parameter uniform convergence of a numerical method for a coupled system of singularly perturbed semilinear reaction diffusion equations with boundary and interior layers abstract in this paper we consider a coupled system of m 2 singularly perturbed semilinear reaction diffusion equations with a discontinuous source term having a discontinuity at a point in the interior of the domain the diffusion term of each equation is multiplied by small singular perturbation parameter but these parameters are assumed to be different in magnitude a numerical method is constructed on a variant of shishkin mesh the approximations generated by this method are shown to be almost second order uniformly convergent with respect to all perturbation parameters numerical results are in support of the theoretical results

parameter uniformly convergent exponential spline difference scheme for singularly perturbed semilinear reaction diffusion problems abstract we consider a dirichlet boundary value problem for singularly perturbed semilinear reaction diffusion equation the problem is discretized using an exponential spline difference scheme derived on the basis of splines in tension on piecewise uniform shishkin type mesh the convergence analysis is given and the method is shown to be almost second order accurate in the discrete maximum norm uniformly in the perturbation parameter e numerical experiments are conducted to demonstrate the theoretical results

numerical integration of non linear two point boundary value problems using iterated deferred corrections i abstract in several recent papers the idea has been proposed of using implicit runge kutta formulae for the approximate integration of non linear two point boundary value problems these formulae often take a very special form so that computational efficiencies are obtained in certain well defined circumstances in this paper we examine the relative merits of these different classes of formulae and discuss their efficient implementation in a deferred correction framework a theoretical comparison of these formulae is important firstly because they call for different computational tasks to be performed and secondly because they are not always applicable to exactly the same classes of problems the most efficient of these formulae are then compared with methods based on finite differences and with methods based on spline collocation for some simple smooth test functions on the basis of operation counts and numerical experimentation it is concluded that runge kutta methods can be generally competitive with the other methods considered

on the solution of block tridiagonal systems of linear algebraic equations having a special structure recently the present author has developed some high order finite difference formulae for the approximate numerical integration of general two point boundary value problems for ordinary differential equations the algebraic equations arising from using these formulae in conjunction with a modified newton iteration scheme are block tridiagonal with an additional special structure efficient algorithms for solving these equations are given and these appear to make high order finite difference schemes particularly attractive for the numerical solution of general two point boundary value problems

fourth order finite difference method for boundary value problems with two small parameters abstract we present a finite difference scheme for a class of linear singularly perturbed boundary value problems with two small parameters the problem is discretized using a bakhvalov type mesh it is proved under certain conditions that this scheme is fourth order accurate and that its error does not increase when the perturbation parameter tends to zero numerical examples are presented which demonstrate computationally the fourth order of the method

adaptive runge kutta methods for nonlinear two point boundary value problems with mild boundary layers abstract a variable mesh deferred correction algorithm based on implicit runge kutta formulae is described for the approximate numerical solution of nonlinear two point boundary value problems a strategy for automatically choosing the variable mesh spacing is described and this seeks to equidistribute an approximation to the global truncation error of the runge kutta formula the facility of being able to use a variable mesh extends considerably the range of applicability of runge kutta methods and in particular allows the possibility of solving rather more difficult problems such as those with mild boundary layers or turning points in an efficient manner an extensive set of numerical results is given to illustrate the various algorithms described and a comparison is made with the deferred correction code of lentini and pereyra appearing in the nag library

nonautonomous impulsive systems with unbounded nonlinear terms in fenner and pinto 1999 23 proved that if the linear impulsive system satisfies the is condition see definition 2 1 with bounded nonlinear term f t x then the perturbed nonlinear impulsive system has a unique bounded solution see theorem a the method used to prove theorem a cannot be applied to the unbounded case in this paper we prove that if f t x e t m then the perturbed nonlinear impulsive system has a unique solution

an almost fourth order uniformly convergent difference scheme for a semilinear singularly perturbed reaction diffusion problem this paper is concerned with a high order convergent discretization for the semilinear reaction diffusion problem varepsilon 2 u b x u 0 for x in 0 1 subject to u 0 u 1 0 where varepsilon in 0 1 we assume that b u x u b 0 2 0 on 0 1 times bbb r 1 which guarantees uniqueness of a solution to the problem asymptotic properties of this solution are discussed we consider a polynomial based three point difference scheme on a simple piecewise equidistant mesh of shishkin type existence and local uniqueness of a solution to the scheme are analysed we prove that the scheme is almost fourth order accurate in the discrete maximum norm uniformly in the perturbation parameter varepsilon we present numerical results in support of this result

completeness ofl1 spaces for measures with values in complex vector spaces we provide a simple and direct proof of the completeness of the space of c valued integrable functions with respect to a vector measure taking its values in a complex frechet space a similar result is established via a quite different argument for closed measures taking their values in more general sequentially complete locally convex hausdorff spaces over c

a uniformly convergent method for a singularly perturbed semilinear reaction diffusion problem with multiple solutions this paper considers a simple central difference scheme for a singularly perturbed semilinear reaction diffusion problem which may have multiple solutions asymptotic properties of solutions to this problem are discussed and analyzed to compute accurate approximations to these solutions we consider a piecewise equidistant mesh of shishkin type which contains o n points on such a mesh we prove existence of a solution to the discretization and show that it is accurate of order n 2 in 2 n in the discrete maximum norm where the constant factor in this error estimate is independent of the perturbation parameter e and n numerical results are presented that verify this rate of convergence

invariant imbedding the box scheme and an equivalence between them we show for general linear two point boundary value problems that the box scheme and invariant imbedding are equivalent in the sense that a specific algorithm for solving the difference equations is valid if and only if an appropriate imbedding is valid a key element in the equivalence is the role played by pivoting to solve the linear equations and switching to get a finite imbedding part of the equivalence proof shows that imbedding is the completion of the box scheme multiple imbeddings are introduced and a lower bound for the critical length problem is obtained

algebra structure for lp of a vector measure abstract in this paper we present new results about the space l p for being a vector measure defined in the borel algebra of a compact abelian group g and satisfying certain property concerning translation of simple functions namely we show that l p is a translation invariant space which can be endowed with an algebra structure via usual convolution product we apply these results to the optimal domain of the fourier transform and convolution operators

a comparison of some global methods for solving two point boundary value problems a numerical comparison is made between three codes which implement global methods for solving two point boundary value problems the codes compared are a collocation method colsys a deferred correction method pasva 3 and a deferred correction method based on mono implicit runge kutta formulae hagron although the difficulty of conducting a fair comparison of complex codes is widely recognised it is nevertheless of great practical importance to try to gauge the relative performance of different codes and the three codes considered in this paper are particularly appropriate for numerical comparison by choosing a test set which includes both smooth and increasingly stiff problems some interesting characteristics of the codes are noted and some recommendations regarding their use are made

high orderp stable formulae for the numerical integration of periodic initial value problems recently there has been considerable interest in the approximate numerical integration of the special initial value problemy f x y for cases where it is known in advance that the required solution is periodic the well known class of stormer cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems an appropriate stability requirement for the numerical integration of periodic problems is that ofp stability however lambert and watson have shown that ap stable linear multistep method cannot have an order of accuracy greater than 2 in the present paper a class of 2 step methods of runge kutta type is discussed for the numerical solution of periodic initial value problems p stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods

a grobman hartman theorem for nonuniformly hyperbolic dynamics abstract we establish a version of the grobman hartman theorem in banach spaces for nonuniformly hyperbolic dynamics we also consider the case of sequences of maps which corresponds to a nonautonomous dynamics with discrete time more precisely we consider sequences of lipschitz maps a m f m such that the linear parts a m admit a nonuniform exponential dichotomy and we establish the existence of a unique sequence of topological conjugacies between the maps a m f m and a m furthermore we show that the conjugacies are holder continuous with holder exponent determined by the ratios of lyapunov exponents with the same sign to the best of our knowledge this statement appeared nowhere before in the published literature even in the particular case of uniform exponential dichotomies although some experts claim that it is well known in this case we are also interested in the dependence of the conjugacies on the perturbations f m we show that it is holder continuous with the same holder exponent as the one for the conjugacies we emphasize that the additional work required to consider the case of nonuniform exponential dichotomies is substantial in particular we need to consider several additional lyapunov norms

e uniformly convergent fitted mesh finite difference methods for general singular perturbation problems abstract we consider general singular perturbation problems of the form cey x a x y x b x y x f x x 0 1 y 0 0 y 1 1 with ce equals to both e and e a x b x f x are positive throughout the interval and 0 1 r the indirect methods those which do not use any acceleration of convergence techniques e g richardson s extrapolation or defect correction etc for such problems on a mesh of shishkin type lead the error as o n 1 ln n where n denotes the total number of sub intervals of 0 1 in this paper we systematically describe a very simple and direct method which reduces the error to o n 2 ln 2 n this method is proved to be e uniformly convergent with the above error bounds on a piecewise uniform mesh of shishkin type the motivation for using this shishkin mesh is inspired by the quotation of stynes m stynes a jejune heuristic mesh theorem comput methods appl math 3 2003 488 492 miller has moved from j j h miller construction of a fem for a singularly perturbed problem in 2 dimensions in numerische behandlung von differentialgleichungen band 2 tagung math forschungsinst oberwolfach 1975 internat ser numer math birkhauser basel vol 31 1976 pp 165 169 the question what scheme should one use on a given mesh to p a farrell a f hegarty j j h miller e o riordan g i shishkin robust computational techniques for boundary layers chapman hall crc new york 2000 what mesh should one use with a given scheme the theoretical estimates have been justified by several numerical examples

a riccati transformation method for solving linear bvps i theoretical aspects a method of solving linear two point boundary value problems bvps for first order systems of ordinary differential equations odes is considered of particular interest is the class of singularly perturbed bvps the method attempts to split the integration of the rapidly increasing decreasing solution components typical for such problems into different initial value problems ivps the case of separated boundary conditions bcs is considered and their ability to detect solution components that eventually increase decrease is exploited by using the riccati transformation the original bvp is then reformulated as three ivps one of them being a matrix riccati equation the riccati equation is nonlinear and a multiple embedding setting must be used the entire method is then fit theoretically into a multiple shooting like framework and the original bvp as a chain of subproblems is studied in this way a generalized decoupling result is presented which enables the study of stability to be brough

a riccati transformation method for solving linear bvps ii computational aspects in part i of this paper siamj numer anal 25 1988 pp 1055 1073 a riccati transformation method for solving linear boundary value problems for ordinary differential equations is presented here the implementation aspects of this method are discussed in particular a double sweep integration process which includes a strategy for solving a matrix riccati differential equation and forming a continuous representation of its solution is described the efficacy of the method is demonstrated on a number of singular perturbation problems particular attention is paid to the choice of scaling for the variables and its effect on the solution method the computational experience while not based upon a refined implementation indicates that the riccati method may well prove to be competitive with the other general methods

an improved uniformly convergent scheme in space for 1d parabolic reaction diffusion systems in this paper the numerical approximation of 1d parabolic singularly perturbed systems with two equations of reaction diffusion type is considered these problems typically exhibit two overlapping boundary layers at both end points of the spatial domain a decomposition of the exact solution into its regular and singular part is established given appropriate bounds for the partial derivatives of the exact solution up to sixth order these bounds are crucial to prove the uniform convergence of a numerical method that combines the classical backward euler method and a hybrid finite difference scheme defined on a special nonuniform mesh condensing in the layer regions the numerical method is uniformly convergent in the discrete maximum norm and it has first and third order of convergence in time and space respectively numerical results for some test problems are showed illustrating in practice the order of convergence theoretically proved

rearrangement invariant spaces of functions on lca groups in this paper the rearrangement invariant hulls and kernels of the classes f f lp and f f l1loc on a noncompact nondiscrete locally compact abelian group are characterized the spacel1loc s of tempered distributions serves as the extended domain of the fourier transform instead of the classical domainl1 l2 we also introduce and study a new function spacep0which arises as the rearrangement invariant hull of some of the classes under consideration

improvement of numerical solution by boundary value technique for singularly perturbed one dimensional reaction diffusion problem a class of singularly perturbed two point boundary value problems bvps for second order ordinary differential equations in self adjoint form arising in the study of chemical catalysis and michaelis menten process in biology is considered in order to solve them a numerical method as in appl math comput 55 1993 31 is proposed the essential idea in this method is to divide the domain of the differential equation into three non overlapping subdomains and solve the given equations over these regions separately as three two point bvps numerically the inner region problems are solved using a fitted operator method uniform numerical methods for problems with initial and boundary layers boole press dublin 1980 whereas the outer region problem is solved using a classical central difference method the boundary conditions at the transition points are obtained using the zero order asymptotic expansion approximation to the solution of the problem this method is well suited for parallel computing and an algorithm for the same is given error estimates are derived for the numerical solution some schemes for self adjoint equations in conservation form are given using newton s method of quasilinearization a class of semilinear problems are also solved numerical experiments are conducted it is found that the present method performs better than the fitted mesh method fitted numerical methods for singular perturbation problems world scientific singapore 1996 and higher order method numeriche mathematik 56 1990 675

on spline basis selection for solving differential equations the suitability of b splines as a basis for piecewise polynomial solution representation for solving differential equations is challenged two alternative local solution representations are considered in the context of collocating ordinary differential equations hermite type and lmonomial both are much easier and shorter to implement and somewhat more efficient than b splines a new condition number estimate for the b splines and hermite type representations is presented one choice of the hermite type representation is experimentally determined to produce roundoff errors at most as large as those for b splines the monomial representation is shown to have a much smaller condition number than the other ones and correspondingly produces smaller roundoff errors especially for extremely nonuniform meshes the operation counts for the two local representations considered are about the same the hermite type representation being slightly cheaper it is concluded that both representations are preferable

uniform convergence and order reduction of the fractional implicit euler method to solve singularly perturbed 2d reaction diffusion problems in this paper we analyze the uniform convergence of a numerical method designed to approximate efficiently the solution of 2d parabolic singularly perturbed problems of reaction diffusion type the method combines a modified fractional implicit euler method to discretize in time and the classical central finite difference scheme on a special nonuniform mesh to discretize in space the resulting fully discrete scheme is uniformly convergent with respect to the diffusion parameter the analysis of the convergence is made by using a two step technique which discretizes first in time and later on in space we show the order reduction phenomenon associated to the fractional implicit euler method which typically appears if the boundary conditions are time dependent and a natural evaluation of them is done an appropriate choice for the boundary conditions is proposed and analyzed in detail proving that the order reduction can be removed some numerical tests show the practical effects of our method as well we compare it with the classical choice for the boundary data in terms of the uniform consistency and the order of uniform convergence of the numerical scheme

the order of accuracy for deferred corrections using uncentered end formulas a result of christiansen and russell siam j numer math 35 1980 pp 21 33 on the order of accuracy of an iterated deferred corrections algorithm of lentini and pereyra siam j numer math 14 1977 pp 91 111 is improved at the same time providing a realistic example for a theoretical framework developed by the author for proving accuracy results for deferred corrections

a variable order deferred correction algorithm for the numerical solution of nonlinear two point boundary value problems abstract we develop two variable order deferred correction algorithms based on finite difference methods for the approximate numerical solution of nonlinear two point boundary value problems the deferred corrections provide a powerful local error estimator and this estimate can be used to refine the mesh if need be in order to obtain a prescribed tolerance mesh refinement is particularly straightforward since our algorithms are truly one step in nature the approach described is particularly appropriate for problems where only low or engineering precision is required and for very large systems of equations where storage space is an extremely important consideration some numerical results are given and a comparison with the method developed by lentini and pereyra and with richardson extrapolation is made

the conditioning of linear boundary value problems investigation is made into the sensitivity of solutions of linear boundary value problems to perturbations of the boundary condition we derive a useful quantity to decide for well or ill conditioning from this it is deduced which kind of requirements the boundary conditions should meet in order to have a well conditioned problem it is shown that this quantity can fruitfully be used to explain why e g the multiple shooting technique is stable in contrast to the single shooting one for certain well posed problems having solutions with different growth behavior the results are sustained by a number of examples

comparison of a higher order method and the simple upwind and non monotone methods for singularly perturbed boundary value problems in this paper we will present a new scheme for discretization of singularly perturbed boundary value problems based on finite difference methods this method is a combination of simple upwind scheme and central difference method on a special non uniform mesh shishkin mesh for the space discretization numerical results show that the convergence of method is uniform with respect to singular perturbation parameter and has a higher order of convergence

a theoretical framework for proving accuracy results for deferred corrections general techniques are described for proving accuracy results for deferred correction solutions to differential equations these techniques apply also to computational estimates of the local discretization error the proofs avoid the necessity of demonstrating the existence of asymptotic expansions of the global error in powers of some meshsize parameter

solid extensions of the cesaro operator on the hardy space h2 d abstract we introduce and study the largest banach space of analytic functions on the unit disc which is solid for the coefficient wise order and to which the classical cesaro operator c h 2 h 2 can be continuously extended while still maintaining its values in h 2 properties of this banach space h c e s 2 as well as a characterization of individual analytic functions which belong to h c e s 2 are presented in addition both the multiplier space of h c e s 2 and the spectrum of c h c e s 2 h c e s 2 are determined

rearrangements of functions on a locally compact abelian group and integrability of the fourier transform abstract we find in this paper the equimeasurable hulls and kernels of some function classes on a locally compact abelian group these classes consist of all functions for which the fourier transform belongs to a given lorentz space on the dual group different special cases of the problems considered in this paper have been originally studied by hardy littlewood hewitt ross cereteli and the author

the close relationships between methods for solving two point boundary value problems the well conditioning of a boundary value problem is shown to be related to bounding two quantities one involving the boundary conditions and the other involving the green s function in the case of a solution dichotomy they are related to known stability results these results easily explain why difficulties arise using superposition and reduced superposition it is shown how multiple shooting overcomes these difficulties by relating its matrix conditioning to the underlying boundary value problem some factorization methods for the multiple shooting method are considered these show the close relationships between multiple shooting the stabilized march and invariant imbedding and one factorization leads to an efficient new way to implement each of these three methods many of the results apply for collocation and finite difference methods too finally it is shown both theroetically and computationally when matrix compactification can lead to difficulty

salem sets in local fields the fourier restriction phenomenon and the hausdorff young inequality abstract we establish the existence of salem sets in the ring of integers of any local field and study the fourier restriction phenomenon on such sets optimal extension of the hausdorff young inequality initially attained for the torus by g mockenhaupt and w ricker is also established in the local field setting

on a fourth order finite difference method for singularly perturbed boundary value problems we present a fourth order finite difference method for singularly perturbed one dimensional reaction diffusion problem the problem is discretized using a bakhvalov type mesh we give a uniform convergence with respect to the perturbation parameter numerical examples are presented which demonstrate computationally the fourth order of the method

optimal rearrangement invariant range for hardy type operators we characterize in the context of rearrangement invariant spaces the optimal range space for a class of monotone operators related to the hardy operator the connection between optimal range and optimal domain for these operators is carefully analyzed

h k trichotomies and asymptotics of nonautonomous difference systems abstract discrete nonautonomous nonlinear systems possessing h k trichotomies are considered we specialize to the case of solutions moving inside and in a neighborhood of an invariant manifold of a special kind which we call h k hyperbolic the aulbach coppel knobloch transformation is then developed and used to state conditions for granting asymptotic equivalence of solutions our results generalize some of aulbach s results in which the manifold considered was made of stationary points and the system taken was autonomous

improvement of numerical solution of selfadjoint singular perturbation problems by incorporation of asymptotic approximations a numerical method for singularly perturbed two point boundary value problems for second order ordinary differential equations without a first derivative term arising in the study of chemical catalysis and michaelis menten process in biology is proposed in this method booster method an asymptotic approximation is incorporated into a suitable finite difference scheme to improve the numerical solution uniform error estimates are derived for this method when implemented in known difference schemes an improvement by a factor of n 1 can be obtained where is the small parameter and n is the order of the asymptotic approximation for a small amount of extra work numerical results are presented in support of this claim

numerical analysis of singularly perturbed nonlinear reaction diffusion problems with multiple solutions semilinear reaction diffusion two point boundary value problems with multiple solutions are considered here the second order derivative is multiplied by a small positive parameter and consequently these solutions can have boundary or interior layers a survey is given of the results obtained in our recent investigations into the numerical solution of these problems on layer adapted meshes

compactness properties of sobolev imbeddings for rearrangement invariant norms compactness properties of sobolev imbeddings are studied within the context of rearrangement invariant norms attention is focused on the extremal situation namely when the imbedding is considered as defined on its optimal sobolev domain with the range space fixed the techniques are based on recent results which reduce the question of boundedness of the imbedding to boundedness of an associated kernel operator of just one variable

a stability criterion for explicit boundary value runge kutta methods abstract a function characterizing the stability of explicit boundary value runge kutta methods for the solution of ordinary differential equations is introduced it can be used to design methods with better stability and also in the selection of mesh points it is based on a generalization of the algebraic stability criterion numerical examples demonstrating the use of the proposed function are given

another uniform convergence analysis technique of some numerical methods for parabolic singularly perturbed problems in this work we analyze the uniform convergence of some monotone finite difference schemes which are used to semidiscretize in space time dependent two dimensional singularly perturbed parabolic problems of convection diffusion or reaction diffusion type the analysis technique combines a suitable semidiscrete maximum principle joint to some techniques used in the study of the convergence of well known numerical schemes developed to solve elliptic singularly perturbed problems we focus our attention to standard central or upwind finite difference schemes defined on special nonuniform meshes of shishkin type nevertheless the technique can be easily extended to other classical discretization methods like for example fitted operator methods we prove that the stiff initial value problems resulting of the spatial semidiscretization processes have a unique solution which converges uniformly with respect to the singular perturbation parameter to obtain an efficient numerical algorithm such initial value problems are discretized by using appropriate time integrators here we have chosen the implicit euler method for doing this obtaining an unconditional and uniformly convergent numerical method in a simple way finally some numerical results are shown in order to illustrate the efficiency of the numerical methods according to the theoretical results

differentials of complex interpolation processes for kothe function spaces we continue the study of centralizers on kothe function spaces and the commutator estimates they generate see 29 our main result is that if x is a super reflexive kothe function space then for every real centralizer on x there is a complex interpolation scale of kothe function spaces through x inducing as a derivative up to equivalence and a scalar multiple thus in a loose sense all real centralizers can be identified with derivatives of complex interpolation processes we apply our ideas in an appendix to show for example that there is a twisted sum of two hilbert spaces which fails to be a umd space

uniform convergence of arbitrary order on nonuniform meshes for a singularly perturbed boundary value problem we consider the numerical solution of a singularly perturbed linear self adjoint boundary value problem assuming that the coefficients of the differential equation are smooth we construct and analyze finite difference methods that converge both with high order and uniformly with respect to the singular perturbation parameter the analysis is done on a locally quasiuniform mesh which permits its extension to the case of adaptive meshes which may be used to improve the solution numerical examples are presented to demonstrate the effectiveness of the method and its low computational cost the convergence obtained in practice satisfies the theoretical predictions

numerical solution of two point boundary value problems abstract this paper presents an original formulation of two point boundary value and eigenvalue problems expressed as a system of first order equations the fundamental difference between the new method and other methods based on a first order approach is the introduction of conditions of an integral character to supplement the simultaneous set of first order equations which are hence never regarded as an initial value problem the consideration of integral conditions leads to establish a class of linear multipoint schemes for the numerical solution of boundary value problems for ordinary differential equations furthermore the global character of the integral conditions nonlocality combined with the block structure of the system of algebraic equations allow dealing with stiff problems by means of the classical procedure of iterative refinement introduced by wilkinson the properties of the numerical schemes are illustrated by the solution of linear and nonlinear problems and by the accurate and efficient determination of some eigensolutions of a difficult problem of hydrodynamic stability the proposed method is conceptually simpler and numerically more convenient than existing initial value methods while still retaining all the advantages of a formulation based on a first order system

on the numerical integration of nonlinear two point boundary value problems using iterated deferred corrections part 2 the development and analysis of highly stable deferred correction formulae iterated deferred correction methods have been very widely used for the numerical solution of general nonlinear two point boundary value problems in ordinary differential equations however there

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

roughness of h k dichotomies abstract in this paper we prove a roughness theorem for a class of dichotomies of linear systems of differential equations the so called h k dichotomies these include a great variety of dichotomies among them the ordinary and exponential dichotomies exponential ordinary dichotomies levinson dichotomies for diagonal systems etc we prove that under general uniform conditions upon the functions h and k these dichotomies are not destroyed by integrable perturbations this result generalizes the property of roughness of ordinary dichotomies under integrable perturbations

a reduction theorem for systems of differential equations with impulse effect in a banach space a reduction theorem for systems of differential equations with impulse effect at fixed moments in a banach space is proven this result allows one to substantially reduce the given system to a much simpler one

high order methods for elliptic and time dependent reaction diffusion singularly perturbed problems the objective of this paper is to construct some high order uniform numerical methods to solve linear reaction diffusion singularly perturbed problems first for 1d elliptic problems based on the central finite difference scheme a new hodie method is defined on a piecewise uniform shishkin mesh using this hodie scheme jointly with a two stage sdirk method we solve a 1d parabolic singularly perturbed problem in both cases we prove that the methods are third order uniform convergent in the maximum norm finally for a 2d parabolic problem of the same type we show numerically that the combination of the hodie scheme with a fractional step rk method gives again a third order uniform convergent scheme

collocation for singular perturbation problems i first order systems with constant coefficients the application of collocation methods for the numerical solution of singularly perturbed ordinary differential equations is investigated collocation at gauss radau and lobatto points is considered for both initial and boundary value problems for first order systems with constant coefficients particular attention is paid to symmetric schemes for boundary value problems these problems may have boundary layers at both interval ends br our analysis shows that certain collocation schemes in particular those based on gauss or lobatto points do perform very well on such problems provided that a fine mesh with steps proportional to the layers width is used in the layers only and a coarse mesh just fine enough to resolve the solution of the reduced problem is used in between ways to construct appropriate layer meshes are proposed of all methods considered the lobatto schemes appear to be the most promising class of methods as they essentially retain their usual superconvergence power for the smooth reduced solution whereas gauss legendre schemes do not br we also investigate the conditioning of the linear systems of equations arizing in the discretization of the boundary value problem for a row equilibrated version of the discretized system we obtain a pleasantly small bound on the maximum norm condition number which indicates that these systems can be solved safely by gaussian elimination with scaled partial pivoting

on the linearization theorem of fenner and pinto this paper reports an improvement of the linearization theorem of fenner and pinto 1999 22 fenner and pinto presented a version of hartman s result they showed that there exists a one to one correspondence between solutions of the linear system and the nonlinear system moreover if h t x denotes the transformation then h t x x is uniformly bounded however no proof of the holder regularity of the transformation h t x appears in fenner and pinto 1999 22 the main objective in this paper is precisely to give a proof of the holder regularity of the transformation h t x namely we show that the conjugating function h t x in the hartman grobman theorem is always holder continuous and has holder continuous inverse moreover we weakened an important assumption in fenner and pinto 1999 22 fenner and pinto obtained the linearization theorem by setting that the whole linear system should satisfy is condition in this paper this assumption is reduced in fact it is enough to assume that the linear system partially satisfies is condition therefore we improve the linearization theorem of fenner and pinto

dichotomies and asymptotic formulas for the solutions of differential equations abstract the new concept of h k dichotomy is studied extensively and is used in asymptotic equivalence problems

stability of a finite difference discretization of a singular perturbation problem abstract a new higher order finite difference scheme is proposed for a linear singularly perturbed convection diffusion problem in one dimension it is shown how the theory of inverse monotone matrices the lorenz decomposition in particular can be applied to the stability analysis of the resulting linear system

defect correction for two point boundary value problems on nonequidistant meshes new finite difference formulae of arbitrary order are derived for the speciai class of second order two point boundary value problems y f x y x a x b variable mesh spacing is possible and the required accuracy is achieved under a very mild mesh condition a naturai defect correction framework is set up to compute the higher order approximations

on stability constants and condition numbers of discretisation methods discretizations lead to systems of algebraic equations of steadily growing size if the accuracy of the approximation is increasing inverse stability constants and condition numbers describe the sensitivity of the solution of these equations with respect to perturbations of the data starting from an abstract result in this paper the inverse stability constants of finite difference and collocation methods for bvodes as well as of ritz galerkin methods for elliptic variational problems are shown to converge to those of their continuous counterparts as the discretization becomes finer and finer for two choices of basis systems the exact asymptotic behavior of the collocation method condition number is given

uniform fourth order difference scheme for a singular perturbation problem the numerical solution of a nonlinear singularly perturbed two point boundary value problem is studied the developed method is based on hermitian approximation of the second derivative on special discretization mesh numerical examples which demonstrate the effectiveness of the method are presented

high order parameter uniform numerical method for singular perturbation problems in this paper we systematically describe how to derive a method of higher order for the numerical solution of singularly perturbed ordinary differential equations first we apply this idea to derive a fourth order method for a self adjoint singularly perturbed two point boundary value problem this method is uniformly convergent on a piecewise uniform mesh of shishkin type after we have developed and analyzed a fourth order method we explain with appropriate details how can one obtain the methods of order higher than four which looks straightforward but has not been seen in the literature so far besides these the fourth order e uniformity in the theoretical estimate has been justified by some numerical experiments

a higher order uniformly convergent method with richardson extrapolation in time for singularly perturbed reaction diffusion parabolic problems a b s t r a c t in this paper we are interested in solving efficiently an initial boundary value singularly perturbed time dependent problem of reaction diffusion type on a priori special mesh we construct a high order uniformly convergent finite difference scheme which combines the implicit euler method to discretize in time together with the richardson extrapolation technique and a hodie scheme to discretize in space the analysis of the uniform convergence splits completely the contribution to the global error of both the time and the space discretizations we show numerical results for different test problems confirming in practice the order of uniform convergence proved

dichotomy and existence of periodic solutions of quasilinear functional differential equations abstract using krasnoselskii s fixed point theorem functional analysis methods and dichotomy theory we study the existence and uniqueness of the periodic solutions of integro differential equations with bounded and unbounded delays

on h k manifolds with asymptotic phase abstract we introduce the notion of h k manifolds and give conditions under which the property of being a manifold with asymptotic phase holds these conditions allow the construction of a transformation of the variational equation of a nonlinear nonautonomous systems with h k dichotomies into an almost decoupled quasilinear system the class of admissiblehandkfunctions is also briefly analyzed