids along an inclined plane is a class of nonlinear fourth order parabolic equations and the maximum principle does not hold directly for applying the classic theory of partial differential equation the paper transforms the fourth order problem into a second order elliptic elliptic system or a second order parabolic elliptic system

existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order abstract this paper is devoted to studying the existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order u t u p 2 u f u in r n with boundary condition u u 0 and initial data u 0 the substantial difficulty is that the general maximum principle does not hold for it the solutions are obtained for both the steady state case and the developing case by the fixed point theorem and the semi discretization method unlike the general procedures used in the previous papers on the subject we introduce two families of approximate solutions with determining the uniform bounds of derivatives with respect to the time and space variables respectively by a compactness argument with necessary estimates we show that the two approximation sequences converge to the same limit i e the solution to be determined in addition the decays of solutions towards the constant steady states are established via the entropy method finally it is interesting to observe that the solutions just tend to the initial data u 0 as p

a matrix function solution for the scaled boundary finite element equation in statics the scaled boundary finite element method is a fundamental solution less boundary element method based on finite elements it leads to semi analytical solutions for displacement and stress fields which permits problems with singularities or in infinite domains to be handled conveniently and accurately however the present eigenvalue method for solving the scaled boundary finite element equation requires additional treatments for multiple eigenvalues with parallel eigenvectors which results from logarithmic terms in the solutions a matrix function solution for the scaled boundary finite element equation in statics is presented in this paper it is numerically stable for multiple or near multiple eigenvalues as it is based on real schur decomposition power functions logarithmic functions and their transitions as occurring in fracture mechanics composites and two dimensional unbounded domains are represented semi analytically no a priori knowledge on the types and orders of singularity are required when simulating stress singularities

steady heat conduction analyses using an interpolating element free galerkin scaled boundary method abstract through use of the improved interpolating moving least squares iimls shape functions in the circumferential direction of the scaled boundary method based on the galerkin approach an interpolating element free galerkin scaled boundary method iefg sbm is developed in this paper for analyzing steady heat conduction problems which weakens the governing differential equations in the circumferential direction and seeks analytical solutions in the radial direction the iimls method exhibits some advantages over the moving least squares approximation and the interpolating moving least squares method because its shape functions possess the delta function property and the involved weight function is nonsingular in the iefg sbm only a nodal data structure on the boundary is required and the primary unknown quantities are real solutions of nodal variables higher accuracy and faster convergence are obtained due to the increased smoothness and continuity of shape functions based on the iefg sbm the steady heat conduction problems with thermal singularities and unbounded domains can be ideally modeled some numerical examples are presented to validate the availability and accuracy of the present method for steady heat conduction analysis

the linear limit of the dipole problem for the thin film equation we investigate self similar solutions of the dipole problem for the one dimensional thin film equation on the half line x ge 0 we study compactly supported solutions of the linear moving boundary problem and show how they relate to solutions of the nonlinear problem the similarity solutions are generally of the second kind given by the solution of a nonlinear eigenvalue problem although there are some notable cases where first kind solutions also arise we examine the conserved quantities connected to these first kind solutions difficulties associated with the lack of a maximum principle and the non self adjointness of the fundamental linear problem are also considered seeking similarity solutions that include sign changes yields a surprisingly rich set of coexisting stable solutions for the intermediate asymptotics of this problem our results include analysis of limiting cases and comparisons with numerical computations

long time asymptotics for strong solutions of the thin film equation in this paper we investigate the large time behavior of strong solutions to the one dimensional fourth order degenerate parabolic equation u t u u xxx x modeling the evolution of the interface of a spreading droplet for nonnegative initial values u 0 x h 1 both compactly supported or of finite second moment we prove explicit and universal algebraic decay in the l 1 norm of the strong solution u x t towards the unique among source type solutions strong source type solution of the equation with the same mass the method we use is based on the study of the time decay of the entropy introduced in 13 for the porous medium equation and uses analogies between the thin film equation and the porous medium equation

hybrid methods for direct integration of special third order ordinary differential equations abstract in this paper we present a new class of direct numerical integrators of hybrid type for special third order ordinary differential equations odes y f x y namely hybrid methods for solving third order odes directly hmtd using the theory of b series order of convergence of the hmtd methods is investigated the main result of the paper is a theorem that generates algebraic order conditions of the methods that are analogous to those of two step hybrid method a three stage explicit hmtd is constructed results from numerical experiment suggest the superiority of the new method over several existing methods considered in the paper

analysis and application of the interpolating element free galerkin iefg method to simulate the prevention of groundwater contamination with application in fluid flow abstract we develop a meshless numerical procedure to simulate the groundwater equation gwe the used technique is based on the interpolating element free galerkin iefg method the interpolating moving least squares imls approximation produces a set of functions such that they are well known as shape functions the iefg technique employs the shape functions of imls approximation the shape functions of imls approximation vanish on the boundary and also they satisfy the property of the kronecker delta function thus dirichlet boundary conditions can be exactly imposed in this paper we check the unconditional stability and convergence of the proposed numerical scheme based on the energy method the numerical results confirm the theoretical analysis

entropy production bound in a dense gas abstract this paper concerns the behavior of the entropy production in a dense gas a bound for this quantity is found in terms of the energy conversion rate between particles and fisher s information integrals describing the system

embedded explicit runge kutta type methods for directly solving special third order differential equations y f x y abstract in this paper three pairs of embedded runge kutta type methods for directly solving special third order ordinary differential equations odes of the form y f x y denoted as rkd methods are presented the first is the rkd4 3 pair which is third order embedded in fourth order method has the property first same as last fsal whereby the last row of the coefficient matrix is equal to the vector output the second method is the rkd5 4 pair followed by the rkd6 5 pair the methods are derived with the strategies such that the higher order methods are very accurate and the lower order methods will give the best error estimates variables stepsize codes are developed based on the methods and used to solve a set of special third order problems numerical results are compared with the existing embedded runge kutta pairs which require the problems to be reduced into a system of first order odes numerical results have clearly shown the advantage and the efficiency of the new rkd pairs

interpolating element free galerkin method for the regularized long wave equation and its error analysis abstract combining the simplified interpolating moving least squares imls method with the galerkin weak form an interpolating element free galerkin iefg method is presented for the numerical study of the regularized long wave rlw equation the trial function in the imls method has one less unknown coefficient than in the traditional mls approximation and the shape functions satisfy the interpolating property at nodes then the iefg method can apply the essential boundary conditions naturally and directly and has high computational accuracy and then this paper studies the error estimates of the iefg method for rlw equation the error estimate of the semi discrete iefg method for the rlw equation on the influence domain radius is first studied and then the convergence rate of the full discrete scheme of the method on the influence domain radius and time step is studied the theoretical results show that the iefg method for rlw equation has good convergence rate for the purpose of demonstration three selected numerical examples are given to demonstrate the method and the mathematical theory of this paper

a numerical and asymptotic study of some third order ordinary differential equations relevant to draining and coating flows some draining or coating fluid flow problems in which surface tension forces are important can be described by third order ordinary differential equations accurate computations are provided here