positive solutions for higher order multi point boundary value problems with nonhomogeneous boundary conditions abstract we study nth order boundary value problems with a nonlinear term f t x subject to nonhomogeneous multi point boundary conditions criteria for the existence of positive solutions of such problems are established conditions are determined by the relationship between the behavior of f t x x near 0 and when compared with the smallest positive characteristic value of an associated linear integral operator this work improves and extends some recent results in the literature for the second order problems the results are illustrated with examples

starting algorithms for low stage order rkn methods when second order differential equations are solved with runge kutta nystrom methods the computational effort is dominated by the cost of solving the nonlinear system that is why it is important to have good starting values to begin the iterations in this paper we consider a type of starting algorithms without additional computational cost we study the general order conditions and the maximum order achieved when the runge kutta nystrom method satisfies some simplifying assumptions

kahlerian twistor spaces the work of r penrose has shown the deep relationship between the conformal structure of minkowski space and the complex geometry of lines in projective three space this transformation applies to a more general class of conformal structures and in particular there is a riemannian version described in 5 which is itself of interest to mathematical physicists basically to each oriented riemannian 4manifold x one associates canonically a 6 manifold z the twistor space of x together with an almost complex structure if this structure is integrable then z becomes a complex 3 manifold with a distinguished family of projective lines on it whose geometry describes the conformal geometry of the manifold x it is natural to ask what sort of complex manifold one obtains by this process in this paper we ask the more specific question which compact 4c manifolds have ka hlerian twistor spaces 1 the answer rather surprisingly is that there are just two examples the 4 sphere s and the complex projective plane p2 c each with their standard homogeneous conformal structures the twistor space of s 4 is the 3 dimensional projective space p3 c and that of p 2 q is the manifold f2 c of flags in c 3 the essential point of the proof of this result is the remark that if z is a twistor space with a kahler metric then the canonical bundle k is negative using vanishing theorems and the riemann roch theorem we then narrow down the possibilities for z to four cases projective space the flag manifold the intersection of two quadrics in p5 and the double covering of p3 branched over a non singular quartic surface the last two candidates cannot be realized however because the value of their euler characteristic is incompatible with their being twistor spaces throughout the proof we make essential use of a real structure on z which reduces considerably the occurrence of singularities at various stages aside from this our methods are the standard ones of algebraic geometry the four examples to which the proof leads us are all fano threefolds if we drop the requirement of compactness on x then it is quite possible for an open set of a fano threefold to be a twistor space in particular we show that the family of conies lying on the intersection of two

global behavior of positive solutions of nonlinear three point boundary value problems we investigate the structure of the positive solution set for nonlinear three point boundary value problems of the form u h t f u 0 u 0 0 u 1 lambdau eta where eta epsilon 0 1 is given lambda epsilon 0 1 n is a parameter f epsilon c 0 infinity 0 infinity satisfies f s 0 for s 0 and h epsilon c 0 1 0 infinity is not identically zero on any subinterval of 0 1 our main results demonstrate the existence of continua of positive solutions of the above problem c 2004 elsevier ltd all rights reserved

symmetric positive solutions to singular system with multi point coupled boundary conditions in this paper we study the existence and multiplicity of symmetric positive solutions for a nonlinear system with multi point coupled boundary conditions the arguments are based upon a specially constructed cone and the fixed point index theorem in cones an example is then given to demonstrate the applicability of our results

higher order boundary value problems with nonhomogeneous boundary conditions abstract we study the following nonlinear boundary value problem with nonhomogeneous multi point boundary condition u 2 n g t f t u t 0 1 u 2 i 0 j 1 m a j u 2 i t j 1 i 2 i u 2 i 1 j 1 m b j u 2 i t j 1 i 2 i 1 i 0 n 1 sufficient conditions are obtained for the problem to have at least one positive solution and to have no solution respectively moreover under certain conditions we prove that there exists a bounded and continuous surface separating r 2 n 0 0 into two disjoint subsets e and n with e such that the problem has at least two positive solutions for each 0 2 n 1 e one positive solution for each 0 2 n 1 and no positive solution for any 0 2 n 1 n

optimal constants in a nonlocal boundary value problem we give improved results on the existence of positive solutions for a nonlinear differential equation with nonlocal boundary conditions that arise as a model of a thermostat we obtain some optimal criteria for the existence of one positive solution which involve the principal eigenvalue of a related linear operator we also determine optimal values of some other constants that are useful in obtaining existence of multiple positive solutions

equivariant deformations of meromorphic actions on compact complex manifolds meromorphicity is the most basic property for holomorphic mathbb c actions on compact complex manifolds we prove that the meromorphicity of mathbb c actions on compact complex manifolds are not necessarily preserved by small deformations if the complex dimension of complex manifolds is greater than two in contrast we also show that the meromorphicity of mathbb c actions on compact complex surface depends only on the topology the first betti number of the surface we construct such examples of dimension greater than two by studying an equivariant deformation of certain complex threefold so called a twistor space

a conic bundle description of moishezon twistor spaces without effective divisors of degree one in k2 moishezon twistor spaces over the connected sum n mathbb cp 2 n geq 4 which do not contain effective divisors of degree one were constructed as deformations of the twistor spaces introduced in leb we study their structure for n geq 4 by constructing a modification which is a conic bundle over mathbb p 2 we show that they are rational in case n 4 we give explicit equations for such conic bundles and use them to construct explicit birational maps between these conic bundles and mathbb p 3

non moishezon twistor spaces of 4 mathbf cp 2 with non trivial automorphism group we show that a twistor space of a self dual metric on 4cp 2 with u 1 isometry is not moishezon iff there is a c orbit biholomorphic to a smooth elliptic curve where the c action is the complexification of the u 1 action on the twistor space it follows that the u 1 isometry has a two sphere whose isotropy group is z 2 we also prove the existence of such twistor spaces in a strong form to show that a problem of campana and kreusler is affirmative even though a twistor space is required to have a non trivial automorphism group

stabilized starting algorithms for collocation runge kutta methods abstract in this paper we propose a technique to stabilize some starting algorithms often used in the newton type iterations appearing when collocation runge kutta methods are applied to solve stiff initial value problems by following the ideas given in 1 we analyze the order classical and stiff of the new starting algorithms and pay special attention to their error amplifying functions from the computational point of view the new algorithms require the solution of an additional linear system per integration step but as shown in the numerical experiments this extra cost is compensated in most of the problems by their better stability properties

multiplicity results for a three point boundary value problem at resonance this paper deals with second order three point boundary value problem u t f t u t t 0 1 u 0 0 u 1 u where f 0 1 r r is continuous 0 and 0 1 are given constants such that 1 we develop the methods of lower and upper solutions by the connectivity properties of the solution set of parameterized families of compact vector fields as applications of these methods we get several multiplicity results for the problems under consideration

algebraic dimension of twistor spaces si l espace twisteur d une variete compacte autoduale est moishezon la classe conforme autoduale contient une metrique de courbure scalaire positive

on the metric structure of non kahler complex surfaces we give a characterization of a locally conformally kahler l c k metric with parallel lee form on a compact complex surface using the kodaira classification of surfaces we classify the compact complex surfaces admitting such structures this gives a classification of sasakian structures on compact three manifolds a weak version of the above mentioned characterization leads to an explicit construction of l c k metrics on all hopf surfaces we characterize the locally homogeneous l c k metrics on geometric complex surfaces and we prove that some inoue surfaces do not admit any l c k metric

the bim code for the numerical solution of odes in this paper we present the code bim based on blended implicit methods j comput appl math 116 2000 41 appl numer math 42 2002 29 recent trends in numerical analysis nova science publ inc new york 2001 pp 81 for the numerical solution of stiff initial value problems for odes we describe in detail most of the implementation strategies used in the construction of the code and report numerical tests comparing the code bim with some of the best codes currently available the numerical tests show that the new code compares well with existing ones moreover the methods implemented in the code are characterized by a diagonal nonlinear splitting which makes its extension for parallel computers very straightforward

solvability of a three point nonlinear boundary value problem for a second order ordinary differential equation in case f x u u ps x p x u pjx u with pk 0 l 4 r locally integrable the boundary value problem l lt 1 2 was studied by kiguradze and lomtatidze i the purpose of this paper is to obtain existence and uniqueness theorems for the boundary value problem 1 1 t 1 2 under natural conditions on f using degree theoretic arguments we note that if u is a solution of 1 1 1 2 then there exists at least one 5 e q l such that u c 0 accordingly the boundary value problem

symmetric solutions of a multi point boundary value problem abstract we apply the fixed point theorem of avery and peterson to the nonlinear second order multi point boundary value problem u t a t f t u t u t t 0 1 u 0 i 1 n i u i u 1 t u t t 0 1 where 0 1 2 n 1 2 i 0 for i 1 n with i 1 n i 1 n 2 we show that under the appropriate growth conditions on the inhomogeneous term symmetric about t 1 2 the problem has triple symmetric solutions

computing smooth solutions of daes for elastic multibody systems abstract multibody systems are considered which involve combinations of rigid and elastic bodies discretizations of the pdes describing the elastic members lead to a semidiscrete system of odes or daes asymptotic methods are introduced which provide a theoretical basis of various known engineering results for the ode case these results are then extended to the dae case by means of suitable local ode representations the recently developed manpak algorithms for computations on implicitly defined manifolds form the basic tools for a computational method which provides consistent approximate solutions of the semidiscrete dae that satisfy all constraints and are close to the smooth motion and an average solution several numerical examples indicate the effectiveness of this asymptotic method for elastic multibody systems

the integration of stiff initial value problems in odes using modified extended backward differentiation formulae abstract a class of modified extended backward differentiation formulae mebdf suitable for the approximate numerical integration of first order systems of stiff ordinary differential equations is introduced the computational aspects of this new class of formulae are examined in detail in particular algorithms for varying both order and stepsize are given and this leads to a variable step variable order process using highly stable formulae of order 2 8 extensive numerical results for the well known detest set are given and on this basis a comparison is made between a code incorporating mebdf a code due to hindmarsh based on conventional backward differentiation formulae and a code due to skeel and kong based on blended linear multistep methods it is shown that on this test set the mebdf code is very reliable and is competitive with the other two codes on a significant class of problems a computer listing of the mebdf code used to obtain the results presented in this paper is available from the author

on asynchronous iterations asynchronous iterations arise naturally on parallel computers if one wants to minimize idle times this paper reviews certain models of asynchronous iterations using a common theoretical framework the corresponding convergence theory and various domains of applications are presented these include nonsingular linear systems nonlinear systems and initial value problems

solvability for a nonlinear second order three point boundary value problem this paper investigates the existence of nontrivial solution for the three point boundary value problem u f t u 0 0 t 1 u 0 0 u 1 u where 0 1 r 1 f c 0 1 r r under certain growth conditions on the nonlinearity f several sufficient conditions for the existence of nontrivial solution are obtained by using leray schauder nonlinear alternative as an application some examples to demonstrate our results are given

nonzero solutions of hammerstein integral equations with discontinuous kernels using the theory of fixed point index we establish new results for the existence of nonzero solutions of integral equations of the form u t gk t s f s u s ds where g is a compact set in rn and k changes sign so positive solutions may not exist f satisfies caratheodory conditions and k may be discontinuous we apply our results to prove the existence of nontrivial solutions of some nonlocal boundary value problems

existence and multiplicity of symmetric positive solutions for three point boundary value problem in this paper we are concerned with the existence and multiplicity of symmetric positive solutions for the following second order three point boundary value problem u t a t f t u t 0 0 t 1 u t u 1 t u 0 u 1 u 1 2 where a 0 1 0 is symmetric on 0 1 and maybe singular at t 0 and t 1 f 0 1 0 0 is continuous and f u is symmetric on 0 1 for all u 0 growth conditions are imposed on f which yield the existence of at least one or at least two positive solutions our proof based on krasnoselskii s fixed point theorem in a cone

optimization of high order diagonally implicit runge kutta methods abstract this article presents constrained numerical optimization of high order linearly and algebraically stable diagonally implicit runge kutta methods after satisfying the desired order conditions undetermined coefficients are optimized with respect to objective functions which consider accuracy stability and computational cost constraints are applied during the optimization to enforce stability properties to ensure a well conditioned method and to limit the domain of the abscissa two promising third order methods are derived using this approach labelled sdirk 3 1 2 2 3 l 14 and sdirk 3 1 4 l sa 5 both optimized schemes have a good balance of properties the relative error norm of the latter the l 2 norm scaled by a function of the number of implicit stages is a factor of two smaller than comparable methods found in the literature variations on these methods are discussed relative to trade offs in their accuracy and stability properties a novel fifth order scheme sdirk 5 1 5 l 02 is derived with a significantly lower relative error norm than the comparable fifth order a stable reference method in addition the optimized scheme is l stable the accuracy and relative efficiency of the runge kutta methods are verified through numerical simulation of van der pol s equation as well as numerical simulation of vortex shedding in the laminar wake of a circular cylinder and in the turbulent wake of a naca 0012 airfoil these results demonstrate the value of numerical optimization for selecting undetermined coefficients in the construction of high order runge kutta methods with a balance between competing objectives

projected implicit runge kutta methods for differential algebraic equations in this paper a new class of numerical methods projected implicit runge kutta methods is introduced for the solution of index 2 hessenberg systems of initial and boundary value differential algebraic equations daes these types of systems arise in a variety of applications including the modeling of singular optimal control problems and parameter estimation for differential algebraic equations such as multibody systems the new methods appear to be particularly promising for the solution of dae boundary value problems where the need to maintain stability in the differential part of the system often necessitates the use of methods based on symmetric discretizations previously defined symmetric methods have severe limitations when applied to these problems including instability oscillation and loss of accuracy the new methods overcome these difficulties for linear problems we define an essential underlying boundary value ode and prove well conditioning of the differential or state space solutio

double solid twistor spaces the case of arbitrary signature in a recent paper 9 we constructed a series of new moishezon twistor spaces which are a kind of variant of the famous lebrun twistor spaces in this paper we explicitly give projective models of another series of moishezon twistor spaces on ncp2 for arbitrary n 3 which can be regarded as a generalization of the twistor spaces of double solid type on 3cp2 studied by kreusler kurke poon and the author similarly to the twistor spaces of double solid type on 3cp2 projective models of the present twistor spaces have a natural structure of double covering of a cp2 bundle over cp1 we explicitly give a defining polynomial of the branch divisor of the double covering whose restriction to fibers is degree four if n 4 these are new twistor spaces to the best of the author s knowledge we also compute the dimension of the moduli space of these twistor spaces differently from 9 the present investigation is based on analysis of pluri half anticanonical systems of the twistor spaces

are high order variable step equistage initializers better than standard starting algorithms in this paper a fourth order starting algorithm for variable step implicit runge kutta methods is developed using the approach of equistage approximation proposed in ima j numer anal 22 2002 153 this starting algorithm is used in combination with the well known radau5 code and numerical results are provided

starting algorithms for irk methods in this paper some classes of starting algorithms for the iterations of irk methods are studied they are of three types according to their additional cost by means of b series the order conditions for them are obtained the maximum order attained by these algorithms and their construction are derived too

on conformally kahler einstein manifolds we prove that any compact complex surface with c1 0 admits an einstein metric which is conformally related to a kahler metric the key new ingredient is the existence of such a metric on the blow up cp2 2cp2 of the complex projective plane at two distinct points

stiff differential equations solved by radau methods radau iia methods are successful algorithms for the numerical solution of stiff differential equations this article describes radau a new implementation of these methods with a variable order strategy the paper starts with a survey on the historical development of the methods and the discoveries of their theoretical properties numerical experiments illustrate the behaviour of the code

stage value predictors and efficient newton iterations in implicit runge kutta methods the prediction of stage values in implicit runge kutta methods is important both for overall efficiency as well as for the design of suitable control strategies for the method the purpose of this paper is to construct good stage value predictors for implicit methods and to verify their behavior in practical computations we show that for stiffly accurate methods of low stage order it is necessary to use several predictors in other words a continuous extension for the method will not yield the best results we also investigate how to gain additional efficiency in the newton iterations used to correct the prediction error this leads to new control strategies with respect to refactorization of jacobians that seek to globally minimize total work per unit time of integration

three solutions for three point boundary value problems abstract in this paper we study the existence of at least three solutions for the three point nonlinear boundary value problems u t a t f u 0 0 t 1 u 0 0 u 1 u where 0 1 0 1 a c 0 1 0 and f c r r without any monotonicity assumptions on the nonlinear term f by using the increasing operator theory and approximation process we prove that the three point boundary value problems has at least three solutions

implementation of high order implicit runge kutta methods abstract in this paper we develop single newton iterative schemes for the solution of the stage equations of some implicit runge kutta methods such as the four stage gauss and radau iia methods and the five stage lobatto iiia formula we also compare the implementation cost of these schemes with the simplified newton iteration and we present some numerical experiments on some well known stiff test problems to show that the proposed iterations are reliable and efficient

a symmetric solution of a multipoint boundary value problem with one dimensional p laplacian at resonance abstract we apply an extension of mawhin s continuation theorem due to ge to show the existence of at least one symmetric solution of the multipoint boundary value problem for the one dimensional p laplacian at resonance p x t f t x t x t t 0 1 subject to the boundary conditions x 0 i 1 n i x i x 1 i 1 n i x i where p s s p 2 s p 1 0 1 2 n 1 2 i i 1 i 1 2 n i 1 n i 1 f 0 1 r 2 r with f t u v f 1 t u v for t u v 0 1 r 2 satisfying the caratheodory conditions

symmetric positive solutions of boundary value problems with integral boundary conditions under conditions weaker than those used by ma we establish various results on the existence and nonexistence of symmetric positive solutions to fourth order boundary value problems with integral boundary conditions the arguments are based upon a specially constructed cone and the fixed point theory in a cone our results improve and extend that obtained in h ma symmetric positive solutions for nonlocal boundary value problems of fourth order nonlinear anal 68 2008 645 651 we illustrate our results by one example which cannot be handled using the existing results

a 17th order radau iia method for package radau applications in mechanical systems a new radau iia code with an order of 17 is programmed for the well known implicit runge kutta program package radau several numerical examples are reported in this article to demonstrate the performance and efficiency of the new algorithm they are real stiff problems obtained from the simulation of mechanical systems

positive solutions of a nonlinear three point boundary value problem in this paper by using the fixed point index theorems we study the existence of at least one or two positive solutions to the three point boundary value problem y t a t f y t 0 0 t 1 y 0 0 y 1 y where 0 1 0 1 as an application we also give some examples to demonstrate our results

order results for mono implicit runge kutta methods the mono implicit runge kutta methods are a subclass of the well known implicit runge kutta methods and have application in the efficient numerical solution of systems of initial and boundary value ordinary differential equations although the efficiency and stability properties of this class of methods have been studied in a number of papers the specific question of determining the maximum order of an s stage mono implicit runge kutta method has not been dealt with in addition to the complete characterization of some subclasses of these methods having a number of stages s leq 5 a main result of this paper is a proof that the order of an s stage mono implicit runge kutta method is at most s 1

starting algorithms for a class of rk methods for index 2 daes when semiexplicit differential algebraic equations are solved with implicit runge kutta methods rk the computational effort is dominated by the cost of solving the nonlinear systems and therefore it is important to have good starting values to begin the iterations for semiexplicit index 2 daes starting algorithms without additional cost for rk methods with regular matrix coefficient were studied in a previous paper however the regularity condition on the matrix coefficient excludes some interesting methods like lobatto iiia and esdirk methods in this paper we study starting algorithms without additional computational cost for a class of runge kutta methods in the case of index 2 daes

multiplicity of symmetric positive solutions for a multipoint boundary value problem with a one dimensional p laplacian abstract in this paper we consider the multipoint boundary value problem for the one dimensional p laplacian p u t q t f t u t u t 0 t 0 1 subject to the boundary conditions u 0 i 1 n i u i u 1 i 1 n i u i where p s s p 2 s p 1 i 0 0 i 1 n i 1 0 1 2 n 1 2 i i 1 i 1 2 n applying a fixed point theorem of functional type in a cone we study the existence of at least three symmetric positive solutions to the above boundary value problem the interesting point is that the nonlinear term f contains the first order derivative explicitly

starting algorithms for gauss runge kutta methods for hamiltonian systems abstract among the symplectic integrators for the numerical solution of general hamiltonian systems implicit runge kutta methods of gauss type rkg play an important role to improve the efficiency of the algorithms to be used in the solution of the nonlinear equations of stages accurate starting values for the iterative process are required in this paper a class of starting algorithms which are based on numerical information computed in two previous steps is studied for two and three stages rkg methods explicit starting algorithms for the stage equations with orders three and four are derived finally some numerical experiments comparing the behaviour of the new starting algorithms with the standard first iterant based on lagrange interpolation of stages in the previous step are presented

a test set for stiff initial value problem solvers in the open source software r package detestset in this paper we present the r package detestset that includes challenging test problems written as ordinary differential equations odes differential algebraic equations daes of index up to 3 and implicit differential equations ides in addition it includes 6 new codes to solve initial value problems ivps the r package is derived from the test set for initial value problem solvers available at http www dm uniba it testset which includes documentation of the test problems experimental results from a number of proven solvers and fortran subroutines providing a common interface to the defining problem functions many of these facilities are now available in the r package detestset which comprises an r interface to the test problems and to most of the fortran solvers the package detestset is free software which is distributed under the gnu general public license as part of the r open source software project

irk methods for dae starting algorithms abstract when semi explicit differential algebraic equations are solved with implicit runge kutta methods the computational effort is dominated by the cost of solving the nonlinear systems that is why it is important to have good starting values to begin the iterations in this paper we study a type of starting algorithms without additional computational cost in the case of index 1 dae the order of the starting values is defined and by using da series and rooted trees we obtain their general order conditions if the rk satisfies some simplified assumptions then the maximum order can be obtained

three positive fixed points of nonlinear operators on ordered banach spaces we generalize the fixed point theorem of leggett williams which is a theorem giving conditions that imply the existence of three fixed points of an operator defined on a cone in a banach space we then show how to apply our theorem to prove the existence of three positive solutions to a second order discrete boundary value problem 2001 elsevier science ltd all rights reserved

control strategies for the iterative solution of nonlinear equations in ode solvers in the numerical solution of odes by implicit time stepping methods a system of nonlinear equations has to be solved each step it is common practice to use fixed point iterations or in the stiff case some modified newton iteration the convergence rate of such methods depends on the stepsize similarly a stepsize change may force a refactorization of the iteration matrix in the newton solver this paper develops new strategies for handling the iterative solution of nonlinear equations in ode solvers these include automatic switching between fixed point and newton iterations investigating the optimal convergence rate with respect to total work per unit step a strategy for when to reevaluate the jacobian a strategy for when to refactorize the iteration matrix coordination with stepsize control examples will be given showing that the new overall strategy works efficiently in particular the new strategy admits a restrained stepsize variation without refactorizations thus permitting the use of a smoother stepsize sequence the strategy is of equal importance for runge kutta and multistep methods

semi lagrangian multistep exponential integrators for index 2 differential algebraic systems implicit explicit imex multistep methods are very useful for the time discretization of convection diffusion pde problems such as the burgers equations and the incompressible navier stokes equations in the latter as well as in pde models of plasma physics and of electromechanical systems semi discretization in space gives rise to differential algebraic dae system of equations often of index higher than 1 in this paper we propose a new class of exponential integrators for index 2 daes arising from the semi discretization of pdes with a dominating and typically nonlinear convection term this class of problems includes the incompressible navier stokes equations the integration methods are based on the backward differentiation formulae bdf and they can be applied without modifications in the semi lagrangian integration of convection diffusion problems the approach gives improved performance at low viscosity regimes

optimal existence criteria for symmetric positive solutions to a three point boundary value problem abstract in this paper we shall consider the following second order three point boundary value problem u t a t f t u t 0 0 t 1 u 0 u 1 u where 0 1 0 1 2 a l 1 0 1 is nonnegative and symmetric on 0 1 f 0 1 0 0 satisfies caratheodory conditions and f u is symmetric on 0 1 for all u 0 by using fixed point index theorems we get some optimal existence criteria for the existence of one or two symmetric solutions which involve the principal eigenvalue of a related linear operator

a sharper condition for the solvability of a three point second order boundary value problem letf 0 1 r2 rbe a function satisfying caratheodory s conditions ande t l1 0 1 let 0 1 r 1 1 be given this paper is concerned with the problem of existence of a solution for the three point boundary value problem formula omitted this problem was studied earlier by gupta ntouyas and tsamatos when 1 and when 1 with 1 in the general case this problem was studied by gupta as a multi point boundary value problem in this paper sharper existence conditions are obtained for the solvability of the above boundary value problem in the general case

equivariant connected sums of compact self dual manifolds due to a recent theorem of taubes self dual conformal structures on compact four dimensional manifolds exist in abundance 43 therefore finding examples in general is no longer an issue in this subject to deepen our understanding of self dual conformal geometry we take the path of trying to understand the possible symmetry groups of this geometry i e the groups of orientation preserving conformal transformations the first systematic approach which constructs a large collection of compact self dual manifolds with a non trivial group of symmetries is lebrun s hyperbolic ansatz 22 the condition of this ansatz is that the symmetry group contains an s l u 1 which acts on the manifold semi freely meaning that the isotropy group at any point of the manifold is either trivial or the entire group s i with this condition on the group action and some favourable topological conditions on the underlying manifold the geometrical equations of self duality on the given four dimensional manifold are reduced to the s 1 monopole equations on hyperbolic 3 space with this setup lebrun explicitly constructed a large number of compact self dual manifolds with semi free s l symmetry including the fubinistudy metric on the complex projective plane ci 2 and a one parameter family of self dual metrics on c c first found in 38 we shall call the self dual metrics on nci 2 n 1 constructed by lebrun s hyperbolic ansatz the lebrun metrics after this very successful ansatz it is natural to try to find new examples which are beyond the scope of that method

evaluating numerical ode dae methods algorithms and software until recently the testing of ode dae software has been limited to simple comparisons and benchmarking the process of developing software from a mathematically specified method is complex it entails constructing control structures and objectives selecting iterative methods and termination criteria choosing norms and many more decisions most software constructors have taken a heuristic approach to these design choices and as a consequence two different implementations of the same method may show significant differences in performance yet it is common to try to deduce from software comparisons that one method is better than another such conclusions are not warranted however unless the testing is carried out under true ceteris paribus conditions moreover testing is an empirical science and as such requires a formal test protocol without it conclusions are questionable invalid or even false we argue that ode dae software can be constructed and analyzed by proven standard scientific techniques instead of heuristics the goals are computational stability reproducibility and improved software quality we also focus on different error criteria and norms and discuss modifications to daspk and radau5 finally some basic principles of a test protocol are outlined and applied to testing these codes on a variety of problems

conformal duality and compact complex surfaces on etudie les structures demi conformement plates sur une variete de riemann a 4 dimensions m g avec l hypothese supplementaire que m est complexe

on an m point boundary value problem for second order ordinary differential equations where q e 0 1 is given we obtain conditions for the existence and uniqueness of a solution for the boundary value problem 2 using the leray schauder continuation theorem 2 we give an example of a three point boundary value problem where the existence condition is not satisfied and no solution exists gupta 3 recently studied the boundary value problem 2 when q 1 our results on the three point boundary value problem 2 extend the results of gupta 3 to the case of general cy see also 4 51 we use the classical spaces c o 11 c o 11 lk o 11 and l o l of continuous k times continuously differentiable measurable real valued functions whose kth power of the absolute value is lebesgue integrable on 0 11 or measurable functions that are essentially bounded

triple solutions for second order three point boundary value problems abstract we establish the existence of at least three positive solutions to the second order three point boundary value problem u f t u 0 u 0 0 u u 1 where 0 lt f 0 1 0 0 is continuous we accomplish this by making growth assumptions on f which can apply to many more cases than the sublinear and superlinear ones discussed in recent works

existence and iteration of n symmetric positive solutions for a singular two point boundary value problem let n be an arbitrary natural number in this paper we consider the existence of n symmetric positive solutions and establish a corresponding iterative scheme for the two point boundary value problem w t h t f w t 0 0 t 1 w 0 w 0 0 1 w 1 0 the main tool is the monotone iterative technique here the coefficient h t is symmetric on 0 1 and may be singular at both end points t 0 and t 1

an efficient implicit runge kutta method for second order systems abstract we will consider the efficient implementation of a fourth order two stage implicit runge kutta method to solve periodic second order initial value problems to solve the resulting systems we will use the factorization of the discretized operator such proposed factorization involves both complex and real arithmetic the latter case is considered here the resulting system will be efficient and small in size it is one fourth the size of systems using normal implicit runge kutta method numerical details and examples will also be presented to demonstrate the efficiency of the method

exponential rosenbrock type methods we introduce a new class of exponential integrators for the numerical integration of large scale systems of stiff differential equations these so called rosenbrock type methods linearize the flow in each time step and make use of the matrix exponential and related functions of the jacobian in contrast to standard integrators the methods are fully explicit and do not require the numerical solution of linear systems we analyze the convergence properties of these integrators in a semigroup framework of semilinear evolution equations in banach spaces in particular we derive an abstract stability and convergence result for variable step sizes this analysis further provides the required order conditions and thus allows us to construct pairs of embedded methods we present a third order method with two stages and a fourth order method with three stages respectively the application of the required matrix functions to vectors are computed by krylov subspace approximations we briefly discuss these implementation issues and we give numerical examples that demonstrate the efficiency of the new integrators

fourth order chebyshev methods with recurrence relation in this paper a new family of fourth order chebyshev methods also called stabilized methods is constructed these methods possess nearly optimal stability regions along the negative real axis and a three term recurrence relation the stability properties and the high order make them suitable for large stiff problems often space discretization of parabolic pdes a new code rock4 is proposed illustrated at several examples and compared to existing programs

an mebdf package for the numerical solution of large sparse systems of stiff initial value problems abstract an efficient algorithm for the numerical integration of large sparse systems of stiff initial value ordinary differential equations odes and differential algebraic equations daes is described the algorithm is constructed by embedding a standard sparse linear algebraic equation solver into a suitably modified mebdf code an important practical application of this algorithm is in the numerical solution of time dependent partial differential equations pdes particularly in two or more space dimensions using the method of lines mol a code based on this algorithm is illustrated by application to several problems of practical interest and its performance is compared to that of the standard code lsodes

variable step implementation of etd methods for semilinear problems abstract spatial discretization of many evolutionary partial differential equations leads to stiff semilinear systems of ordinary differential equations exponential integrators solve exactly the linear part which is assumed to be responsible of the stiffness of the problem while the nonlinear terms are approximated numerically in this paper a variable step implementation of the exponential time differencing methods of multistep type is considered a local error estimate of exponential type whose computation is for free is introduced and some implementation issues for diagonalizable matrices are addressed numerical results showing the behaviour of the proposed variable step implementation are provided too

parallel iterative linear solvers for multistep runge kutta methods abstract this paper deals with solving stiff systems of differential equations by implicit multistep runge kutta mrk methods for this type of methods nonlinear systems of dimension sd arise where s is the number of runge kutta stages and d the dimension of the problem applying a newton process leads to linear systems of the same dimension which can be very expensive to solve in practice with a parallel iterative linear system solver especially designed for mrk methods we approximate these linear systems by s systems of dimension d which can be solved in parallel on a computer with s processors in terms of jacobian evaluations and lu decompositions the k steps stage mrk applied with this technique is on s processors equally expensive as the widely used k step backward differentiation formula on 1 processor whereas the stability properties are better than that of bdf a simple implementation of both methods shows that for the same number of newton iterations the accuracy delivered by the new method is higher than that of bdf

adaptive time stepping and computational stability we investigate the effects of adaptive time stepping and other algorithmic strategies on the computational stability of ode codes we show that carefully designed adaptive algorithms have a most significant impact on computational stability and reliability a series of computational experiments with the standard implementation of dassl and a modified version including stepsize control based on digital filters is used to demonstrate that relatively small algorithmic changes are able to extract a vastly better computational stability at no extra expense the inherent performance and stability of dassl are therefore much greater than the standard implementation seems to suggest

remarks on a non local boundary value problem we remark that results on existence of symmetric positive solutions obtained in some recent papers can in fact be deduced as special cases of some known results we also correct the values of the principal eigenvalue previously given in some examples

an integration scheme for stiff solid gas reactor models abstract many dynamic models encounter numerical integration problems because of a large span in the dynamic modes in this paper we develop a numerical integration scheme for systems that include a gas phase and solid and liquid phases such as a gas solid reactor the method is based on neglecting fast dynamic modes and exploiting the structure of the algebraic equations the integration method is suitable for a large class of industrially relevant systems the methodology has proven remarkably efficient it has in practice performed excellently and been a key factor for the success of the industrial simulator for electrochemical furnaces for ferro alloy production

a time adaptive finite volume method for the cahn hilliard and kuramoto sivashinsky equations this paper presents a complete finite volume method for the cahn hilliard and kuramoto sivashinsky type of equations the spatial discretization is high order accurate and suitable for general unstructured grids the time integration is addressed by means of implicit an implicit explicit fourth order runge kutta schemes with error control and adaptive time stepping the outcome is a practical accurate and efficient simulation tool which has been successfully applied to accuracy tests and representative simulations the use of adaptive time stepping is of paramount importance in problems governed by the cahn hilliard model an adaptive method may be several orders of magnitude more efficient than schemes using constant or heuristic time steps in addition to driving the simulations efficiently the time adaptive procedure provides a quantitative not just qualitative characterization of the rich temporal scales present in phase separation processes governed by the cahn hilliard phase field model

positive solutions for singular second order three point boundary value problems abstract in this paper the singular three point boundary value problem u t h t f t u 0 0 t 1 u 0 0 u 1 u where 0 1 0 1 is considered under some conditions concerning the first eigenvalue corresponding to the relevant linear operator h t is allowed to be singular at t 0 1 and f may be singular at u 0 the existence of a positive solution is obtained by a fixed index point

an improved approximate newton method for implicit runge kutta formulas abstract implicit runge kutta irk methods such as the s stage radau iia method with s 3 5 or 7 for solving stiff ordinary differential equation systems have excellent stability properties and high solution accuracy orders but their high computing costs in solving their nonlinear stage equations have seriously limited their applications to large scale problems to reduce such a cost several approximate newton algorithms were developed including a commonly used one called the simplified newton method in this paper a new approximate jacobian matrix and two new test rules for controlling the updating of approximate jacobian matrices are proposed yielding an improved approximate newton method theoretical and numerical analysis show that the improved approximate newton method can significantly improve the convergence and performance of the simplified newton method