positive solutions for a class of multiparameter ordinary elliptic systems abstract using a fixed point theorem due to m a krasnosel skii the upper lower solutions method and fixed point index theory we study existence non existence and multiplicity of positive solutions for a class of systems of second order ordinary differential equations we give a new notion of superlinearity for nonlinearities observe that the nonlinearities may even be in some sense singular as an application we show multiplicity results for a class of semilinear elliptic systems in both bounded annular domains and exterior domains in addition we apply our results to fourth order boundary value problems

stochastic homogenization of certain nonconvex hamilton jacobi equations abstract in this paper we prove the stochastic homogenization of certain nonconvex hamilton jacobi equations the nonconvex hamiltonians which are generally uneven and inseparable are generated by a sequence of quasiconvex hamiltonians and a sequence of quasiconcave hamiltonians through the min max formula we provide a monotonicity assumption on the contact values between those stably paired hamiltonians so as to guarantee the stochastic homogenization

emden fowler equations involving critical exponents on considere le probleme y t k y 2 y 2k 3 0 t 0 pour t 2 0 0 soit t inf t 0 y 0 sur t alors t 0 0 on etudie comment la position t du premier zero de y t varie avec

dense output for extrapolation methods this paper is concerned with dense output formulas for extrapolation methods for ordinary differential equations in particular the extrapolated explicit euler method the gbs method for non stiff equations and the extrapolated linearly implicit euler method for stiff and differential algebraic equations are considered existence and uniqueness questions for dense output formulas are discussed and an algorithmic description for their construction is given several numerical experiments illustrate the theoretical results

natural runge kutta and projection methods recently the author defined the class of natural runge kutta methods and observed that it includes all the collocation methods the present paper is devoted to a complete characterization of this class and it is shown that it coincides with the class of the projection methods in some polynomial spaces

uniqueness and nonuniqueness for positive radial solutions of δu f u r 0 on considere le probleme de l unicite pour les solutions positives du probleme de dirichlet non lineaire u f u x 0 dans u 0 est une boule ou un anneau de r n et f 0 est superlineaire en u et satisfaisait f 0 x 0

local superlinearity for elliptic systems involving parameters abstract we study the existence non existence and multiplicity of positive solutions for a class of systems of second order ordinary differential equations using the fixed point theorem of cone expansion compression type the upper lower solutions method and degree arguments we apply our abstract results to study semilinear elliptic systems in bounded annular domains with non homogeneous boundary conditions here the nonlinearities satisfy local superlinear assumptions

homogenization of hamilton jacobi bellman equations with respect to time space shifts in a stationary ergodic medium we consider a family fu t x g 0 of solutions to the equation u t c u 2ch t x ru d0 with the terminal datau t x du x assuming that the dependence of the hamiltonian h t x p on time and space is realized through shifts in a stationary ergodic random medium and that h is convex inp and satisfies certain growth and regularity conditions we show the almost sure locally uniform convergence in time and space of u t x as 0 to the solution u t x of a deterministic averaged equation u t c h ru d 0 u t x d u x the effective hamiltonian h is given by a variational formula

an eulerian particle level set method for compressible deforming solids with arbitrary eos in this study an eulerian finite volume method is developed for the numerical simulation of elastic plastic response of compressible solid materials with arbitrary equation of state eos under impact loading the governing equations of mass momentum and energy along with evolution equations for deviatoric stresses are solved in eulerian conservation law form since the position of material boundaries is determined implicitly by eulerian schemes the solution procedure is split into two separate subproblems which are solved sequentially at each time step first the conserved variables are evolved in time with appropriate boundary conditions at the material interfaces in the present work a fourth order central weighted essentially non oscillatory shock capturing method that was developed for gas dynamics has been extended to high strain rate solids problems in this method fluxes are determined on a staggered grid at places where solution is smooth as a result the method does not rely on the solution of riemann problems but enjoys the flexibility of using any type of eos boundary conditions at material interfaces are also treated by a special ghost cell approach then in the second subproblem the position of material interfaces is advanced to the new time using a particle level set method a fifth order godunov type central scheme is used to solve the hamilton jacobi equation of level sets in two space dimensions the capabilities of the proposed method are evaluated at the end by comparing numerical results with the experimental results and the reported benchmark solutions for the taylor rod impact spherical groove jetting and void collapse problems copyright 2009 john wiley sons ltd

derivation of continuous explicit two step runge kutta methods of order three we describe a construction of continuous extensions to a new representation of two step runge kutta methods for ordinary differential equations this representation makes possible the accurate and reliable estimation of local discretization error facilitates the efficient implementation of these methods in variable stepsize environment and adapts readily to the numerical solution of a class of delay differential equations a number of numerical tests carried out on the obtained methods of order 3 with quadratic interpolants show their efficiency and robust performance which allow them to compete with the state of the art dde23 code from matlab

iterated collocation methods for volterra integral equations with delay arguments in this paper we give a complete analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation for volterra integral equations with constant delay this analysis includes continuous collocation based volterra runge kutta methods as well as iterated collocation methods and their discretizations

nonlinear elliptic problems with critical exponent in shrinking annuli soit dans r n n 2 l anneau a x p 0 dans a u 0 sur a et u u n 2 n 2 0 u 0 dans a u n 0 quand x u 0 quand x r

stability analysis of one step methods for neutral delay differential equations in this paper stability properties of one step methods for neutral functional differential equations are investigate stability regions are characterized for runge kutta methods with respect to the linear test equation begin gathered y left t right ay left t right by left t tau right cy left t tau right t geqq 0 hfill y left t right g left t right tau leqq t leqq 0 hfill end gathered 0 where a b andc are complex parameters in particular it is shown that everya stable collocation method for ordinary differential equations can be extended to a method for neutrals delay differential equations with analogous stability properties the so called np stable method we also investigate how the approximation to the derivative of the solution affects stability properties of numerical methods for neutral equations

numerical solution of threshold problems in epidemics and population dynamics a new algorithm is proposed for the numerical solution of threshold problems in epidemics and population dynamics these problems are modeled by the delay differential equations where the delay function is unknown and has to be determined from the threshold conditions the new algorithm is based on embedded pair of continuous runge kutta method of order p 4 and discrete runge kutta method of order q 3 which is used for the estimation of local discretization errors combined with the bisection method for the resolution of the threshold condition error bounds are derived for the algorithm based on continuous one step methods for the delay differential equations and arbitrary iteration process for the threshold conditions numerical examples are presented which illustrate the effectiveness of this algorithm

natural continuous extensions of runge kutta methods the present paper develops a theory of natural continuous extensions nces for the discrete approximate solution of an ode given by a runge kutta process these nces are defined in such a way that the continuous solutions furnished by the one step collocation methods are included

existence of positive nonradial solutions for nonlinear elliptic equations in annular domains we study the existence of positive nonradial solutions of equation a f u 0 in sia u 0 on d la where s x 6 w a 2 and is positive and superlinear at both 0 and oo we use a bifurcation method to show that there is a nonradial bifurcation with mode k at ak 0 1 for any positive integer k if is subcritical and for large k if is supercritical when is subcritical then a nehari type variational method can be used to prove that there exists a e 0 1 such that for any a e a 1 the equation has a nonradial solution on 0 a

order barriers and characterizations for continuous mono implicit runge kutta schemes the mono implicit runge kutta mirk schemes a subset of the family of implicit runge kutta irk schemes were originally proposed for the numerical solution of initial value odes more than fifteen years ago dur ing the last decade a considerable amount of attention has been given to the use of these schemes in the numerical solution of boundary value ode prob lems where their efficient implementation suggests that they may provide a worthwhile alternative to the widely used collocation schemes recent work in this area has seen the development of some software packages for bound ary value odes based on these schemes unfortunately these schemes lead to algorithms which provide only a discrete solution approximation at a set of mesh points over the problem interval while the collocation schemes provide a natural continuous solution approximation the availability of a continuous solution is important not only to the user of the software but also within the code itself for example in estimation of errors defect control mesh selection and the provision of initial solution estimates for new meshes an approach for the construction of a continuous solution approximation based on the mirk schemes is suggested by recent work in the area of continuous extensions for explicit runge kutta schemes for initial value odes in this paper we describe our work in the investigation of continuous versions of the mirk schemes i we give some lower bounds relating the stage order to the minimal number of stages for general continuous irk schemes ii we establish lower bounds on the number of stages needed to derive continuous mirk schemes of orders 1 through 6 and iii we provide characterizations of these schemes having a minimal number of stages for each of these orders

numerical solution of functional differential integral and integro differential equations this paper describes a numerical method based on lagrange interpolation and chebyshev interpolation to treat functional integral equations of volterra type and fredholm type also the method can be extended to functional differential and integro differential equations various numerical examples are treated

continuous parallel block methods and their applications continuous numerical methods have many applications in the numerical solution of discontinuous ordinary differential equations delay differential equations delay differential algebraic equations neutral delay differential equations integro differential equations etc this paper deals with a continuous extension for the discrete approximate solution of ordinary differential equations generated by a class of parallel block methods existence and uniqueness for the continuous extension are discussed convergence and absolute stability of the continuous parallel block methods for ordinary differential equations are studied as applications we adopt the continuous parallel block methods to solve delay differential equations neutral delay differential equations and delay differential algebraic equations and obtain sufficient and necessary conditions for the continuous parallel block methods to be asymptotically stable several numerical experiments are given to illustrate the performance of the continuous parallel block methods

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

order barriers for continuous explicit runge kutta methods in this paper we deal with continuous numerical methods for solving initial value problems for ordinary differential equations the need for which occurs frequently in applications whereas most of the commonly used multistep methods provide continuous extensions by means of an interpolant which is available without making extra function evaluations this is not always the case for one step methods we consider the class of explicit runge kutta methods and provide theorems used to obtain lower bounds for the number of stages required to construct methods of a given unifonn order p these bounds are similar to the butcher barriers known for the discrete case and are derived up to order p 5 as far as we know the examples we present of 8 stage continuous runge kutta methods of uniform order 5 are the first of their kind

stochastic homogenization of hamilton jacobi bellman equations we study the homogenization of some hamilton jacobi bellman equations with a vanishing second order term in a stationary ergodic random medium under the hyperbolic scaling of time and space imposing certain convexity growth and regularity assumptions on the hamiltonian we show the locally uniform convergence of solutions of such equations to the solution of a deterministic effective first order hamilton jacobi equation the effective hamiltonian is obtained from the original stochastic hamiltonian by a minimax formula our homogenization results have a large deviations interpretation for a diffusion in a random environment c 2006 wiley periodicals inc

error estimates and convergence rates for the stochastic homogenization of hamilton jacobi equations we present exponential error estimates and demonstrate an algebraic convergence rate for the homogenization of level set convex hamilton jacobi equations in i i d random environments the first quantitative homogenization results for these equations in the stochastic setting by taking advantage of a connection between the metric approach to homogenization and the theory of first passage percolation we obtain estimates on the fluctuations of the solutions to the approximate cell problem in the ballistic regime away from flat spot of the effective hamiltonian in the sub ballistic regime on the flat spot we show that the fluctuations are governed by an entirely different mechanism and the homogenization may proceed without further assumptions at an arbitrarily slow rate we identify a necessary and sufficient condition on the law of the hamiltonian for an algebraic rate of convergence to hold in the sub ballistic regime and show under this hypothesis that the two rates may be merged to yield comprehensive error estimates and an algebraic rate of convergence for homogenization our methods are novel and quite different from the techniques employed in the periodic setting although we benefit from previous works in both first passage percolation and homogenization the link between the rate of homogenization and the flat spot of the effective hamiltonian which is related to the nonexistence of correctors is a purely random phenomenon observed here for the first time

some inverse problems in periodic homogenization of hamilton jacobi equations we look at the effective hamiltonian overline h associated with the hamiltonian h p x h p v x in the periodic homogenization theory our central goal is to understand the relation between v and overline h we formulate some inverse problems concerning this relation such types of inverse problems are in general very challenging in this paper we discuss several special cases in both convex and nonconvex settings

stochastic homogenization of nonconvex hamilton jacobi equations in one space dimension abstract we prove stochastic homogenization for a general class of coercive nonconvex hamilton jacobi equations in one space dimension some properties of the effective hamiltonian arising in the nonconvex case are also discussed

natural continuous extensions of runge kutta methods for volterra integral equations of the second kind and their applications we consider a very general class of runge kutta methods for the numerical solution of volterra integral equations of the second kind which includes as special cases all the more important methods which have been considered in the literature the main purpose of this paper is to define and prove the existence of the natural continuous extensions nce s of runge kutta methods i e piecewise polynomial functions which extend the approximation at the grid points to the whole interval of integration the particular properties required of the nce s allow us to construct the tail approximations which are quite efficient in terms of kernel evaluations

runge kutta methods of dynamic iteration for index 2 differential algebraic equations this paper investigates a few forms of runge kutta methods based on dynamic iteration for index 2 differential algebraic equations in particular the convergence of iterative runge kutta methods is proven for the complex systems the iterative processes allow implementation with different meshes which can help improve the accuracy and the stability properties of the proposed method finally the iterative schemes how to appropriately implement on the parallel computers are also involved

homogenization for stochastic hamilton jacobi equations homogenization asks whether average behavior can be discerned from partial differential equations that are subject to high frequency fluctuations when those fluctuations result from a dependence on two widely separated spatial scales we prove homogenization for certain stochastic hamilton jacobi partial differential equations the idea is to use the subadditive ergodic theorem to establish the existence of an average in the infinite scale separation limit in some cases we also establish a central limit theorem

one step methods of any order for neutral functional differential equations a new class of one step methods for the numerical solution of neutral functional differential equations is considered these methods can be taken to have arbitrarily high order this should be compared with other one step methods such as those of tavernini for which the obvious modification to neutral functional differential equations leads only to methods of order one

an elliptic problem with critical growth in domains with shrinking holes we study an elliptic problem involving critical sobolev exponent in domains with small holes we prove the existence of solutions which blow up like a volcano near the centre of each hole c 2003 elsevier inc all rights reserved

automatic selection of the initial step size for an ode solver abstract the choice of initial step size is critical for the reliable numerical solution of the initial value problem for a system of ordinary differential equations automatic selection of this step size may lead to a more robust and efficient integration than its provision by a user and is always more convenient it is especially important for the reliability of an ode solver used as a module in a larger software package previous approaches to making the selection are combined with some new ideas to produce an effective scheme for the automatic choice of the initial step size numerical results illustrate the roles played by the individual phases of the algorithm and show that the whole algorithm is both robust and efficient

natural continuous extensions of runge kutta methods for volterra integrodifferential equations in this paper we deal with a very general class of runge kutta methods for the numerical solution of volterra integrodifferential equations our main contribution is the development of the theory of natural continuous extensions nces i e piecewice polynomial functions which interpolate the values given by the rk method at the mesh points the particular features of these nces allow us to construct tail approximations which are quite efficient since they require a minimal number of kernel evaluations

stochastic homogenization of nonconvex hamilton jacobi equations a counterexample this paper provides a counterexample to hamilton jacobi homogenization in the nonconvex case for general stationary ergodic environments

differential algebraic equations with after effect in this paper we are concerned with the solution of delay differential algebraic equations these are differential algebraic equations with after effect or constrained delay differential equations the general semi explicit form of the problem consists of a set of delay differential equations combined with a set of constraints that may involve retarded arguments even simply stated problems of this type can give rise to difficult analytical and numerical problems the more tractable examples can be shown to be equivalent to systems of delay or neutral delay differential equations our purpose is to highlight some of the complexities and obstacles that can arise when solving these problems and to indicate problems that require further research

a cweno type central upwind scheme for ideal mhd equations a simple fourth order central upwind scheme based on central weighted essentially non oscillatory cweno reconstruction is proposed in this paper for computing the approximate solutions of one and two dimensional ideal magnetohydrodynamics mhd equations with high resolution since the non uniform width of the different local riemann fans is calculated more accurately the central upwind schemes enjoy a much smaller numerical viscosity as well as the staggering between two neighboring sets of grids is avoided due to the central upwind scheme is combined with the fourth order cweno reconstruction the scheme we present has the non oscillatory behavior the numerical results show the desired accuracy high resolution and robustness of our method

min max formulas and other properties of certain classes of nonconvex effective hamiltonians this paper is the first attempt to systematically study properties of the effective hamiltonian overline h arising in the periodic homogenization of some coercive but nonconvex hamilton jacobi equations firstly we introduce a new and robust decomposition method to obtain min max formulas for a class of nonconvex overline h secondly we analytically and numerically investigate other related interesting phenomena such as quasi convexification and breakdown of symmetry of overline h from other typical nonconvex hamiltonians finally in the appendix we show that our new method and those a priori formulas from the periodic setting can be used to obtain stochastic homogenization for the same class of nonconvex hamilton jacobi equations some conjectures and problems are also proposed

solutions of semilinear elliptic problems in shrinking annuli the positive radially symmetric solutions of semilinear dirichlet problems in annuli is studied the inner radius of which is sufficiently small by means of a phase plane analysis their asymptotic behaviour is computed as the inner radius shrinks it is of particular interest in the cases where the dirichlet problem for the sphere has no non trivial solution this work extends results of bandle and peletier math ann 280 1988 1 19

stability properties of interpolants for runge kutta methods in solving initial value problems for ordinary differential equations by means of a discrete method we often need to know the solution on a set of points that differs from the grid systems with driving equation dense output discontinuity problems delay equations etc the nonnodal approximations are generally obtained by local interpolation and different features can be requested from the interpolants according to the problem treated in this paper we introduce the concept of stable interpolant for runge kutta methods roughly speaking a stable interpolant maintains the stability properties of the runge kutta method itself the lack of such stability makes the continuous extension not reliable especially in solving stiff equations

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

a rigidity result for effective hamiltonians with 3 mode periodic potentials abstract we continue studying an inverse problem in the theory of periodic homogenization of hamilton jacobi equations proposed in 14 let v 1 v 2 c r n be two given potentials which are z n periodic and h 1 h 2 be the effective hamiltonians associated with the hamiltonians 1 2 p 2 v 1 1 2 p 2 v 2 respectively a main result in this paper is that if the dimension n 2 and each of v 1 v 2 contains exactly 3 mutually non parallel fourier modes then h 1 h 2 v 1 x v 2 x c x 0 for all x t 2 r 2 z 2 for some c q 0 and x 0 t 2 when n 3 the scenario is slightly more subtle and a complete description is provided for any dimension these resolve partially a conjecture stated in 14 some other related results and open problems are also discussed

exact and approximate correctors for stochastic hamiltonians the 1 dimensional case we perform a qualitative investigation of critical hamilton jacobi equations with stationary ergodic hamiltonian in dimension 1 we show the existence of approximate correctors give characterizing conditions for the existence of correctors provide lax type representation formulae and establish comparison principles the results are applied to look into the corresponding effective hamiltonian and to study a homogenization problem in the analysis a crucial role is played by tools from stochastic geometry such as for instance closed random stationary sets