a uniformly convergent difference scheme for a semilinear singular perturbation problem we present a difference scheme for solving a semilinear singular perturbation problem with any number of turning points of arbitrary orders it is shown that a solution of the scheme converges uniformly in a perturbation parameter to that of the continuous problem

difference schemes for the singularly perturbed sobolev periodic boundary problem a finite difference method is suggested and analyzed for evolution equations of sobolev type in a single space variable and with boundary layers the convergence and error estimates for an exponentially fitted difference scheme in an equidistant mesh are obtained the numerical methods discussed here are extremely robust in the sense that they have good convergence properties not only for small values of the perturbation parameter but also for moderate and large values

an analysis of the box and trapezoidal schemes for linear singularly perturbed boundary value problems stability and convergence results are derived for the box and trapezoidal schemes applied to boundary value problems for linear singularly perturbed first order systems of o d e s without turning points

singularly perturbed finite element methods this paper introduces a new piecewise linear finite element which is designed to handle singularly perturbed ordinary differential equations both pointwise and global estimates which are independent of the perturbation parameter are obtained

a boundary value technique for singular perturbation problems abstract a new way to solve singular perturbation problems is introduced it is designed for the practicing engineer or applied mathematician who needs a practical tool for these problems easy to use modest problem preparation and ready computer implementation as with other methods the original problem is partitioned into inner and outer solution differential equation systems however asymptotic solution techniques are not employed the method is distinguished by the following facts the inner solution problem is solved as a two point boundary problem where the terminal boundary conditions are supplied by the solution of the outer solution problem in turn the outer solution differential equations are solved as an initial value system where the initial conditions are supplied by the inner solution at its terminal point the method is iterative on the terminal point of the inner solution problem three numerical examples are included

experiments with a local adaptive grid h refinement for the finite difference solution of bvps in singularly perturbed second order odes abstract despite many years of the development of adaptive grid methods there is still a need for re evaluation tuning and comparisons of the various approaches to grid adaptation in this study we seek to identify the most reliable and computationally inexpensive variant of the iterative grid refinement de refinement algorithm for the finite difference solution of singularly perturbed bvps in second order odes that is also easily extendable to evolutionary pdes conventional three point discretisations of the first and second derivatives are used errors of the discrete solution and its first derivative are simultaneously controlled the grid refinement is accomplished by adding nodes at inter nodal positions and de refinement by removing previously added nodes in reverse order computational experiments are performed for 15 examples of odes assuming several alternative choices of the a posteriori error estimators grid refinement indicators and regridding strategies the most satisfactory results are obtained by combining deferred approach to error estimation based on higher order reference derivative approximations for truncation error evaluation interpolation based indicator for grid refinement and regridding strategy that uses the mean indicator value as the indicator threshold for refinement

nonstandard finite difference equations for odes and 1 d pdes based on piecewise linearization a method for the solution of initial and boundary value problems in nonlinear ordinary differential equations and for one dimensional partial differential equations which provides c 1 solutions is presented the method is based on the linearization of the differential equation in intervals which contain only two grid points and provides three point nonstandard finite difference equations for the nodal amplitudes the method is applied to steady reaction diffusion equations two point singularly perturbed boundary value problems and the steady burgers equation and compared with standard finite difference and finite element formulations for one dimensional partial differential equations the temporal derivatives are first discretized and the resulting ordinary differential equation accounts for both the temporal and spatial stiffnesses and is solved by means of piecewise linearization since the linearization includes a jacobian matrix it may be easily employed to refine the mesh where steep gradients occur

chord arc surfaces with small constant ii good parameterizations abstract in a companion paper we introduced a notion for hypersurfaces in r d 1 that is analogous to the chord arc condition with small constant for planar curves in this paper we address the problem of finding good parameterizations for these surfaces in particular we show that these surfaces must be homeomorphic to r d and we build parameterizations with l p bounds on the gradient and the gradient of the inverse with p as the chord arc constant goes to 0 when d 2 we use the uniformization theorem to find a quasisymmetric parameterization we do not know how to build bilipschitz parameterizations when d 2 or quasisymmetric parameterizations when d 3 the author conjectures that they do not exist in general however we can build bilipschitz reflections in all dimensions and these reflections will satisfy certain extra estimates in terms of carleson measures

a spline method for second order singularly perturbed boundary value problems we consider a difference scheme based on cubic spline in compression for second order singularly perturbed boundary value problem of the form ey p x y q x y r x y a 0 y b 1 the method is shown to have second and fourth order convergence depending on the choice of parameters 1 and 2 involved in the method the method is tested on an example and the results found to be in agreement with the theory

numerical patching method for singularly perturbed two point boundary value problems using cubic splines a numerical patching method is presented for solving singularly perturbed two point boundary value problems with the boundary layer at one end left or right point the method is distinguished by the following fact the original singularly perturbed two point boundary value problem is divided into two problems namely inner and outer region problems the terminal boundary condition is obtained from the solution of the reduced problem using general stretching transformation a modified inner region problem is constructed then both inner region problem and outer region problems are solved as two point boundary value problems by employing cubic splines several linear and non linear problems are solved to demonstrate the applicability of the method

a note on the computation of the outflow derivative for singular perturbation problems a method for computing the outflow derivative of singular perturbation problems is derived the method essentially involves integrating the differential equation numerically and numerical experiments indicate the technique is very accurate when the problem is convection dominated

quantitative stability for the first dirichlet eigenvalue in reifenberg flat domains in rn abstract in this paper we prove that if and are close enough for the complementary hausdorff distance and their boundaries satisfy some geometrical and topological conditions then 1 1 c n where 1 resp 1 is the first dirichlet eigenvalue of the laplacian in resp and is the lebesgue measure of the symmetric difference here the constant 1 could be taken arbitrary close to 1 but strictly less and c is a constant depending on a lot of parameters including dimension n and some geometric properties of the domains

a consistent and accurate procedure for obtaining difference equations from differential equations a general consistent and accurate procedure for obtaining difference equations from differential equations is presented and illustrated by several examples the procedure is an extension of the integral method and incorporates what is known of the solution to the differential equation in the formation of the corresponding difference equation for the same accuracy large increases in grid size over that obtained by standard taylor s series approximations are possible both boundary value and initial value problems are treated

dahlberg s theorem in higher co dimension abstract in 1977 the celebrated theorem of b dahlberg established that the harmonic measure is absolutely continuous with respect to the hausdorff measure on a lipschitz graph of dimension n 1 in r n and later this result has been extended to more general non tangentially accessible domains and beyond in the present paper we prove the first analogue of dahlberg s theorem in higher co dimension on a lipschitz graph of dimension d in r n d n 1 with a small lipschitz constant we construct a linear degenerate elliptic operator l such that the corresponding harmonic measure l is absolutely continuous with respect to the hausdorff measure on more generally we provide sufficient conditions on the matrix of coefficients of l which guarantee the mutual absolute continuity of l and the hausdorff measure

uniformization of two dimensional metric surfaces we establish uniformization results for metric spaces that are homeomorphic to the euclidean plane or sphere and have locally finite hausdorff 2 measure applying the geometric definition of quasiconformality we give a necessary and sufficient condition for such spaces to be qc equivalent to the euclidean plane disk or sphere moreover we show that if such a qc parametrization exists then the dilatation can be bounded by 2 as an application we show that the euclidean upper bound for measures of balls is a sufficient condition for the existence of a 2 qc parametrization this result gives a new approach to the bonk kleiner theorem on parametrizations of ahlfors 2 regular spheres by quasisymmetric maps

method of reduction of order for solving singularly perturbed two point boundary value problems in this paper a method of reduction of order is proposed for solving singularly perturbed two point boundary value problems with a boundary layer at one end point it is distinguished by the following fact the original singularly perturbed boundary value problem is replaced by a pair of initial value problems classical fourth order runge kutta method is used to solve these initial value problems several linear and non linear singular perturbation problems have been solved and the numerical results are presented to support the theory

bi lipschitz parameterization of surfaces we consider surfaces z homeomorphic to the plane with complete possibly singular riemannian metrics if we have z k 0 and c 0 this result implies a conjecture by j fu

an error analysis for the finite element method applied to convection diffusion problems abstract this paper analyzes the finite element method applied to a convection diffusion model problem linear elements are used for the trial space the error is measured in a norm closely related to the l sup p norm when the test space is composed of linear elements with parabolic upwinding the method is shown to be optimal when the input data is piecewise smooth a condition which is usually observed in practice without these smoothness assumptions the method is shown to be non optimal even if the class of test spaces is extended to include any elements which have a shape independent of the mesh size author

heterogeneous domain decomposition for singularly perturbed elliptic boundary value problems a heterogeneous domain decomposition method is presented for the numerical solution of singularly perturbed elliptic boundary value problems the method which is parallelizable at various levels uses several ideas of asymptotic analysis the subdomains match the domains of validity of the local inner and outer asymptotic expansions and cut off functions are used to match solutions in neighboring subdomains the positions of the interfaces as well as the mesh widths depend on the small parameter varepsilon on the subdomains iterative solution techniques are used which may vary from one subdomain to another the global convergence rate depends on varepsilon it generally increases like some power of log varepsilon 1 1 as varepsilon downarrow 0 the method is illustrated on several two dimensional singular perturbation problems

quasiconformal planes with bi lipschitz pieces and extensions of almost affine maps abstract a quasiplane f v is the image of an n dimensional euclidean subspace v of r n 1 n n 1 under a quasiconformal map f r n r n we give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi lipschitz n manifold and for a quasiplane to have big pieces of bi lipschitz images of r n one main novelty of these results is that we analyze quasiplanes in arbitrary codimension n n to establish the big pieces criterion we prove new extension theorems for almost affine maps which are of independent interest this work is related to investigations by tukia and vaisala on extensions of quasisymmetric maps with small distortion

a seventh order numerical method for singular perturbation problems in this paper a seventh order numerical method is presented for solving singularly perturbed two point boundary value problems with a boundary layer at one end point the two point boundary value problem is transformed into general first order ordinary differential equation system a discrete approximation of a seventh order compact difference scheme is presented for the first order system an asymptotically equivalent first order equation of the original singularly perturbed two point boundary value problem is obtained from the theory of singular perturbations it is used in the seventh order compact difference scheme to get a two term recurrence relation and is solved several linear and nonlinear singular perturbation problems have been solved and the numerical results are presented to support the theory

a review on singularly perturbed differential equations with turning points and interior layers singular perturbation problems with turning points arise as mathematical models for various physical phenomena the problem with interior turning point represent one dimensional version of stationary convection diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin boundary turning point problems on the other hand arise in geophysics and in modeling thermal boundary layers in laminar flow in this paper we review some existing literature on asymptotic and numerical analysis of singularly perturbed turning point and interior layer problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye towards the goal of developing general methods to solve such problems since major work in this area started after 1970 so this paper limits its coverage to the work done by numerous researchers between 1970 and 2011

effective reifenberg theorems in hilbert and banach spaces a famous theorem by reifenberg states that closed subsets of mathbb r n that look sufficiently close to k dimensional at all scales are actually c 0 gamma equivalent to k dimensional subspaces since then a variety of generalizations have entered the literature for a general measure mu in mathbb r n one may introduce the k dimensional jones beta numbers of the measure where beta k mu x r quantifies on a given ball b r x how closely in an integral sense the support of the measure is to living inside a k dimensional subspace recently it has been proven that if these beta numbers satisfy the uniform summability estimate int 0 2 beta k mu x r 2 frac dr r m then mu must be rectifiable with uniform measure bounds note that one only needs the square of the beta numbers to satisfy the summability estimate this power gain has played an important role in the applications for instance in the study of singular sets of geometric equations one may also weaken these pointwise summability bounds to bounds which are more integral in nature the aim of this article is to study these effective reifenberg theorems for measures in a hilbert or banach space for hilbert spaces we see all the results from mathbb r n continue to hold with no additional restrictions for a general banach spaces we will see that the classical reifenberg theorem holds and that a weak version of the effective reifenberg theorem holds in that if one assumes a summability estimate int 0 2 beta k mu x r 1 frac dr r m without power gain then mu must again be rectifiable with measure estimates improving this estimate in order to obtain a power gain turns out to be a subtle issue for k 1 we will see for a uniformly smooth banach space that if int 0 2 beta 1 mu x r alpha frac dr r m alpha 2 where alpha is the smoothness power of the banach space then mu is again rectifiable with uniform measure estimates

asymptotic and numerical analysis of singular perturbation problems a survey this paper is intended to be a brief survey of the asymptotic and numerical analysis of singular perturbation problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye toward the goal of developing general methods to solve such problems a summary of the results of some recent methods is presented and this leads to conclusions and recommendations about what methods to use on singular perturbation problems finally some areas of current research are indicated a bibliography of about 130 items is provided

large time behaviors of solutions to an inflow problem in the half space for a one dimensional system of compressible viscous gas the inflow problem for a one dimensional compressible barotropic flow on the half line r 0 is investigated not only classical waves but also the new wave which is called the boundary layer solution arise large time behaviors of the solutions to be expected have been classified in terms of the boundary values by a matsumura inflow and outflow problems in the half space for a one dimensional isentropic model system of compressible viscous gas to appear in proceedings of ims conference on differential equations from mechanics hong kong 1999 in this paper we give the rigorous proofs of the stability theorems on both the boundary layer solution and a superposition of the boundary layer solution and the rarefaction wave

perturbation analysis of the limit cycle of the free van der pol equation a power series expansion in the damping parameter varepsilon of the limit cycle u t varepsilon of the free van der pol equation ddot u varepsilon u 2 1 dot u u 0 is constructed and analyzed coefficients in the expansion are computed up to o varepsilon 24 in exact rational arithmetic using the symbolic manipulation system macsyma and up to o varepsilon 163 using a fortran program the series is analyzed using pade approximants the convergence of the series for the maximum amplitude of the limit cycle is limited by two pairs of complex conjugate singularities in the complex varepsilon plane these singularities are the same as those which limit the convergence of the series expansion of the frequency of the limit cycle a new expansion parameter is introduced which maps these singularities to infinity and leads to a new expansion for the amplitude which converges for all real values of varepsilon amplitudes computed from this transformed series agree very we

numerical techniques for a class of singular perturbation problems two numerical techniques for solving a class of linear singularly perturbed two point boundary value problems with a boundary layer on the left end of the underlying interval are presented these are distinguished by the following fact the original second order differential equation is replaced by an approximate first order differential equation with a small deviating argument and then solved efficiently by employing finite differences to obtain the three term recurrence relationship the proposed techniques are iterative on the deviating argument some numerical experiments have been included to demonstrate the applicability of these techniques

new monotonicity formulas for ricci curvature and applications i we prove three new monotonicity formulas for manifolds with a lower ricci curvature bound and show that they are connected to rate of convergence to tangent cones in fact we show that the derivative of each of these three monotone quantities is bounded from below in terms of the gromov hausdorff distance to the nearest cone the monotonicity formulas are related to the classical bishop gromov volume comparison theorem and perelman s celebrated monotonicity formula for the ricci flow we will explain the connection between all of these moreover we show that these new monotonicity formulas are linked to a new sharp gradient estimate for the green function that we prove this is parallel to the fact that perelman s monotonicity is closely related to the sharp gradient estimate for the heat kernel of li yau in cm4 one of the monotonicity formulas is used to show uniqueness of tangent cones with smooth cross sections of einstein manifolds finally there are obvious parallelisms between our monotonicity and the positive mass theorem of schoen yau and witten

harmonic measure on locally flat domains we will review work with tatiana toro yielding a characterizati on of those domains for which the harmonic measure has a density whose logarithm has vanishing mean oscillation

chord arc surfaces with small constant i abstract we introduce a notion for hypersurfaces in r d 1 that is analogous to the chord arc condition with small constant for curves in the plane we give various equivalent operator theoretic function theoretic and geometrical characterizations of these surfaces we not know any nice potential theoretic characterizations in a companion paper we address the issue of finding good parameterizations for these surfaces

an exponentially fitted finite difference method for singular perturbation problems in this paper an exponentially fitted finite difference method is presented for solving singularly perturbed two point boundary value problems with the boundary layer at one end left or right point a fitting factor is introduced in a tridiagonal finite difference scheme and is obtained from the theory of singular perturbations thomas algorithm is used to solve the system the stability of the algorithm is investigated several linear and nonlinear problems are solved to demonstrate the applicability of the method it is observed that the present method approximates the exact solution very well

numerical treatment of singularly perturbed two point boundary value problems we propose a method for numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval this is a practical method and can be easily implemented on a computer the original problem is divided into inner and outer region differential equation systems the reduced problem is solved to obtain the terminal boundary condition then a new inner region problem is created and solved as a two point boundary value problem tpbvp in turn the outer region problem is also solved as a tpbvp both these tpbvps are efficiently treated by employing a slightly modified classical finite difference scheme coupled with discrete invariant imbedding algorithm to obtain the numerical solutions the stability of some recurrence relations involved in the algorithm is investigated the proposed method is iterative on the terminal point some numerical examples are included and the computational results are compared with exact solutions it is observed that the accuracy predicted can always be achieved with very little computational effort

analysis aspects of willmore surfaces a new formulation for the euler lagrange equation of the willmore functional for immersed surfaces in m is given as a nonlinear elliptic equation in divergence form with non linearities comprising only jacobians letting vec h be the mean curvature vector of the surface our new formulation reads mathcal l vec h 0 where mathcal l is a well defined locally invertible self adjoint elliptic operator several consequences are studied in particular the long standing open problem asking for a meaning to the willmore euler lagrange equation for immersions having only l 2 bounded second fundamental form is now solved the regularity of weak willmore immersions with l 2 bounded second fundamental form is also established its proof relies on the discovery of conservation laws which are preserved under weak convergence a weak compactness result for willmore surfaces with energy less than 8 the li yau condition ensuring the surface is embedded is proved via a point removability result established for wilmore surfaces in m thereby extending to arbitrary codimension the main result in ks3 finally from this point removability result the strong compactness of willmore tori below the energy level 8 is proved both in dimension 3 this had already been settled in ks3 and in dimension 4

the method of inner boundary condition a new approach for solving singular perturbation problems abstract the object of this paper is to present a new approach based on the method of inner boundary condition for solving singular perturbation problems the original problem is partitioned into inner and outer region differential equation systems asymptotic expansion is used to obtain the terminal boundary condition using an appropriate transformation a new inner region problem is obtained and solved as a two point boundary value problem the derivative boundary condition at the terminal point is then derived from the solution of the inner region problem using this condition the outer region problem is efficiently solved by employing the classical finite difference scheme the proposed method is iterative on the terminal point some numerical examples have been solved to demonstrate the efficiency of the method

the analytical and approximate solutions of ϵy yy abstract the analytical and approximate soluions of y yy are developed and compared and the analytical solution is evaluated numerically the analytical soluion has several interesting properties i it possesses three different types of solutions constant exponential and tangent ii for a two point boundary value problem it has an undecidability aspect relative to which solution type holds iii for the tangent solution type an infinite number of solutions exist both continuous and discontinuous iv even with the analytical solution known its numerical evaluation can still be difficult to generate

numerical methods for singular perturbation problems consider the two point boundary value problem for a stiff system of ordinary differential equations an adaptive method to solve these problems even when turning points are present is discussed

adaptive solution of bvps in singularly perturbed second order odes by the extended numerov method combined with an iterative local grid h refinement the extended numerov scheme of chawla is one of high order compact finite difference discretisations applicable to bvps in second order odes and to pdes the use of such discretisations in connection with adaptive grids remains largely unexplored in order to make up this deficiency we study the combination of the extended numerov discretisation with the iterative local adaptive grid h refinement previously tested by the present author at the assumption of the conventional finite difference discretisation a detailed formalism enabling a posteriori error analysis of the extended numerov scheme is developed together with several alternative choices for error estimators grid refinement indicators and regridding strategies the adaptive algorithms obtained are examined in calculations on a collection of 15 examples of singularly perturbed odes including equations relevant for electrochemistry the most satisfactory algorithm involves deferred approach to error estimation based on virtual grid coarsening for truncation error evaluation local discretisation errors as refinement indicators and regridding strategy that uses the mean indicator values as the indicator thresholds for refinement the extended numerov scheme proves more efficient than the conventional discretisation when used together with the iterative h refinement

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

numerical integration method for general singularly perturbed two point boundary value problems in this paper a numerical integration method is presented for solving general singularly perturbed two point boundary value problems the original second order differential equation is replaced by an approximate first order differential equation with a small deviating argument then the trapezoidal formula is used to obtain the three term recurrence relationship the proposed method is iterative on the deviating argument to demonstrate the applicability of the method we have solved several model linear and non linear examples with left end boundary layer or right end boundary layer or an internal layer or two boundary layers and presented the computational results

singular perturbation analysis of boundary value problems for differential difference equations a study is made of a class of boundary value problems for linear second order differential difference equations in which the highest order derivative is multiplied by a small parameter consideration of the exact solution of simple model equations provides insight into the appropriate use of singular perturbation techniques for more general problems the resulting analysis leads to several novel features which are not present in problems without differences

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

analysis of difference schemes for a stationary shock problem we consider the limit function u of the solutions u varepsilon of a singularly perturbed boundary value problem in general u will be discontinuous two difference approximations of the boundary value problem are studied and the discrete values for varepsilon 0 h to 0 are compared with u error estimates of order o h and o h 2 are proved and the sharpness of numerical layers is established

non standard difference schemes for singular perturbation problems revisited non standard finite difference techniques for solving various linear two point singularly perturbed boundary value problems are considered they consist of replacing the original differential equation by a related differential equation with a small deviating argument this related problem is then discretized using finite differences this discretized problem is solved using standard matrix methods the accuracy and robustness of these numerical schemes are then carefully investigated and compared to standard centered finite differences in particular earlier claims for the superiority of such schemes over standard finite differences are shown to be invalid

nonlinear singular perturbation problems and one sided difference schemes we consider the boundary value problem varepsilon y a y y b x y f x 1 leqq x leqq 1 for y 1 and y 1 given and b x 0 equiv 0 b y x y geqq 0 we construct a simple difference scheme approximating varepsilon u xx a u u x b x u f x u t for t geqq 0 1 leqq x leqq 1 with the same boundary conditions and an arbitrary initial guess as the numerical artificial time approaches infinity we prove convergence to a unique steady state solution as the mesh size delta x and the parameter varepsilon approach zero in any order we prove convergence to the solution of the corresponding o d e we use one sided differences to approximate the transport term the method applies equally well to multidimensional analogues we present numerical results verifying the theory

monotone difference schemes for singular perturbation problems we consider numerical solutions of the boundary value problem varepsilon y f y b x y 0 0 leqq x leqq 1 y 0 a y 1 b varepsilon 0 b y geqq delta 0 with monotone difference schemes on nonuniform grids we prove general convergence results and show that the engquist osher monotone scheme will reproduce essential properties of the true solution for any grid for the inversion of the nonlinear scheme we suggest implicit time relaxation with variable time steps

one sided difference approximations for nonlinear conservation laws we analyze one sided or upwind finite difference approximations to hyperbolic partial differential equations and in particular nonlinear conservation laws second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions we show that no such stable approximation of order higher than two is possible these one sided schemes have desirable properties for shock calculations we show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance new and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles existence uniqueness and stability of discrete shocks numerical examples are given

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

a numerical integration method for solving singular perturbation problems we present a numerical integration method for solving linear singularly perturbed two point boundary value problems with a boundary layer on the left end of the underlying interval we replace the original singular perturbation problem by an approximate first order differential equation with a small deviating argument we use simpson s formula for the numerical integration of the first order differential equation with a small deviating argument and then use taylor series and finite differences to obtain a three term recurrence relationship the proposed method is iterative on the deviating argument some numerical experiments have been included to demonstrate the applicability of the method

a booster method for singular perturbation problems arising in chemical reactor theory a numerical method for singularly perturbed two point boundary value problems bvps for second order ordinary differential equations odes arising in chemical reactor theory is proposed in this method an asymptotic approximation is incorporated into a suitable finite difference scheme to improve the numerical solution uniform error estimates are derived when implemented in known difference schemes numerical results are presented in support of this claim

energy quantization for willmore surfaces and applications we prove a bubble neck decomposition together with an energy quantization result for sequences of willmore surfaces into r m with uniformly bounded energy and nondegenerating conformal type we deduce the strong compactness of willmore closed surfaces of a given genus modulo

a smooth locally analytical technique for singularly perturbed two point boundary value problems a locally analytical method for singularly perturbed two point boundary value problems with internal and boundary layers and with turning points is presented the method is based on the linearization of ordinary differential equations in nonoverlapping intervals and results in linear constant coefficients ordinary differential equations which can be integrated analytically thus yielding piecewise analytical solutions by imposing continuity conditions at the end points of each interval plus a smoothness condition at the common end point of two adjacent intervals a global smooth solution is obtained the accuracy of the globally smooth locally analytical method is assessed by comparisons with exact and approximate solutions of several singularly perturbed problems with internal and boundary layers it is shown that the smooth locally analytical method is more precise than second order accurate finite difference discretizations it is also shown that the accuracy of the smooth locally analytical method depends on the kind of nonlinearity and inhomogeneities of singularly perturbed ordinary differential equations but is always higher than that of exponentially fitted techniques based on the local solution of advection diffusion operators

a highly anisotropic nonlinear elasticity model for vesicles ii derivation of the thin bilayer bending theory we study the thin shell limit of the nonlinear elasticity model for vesicles introduced in part i we consider vesicles of width 2 varepsilon downarrow 0 with elastic energy of order varepsilon 3 in this regime we show that the limit model is a bending theory for generalized hypersurfaces namely co dimension one oriented varifolds without boundary up to a positive factor the limit functional is the willmore energy in the language of it gamma convergence we establish a compactness result a lower bound result and the matching upper bound in the smooth case

shooting method for the solution of singularly perturbed two point boundary value problems having less severe boundary layer singularly perturbed two point boundary value problems bvps having less severe boundary layers are considered a method is presented in this paper to solve these problems numerically the domain of definition of the differential equation is divided into two subintervals in one subinterval inner region an initial value problem ivp obtained from the given bvp is solved by an exponentially fitted difference efd scheme whereas a classical upwind scheme is used to solve the bvp in the other subinterval outer region in order to obtain initial and boundary conditions an asymptotic approximation is used error estimates are derived for the numerical solution the implementation of the method on parallel architectures is discussed numerical results are also presented in support of the proposed method

an analysis of a uniformly accurate difference method for a singular perturbation problem it will be proven that an exponential tridiagonal difference scheme when applied with a uniform mesh of size h to euxx b x ux f x for o o b and f smooth e in 0 1 and u o and u i given is uniformly second order accurate i e the maximum of the errors at the grid points is bounded by ch2 with the constant c independent of h and e this scheme was derived by el mistikawy and werle by a c patching of a pair of piecewise constant coefficient approximate differential equations across a common grid point the behavior of the approximate solution in between the grid points will be analyzed and some numerical results will also be given

a rational spectral collocation method for third order singularly perturbed problems a new numerical method is developed for solving a class of third order singularly perturbed boundary value problems first of all the given problem is transformed into a system of two ordinary differential equations odes subject to suitable initial and boundary conditions then the rational spectral collocation method in barycentric form with sinh transformation is applied to solve the system of odes according to the asymptotic analysis the location and width of boundary layer of the given problem which are chosen as parameters in the sinh transformation can be determined ample numerical experiments are presented to illustrate the computational efficiency and accuracy of the our method

on some difference schemes for singular singularly perturbed boundary value problems singularly perturbed boundary value ordinary differential problems are considered where the problem defining the reduced solution is singular for numerical approximation families of symmetric difference schemes which are equivalent to certain collocation schemes based on gauss and lobatto points are used convergence results previously obtained for the regular singularly perturbed case are extended while gauss schemes are extended with no change lobatto schemes require a small modification in the mesh selection procedure with meshes as prescribed in the text highly accurate solutions can be obtained with these schemes for singular singularly perturbed problems at a very reasonable cost this is demonstrated by examples

an asymptotic numerical method for singularly perturbed robin problems i in this paper singularly perturbed robin type problem for second order ordinary differential equations odes is considered here an asymptotic numerical method is suggested for this type of problems in this method an asymptotic approximate solution and solution obtained by a numerical method which involves an exponentially fitted difference efd scheme are combined suitably to yield solution of the present problem more precisely a transition point is chosen inside the interval of integration where the solution of the reduced problem is evaluated which will be taken as a boundary value for the boundary layer region inner region problem we perform iteration here by moving the transition point towards right hand side slowly until the solution gets stabilized in the outer region rest of the interval the solution of the reduced problem is taken as an approximation to the original problem error estimates are provided and numerical examples are presented

rectifiable sets and the traveling salesman problem let k c c be a bounded set in this paper we shall give a simple necessary and sufficient condit ion for k to lie in a rectifiable curve we say that a set is a rectifiable curve if it is the image of a finite interval under a lipschitz mapping recall that for a connected set f c c f is a rectifiable curve not necessarily simple if and only if l f where l denotes one dimensional hausdorff measure this classical result follows from the fact that on any finite graph there is a tour which covers the entire graph and which crosses each edge but not necessarily each vertex at most twice if k is a finite set then we are essentially reduced to the classical traveling salesman problem tsp compute the length of the shortest hami l ton ian cycle which hits all points of k this is the same up to a constant multiple as asking for the inf imum of l f where f is a curve k c f such a f is called a spanning tree in tsp theory for infinite sets k we cannot hope in general to have k be a subset of a jordan curve what we should therefore look at is connected sets which conta in k let fmi n be the shortest minimal spanning tree then we cannot possibly solve our problem for sets k of infinite cardinality if we cannot find f i f c o fmin for any finite set k here and throughout the paper c co c1 c o etc denote various universal constants while there are several algorithms for computing l fml these algorithms work for finite graphs satisfying the triangle inequality and do not use the euclidean properties of k see 13 for an excellent discussion of some of these algorithms therefore these methods cannot solve our problem for general infinite k we present a method which is a minor modification of a well known algorithm farthest insert ion see 13 which yields a f with i f c o l fmi the farthest insert ion algorithm has been extensively studied with large numerical calculations on computers and is experimentally good in the sense that the f produced satisfy i f c o l f i for all examples which have