exponentially fitted spline in compression for the numerical solution of singular perturbation problems abstract some exponentially fitted difference schemes for singularly perturbed two point boundary value problems are derived using spline in compression these schemes are second order accurate numerical examples are given in support of the theoretical results

third order variable mesh cubic spline methods for singularly perturbed boundary value problems abstract third order variable mesh methods based on cubic spline approximation for nonlinear singularly perturbed boundary value problems of the form ey f x y y a y b are presented the convergence analysis is given and the method is shown to have third order convergence several tet examples are solved to demonstrate the efficiency of the method

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

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

an adaptive uniformly convergent numerical method for a semilinear singular perturbation problem a singularly perturbed semilinear two point boundary value problem is considered without any restriction on its turning points a difference scheme is presented for solving this problem on an arbitrary locally quasi uniform mesh it is shown that the solution of the scheme is first order accurate uniformly in the perturbation parameter in a discrete l 1 norm numerical results are presented for an adaptive method based on the scheme

digital image authentication and recovery recently several semi fragile watermarking approaches with the additional capability of image recovery have been proposed however the security robustness and image recovery aspect of these approaches have certain shortcomings in this paper a novel semi fragile watermarking framework using integer transform based information embedding and extraction is proposed which allows accurate authentication and recovery of the image it is based on integer wavelet transform with improved security against collage attack enhanced robustness and capability of producing better quality recovered image security is enhanced by correlating the to be embedded watermark with the approximation subband of wavelet transform similarly no unprotected area is left for attacks on the image either in spatial or transform domain robustness is enhanced by using the idea of embedding in largest coefficient inside a group and correlating it with the quantized version of the mean of the group in particular the recovery approach is improved by introducing lossless compression and bch coding of the integer dct based low pass version of the cover image itself alteration sensitivity is improved compared to traditional block based approaches and thus the proposed approach can concisely determine the regions where the integrity verification fails experimental comparisons with existing approaches validate the usefulness of the proposed multiple semi fragile watermarking approach

the wave equation approach for solving inverse eigenvalue problems of a multi connected region in r3 with robin conditions the asymptotic expansion for t of the trace of the wave kernel email protected t email protected u 1exp email protected 1 2 u where m u u 1 are the eigenvalues of the negative laplacian email protected email protected 3 b 1 x b 2 in the x 1 x 2 x 3 space where i 1 and

adaptive lossless steganographic scheme with centralized difference expansion in this paper a novel adaptive lossless data hiding scheme is presented that is capable of offering greater embedding capacity than the existing schemes unlike the fixed hiding capacity each block provides in most of the currently available lossless data hiding approaches the proposed method utilizes a block based lossless data embedding algorithm where the quantity of the hidden information each block bears is variable to both reduce the image distortion and increase the hiding capacity the payload of each block depends on its cover image complexity due to the fact that schemes with difference expansion tend to damage the image quality seriously in the edge areas in the proposed scheme smoother areas are chosen to conceal more secret bits this way a better balance can be reached between the embedding ratio and the stego image quality in addition when recovered the cover image can came back to its old self to the last bit without any distortion at all experimental results as this paper will show have demonstrated that the proposed method is capable of hiding more secret data while maintaining imperceptible stego image quality degradation

region level image authentication using bayesian structural content abstraction image authentication ia verifies the integrity of image content by detecting malicious modifications a good ia system should be able to tolerate noncontent changing operations ncos robustly and detect content changing operations cos sensitively most existing ia methods realize either bit level or pixel level authentication thus they can tolerate only particular and limited kinds of ncos in this paper we propose an unsupervised region level ia scheme named bayesian structural content abstraction basca which is capable of tolerating a wide and dynamic range of ncos and can sensitively detect real cos we model image structural content using the net structured markov pixon random field ns mprf from which we derive the size controllable basca signature furthermore to support dynamic nco co partition we present an analogous mean shift algorithm to iteratively optimize the basca signature in the user defined nco space both theoretical analysis and experimental results demonstrate that our basca scheme has much less false positive and comparable false negative probability as compared to state of the art ia methods

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

an image fragile watermark scheme based on chaotic image pattern and pixel pairs fragile watermarking techniques for digital content have been studied in the past few years fragile watermarks are used to determine if a piece of watermarked digital content has been tampered and distinguish tampered areas from non tampered areas without referring to the original digital content in this paper a general framework for fragile watermark is proposed and then a novel fragile watermarking scheme for image authentication is presented the embedding process of fragile watermark starts from computing the difference image between the host image and its chaotic pattern followed by mapping the difference image into a binary image the binary image is then inserted into the least significant bit lsb bitplane of the host image in addition chaotic map is used to generate the chaotic pattern image which can be used as secret key to improve the security of watermark algorithm due to employing permutation transform and chaotic image pattern the corresponding position relation is broken between pixels in the watermarked image and the watermark simulation results and performance analysis show that the presented method is fast secure and capable of detecting and localizing modification

combinational image watermarking in the spatial and frequency domains in order to provide more watermarks and to minimize the distortion of the watermarked image a novel technique using the combinational spatial and frequency domains is presented in this paper the splitting of the watermark image into two parts respectively for spatial and frequency insertion relies on the user s preference and data importance experimental results provide the comparisons when different sized watermarks are embedded into a grayscale image the proposed combinational image watermarking possesses the following advantages more watermark data can be inserted into the host image so that the capacity is increased the splitting of the watermark into two parts makes the degree of protection double the splitting strategy can be designed even more complicated to be unable to compose furthermore to enhance robustness a random permutation of the watermark is used to defeat the attacks of signal processing such as image crops

higher dimensional inverse problem of the wave equation for a general multi connected bounded domain with a finite number of smooth mixed boundary conditions the spectral function m t j 1 exp it m j 1 2 where m j j 1 are the eigenvalues of the negative laplacian d 3 u 1 3 x u 2 in the x 1 x 2 x 3 space is studied for small t for a variety of bounded domains where

e uniformly convergent fitted methods for the numerical solution of the problems arising from singularly perturbed general ddes we consider some problems arising from singularly perturbed general differential difference equations first we construct in a new way and analyze a fitted operator finite difference method fofdm which is first order e uniformly convergent with the aim of having just one function evaluation at each step attempts have been made to derive a higher order method via shishkin mesh to which we refer as the fitted mesh finite difference method fmfdm this fmfdm is a direct method and e uniformly convergent with the nodal error as o n 2ln 2n which is an improvement over the existing direct methods i e 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 that lead the error as o n 1lnn where n denotes the total number of sub intervals of 0 1 comparative numerical results are presented in support of the theory

comparative study of singularly perturbed two point bvps via fitted mesh finite difference method b spline collocation method and finite element method abstract the objective of this paper is to present a comparative study of fitted mesh finite difference method b spline collocation method and finite element method for general singularly perturbed two point boundary value problems due to the small parameter the boundary layer arises we have taken a piecewise uniform fitted mesh to resolve the boundary layer and we have shown that fitted mesh finite difference method has uniform first order convergence b spline collocation method has almost second order uniform convergence and ritz galerkin method also has almost second order uniform convergence two test examples have been solved to compare the maximum absolute error and rate of convergence of the methods

a framework for evaluating the data hiding capacity of image sources an information theoretic model for image watermarking and data hiding is presented in this paper previous theoretical results are used to characterize the fundamental capacity limits of image watermarking and data hiding systems capacity is determined by the statistical model used for the host image by the distortion constraints on the data hider and the attacker and by the information available to the data hider to the attacker and to the decoder we consider autoregressive block dct and wavelet statistical models for images and compute data hiding capacity for compressed and uncompressed host image sources closed form expressions are obtained under sparse model approximations models for geometric attacks and distortion measures that are invariant to such attacks are considered

approximate method for the numerical solution of singular perturbation problems we present an approximate method for the numerical solution of linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval it is motivated by the asymptotic behavior of singular perturbation problems the original problem is divided into inner and outer region problems 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 in turn the outer region problem is also modified and the resulting problem is efficiently treated by employing the trapezoidal formula coupled with discrete invariant imbedding algorithm the proposed method is iterative on the terminal point some numerical experiments have been included to demonstrate its applicability

on solutions of some unsteady flows over a continuous moving porous flat surface abstract a singular perturbation solution for hydromagnetic boundary layer flow over a continuously moving surface with large suction is obtained and compared with its exact solution also the solutions are obtained for finite and large values of t time and for various asymptotic cases flow of this type represents a new class of boundary layer problems with solutions substantially different from those for boundary layer flow on a flat surface of finite length surprisingly the inner solution of the problem is valid even in the outer region and the effect of magnetic field on the flow field is almost negligible furthermore it is shown that a straightforward perturbation expansion valid for all t time and y space coordinate for small suction is not possible because of the infinite condition on y

higher order uniformly convergent methods for singular perturbation problems abstract until now the best way to construct high order uniformly convergent schemes for singular perturbation problems was to apply exponentially fitted compact difference schemes we propose the following approach firstly solve an auxiliary problem with piecewise or nearly piecewise constant coefficients secondly improve the approximation iteratively using the defect correction idea and piecewise polynomial approximations of higher order an important advantage of our approach lies in the fact that it is possible to start from a classical or weak formulation of a boundary value problem therefore the approach is useful for singular perturbations related to ordinary as well as partial differential equations

a novel image watermarking scheme based on amplitude attack this paper presents a novel image watermark scheme based on the characteristics of amplitude attacks first of all the analysis of amplitude attacks is performed then the amplitude attacks are categorized into three types positive attack negative attack and random attack finally based on these attack types we design three corresponding detectors the detectors remove the unreliable discrete cosine transform dct coefficients in the attacked image so that only reliable coefficients are used for the extraction of the watermark embedded the experimental results indicate that the proposed technique improves robustness significantly without sacrificing the false alarm rate

a computational method for solving singular perturbation problems using exponentially fitted finite difference schemes abstract a computational method is presented for solving linear singularly perturbed two point boundary value problems with a boundary layer on the left end of the interval the original problem is divided into inner and outer region problems the zeroth order asymptotic expansion is used to obtain the terminal boundary condition then a new inner region problem irp is created and solved as a two point boundary value problem tpbvp in turn the outer region problem orp is also solved as a tpbvp both these problems tpbvps are efficiently solved by employing uniform and optimal exponentially fitted finite difference schemes

image watermarking an evolution to content based approaches the paper is focused on the evolution of the watermarking schemes from pixel based approaches to content based methods firstly the field of watermarking is presented its implication in the real word is outlined the second section gives a brief overview of different approaches in the spatial and the frequential domain we then present different content based approaches used to improve the invisibility or the synchronisation of the mark the last section describes our content based scheme the synchronisation of the mark after image processing lies in the use of feature point detectors the image is decomposed into a set of triangles and each triangle is individually marked c 2001 pattern recognition society published by elsevier science ltd all rights reserved

can one hear the dimension of a fractal we consider the spectrum of the laplacian in a bounded open domain of n with a rough boundary i e with possibly non integer dimension and we discuss a conjecture by m v berry generalizing weyl s conjecture then using ideas mark kac developed in his famous study of the drum we give upper and lower bounds for the second term of the expansion of the partition function the main thesis of the paper is to show that the relevant measure of the roughness of the boundary should be based on minkowski dimensions and on minkowski measures rather than on haussdorff ones

hearing the shape of a general convex domain abstract the spectral function t j 1 exp t j where j j 1 are the eigenvalues of the laplace operator i 1 2 x i 2 in the x 1 x 2 plane is studied for a general convex domain together with an impedance condition on a part of its boundary and another impedance condition on the remaining part of that boundary

asymptotic expansion for the initial value problem of the sunflower equation abstract in this paper we consider the initial value problem ivp for the sunflower equation ex t ax t b sin x t e 0 with x t t t e 0 x 0 y0 we prove that the solution of the above mentioned ivp has an asymptotic expansion uniformly valid on e for e sufficiently small

continuous and numerical analysis of a multiple boundary turning point problem a singularly perturbed boundary value problem with a multiple turning point at a boundary is considered a representation of the solution is given and it is used in the construction of a uniform finite difference scheme the scheme is a first order exponentially fitted one an improved modification on a special discretization mesh is given

a finite element method for a singularly perturbed boundary value problem we examine the problem eu a x u b x u f x for 0 0 b x 2 4e 0 a b andf inc 2 0 1 e in 0 1 u 0 andu 1 given using finite elements and a discretized green s function we show that the el mistikawy and werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem i e the nodal errors are bounded bych 2 wherec is independent ofh and e with a natural choice of trial functions uniform first order accuracy is obtained in thel 0 1 norm on choosing piecewise linear trial functions hat functions uniform first order accuracy is obtained in thel 1 0 1 norm

optimal order nonnested multigrid methods for solving finite element equations iii on degenerate meshes in this paper we consider several model problems where finite element triangular meshes with arbitrarily small angles high aspect ratios are utilized to deal with anisotropy interfaces or singular perturbations the constant rate independent of the number of unknowns the smallest angle the interface discontinuity the singular perturbation parameter etc convergence of some special nonnested multigrid methods for solving the finite element systems on such degenerate meshes will be proved numerical data are provided to support the analysis in each case

on hearing the shape of rectilinear regions from complete knowledge of the eigenvalues of the negative laplacian on a bounded domain one may extract information on the geometry and the boundary conditions by analyzing the asymptotic expansion of a spectral function explicit calculations are performed for an equilateral triangular domain with dirichlet or neumann boundary conditions yielding in particular the corner angle terms in three dimensions some applications to eigenvalue problems for an equilateral triangular prism are dealt with including the solid vertex terms

uniform high order difference schemes for a singularly perturbed two point boundary value problem on construit une famille de schemas aux differences finies uniformement precis pour le probleme du type eu a x u b x u f x on donne une analyse complete de l erreur de discretisation basee sur les resultats de stabilite on presente des essais numeriques

hierarchical watermarking for secure image authentication with localization several fragile watermarking schemes presented in the literature are either vulnerable to vector quantization vq counterfeiting attacks or sacrifice localization accuracy to improve security using a hierarchical structure we propose a method that thwarts the vq attack while sustaining the superior localization properties of blockwise independent watermarking methods in particular we propose dividing the image into blocks in a multilevel hierarchy and calculating block signatures in this hierarchy while signatures of small blocks on the lowest level of the hierarchy ensure superior accuracy of tamper localization higher level block signatures provide increasing resistance to vq attacks at the top level a signature calculated using the whole image completely thwarts the counterfeiting attack moreover sliding window searches through the hierarchy enable the verification of untampered regions after an image has been cropped we provide experimental results to demonstrate the effectiveness of our method

digital watermarking based on chaotic map and reference register in this paper we present a new digital watermarking technique based on chaotic map and reference register rr we develop a block based chaotic map which outperforms the traditional one by breaking local spatial similarity to increase the amount of significant coefficients in the transformed image a rr is then employed to locate specific coefficients of a container efficiently for watermark embedding and extraction we demonstrate the superiority of the proposed scheme in terms of better embedding capacity and lower message error rate furthermore we show the proposed algorithm works well under some image processing distortions such as the jpeg compression gaussian noise and low pass filter

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

an analysis of a uniformly convergent finite difference finite element scheme for a model singular perturbation problem on etablit la convergence uniforme de la discretisation d un probleme a perturbation singuliere on utilise quelques resultats de stabilite et un theoreme de representation des solutions

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

an efficient image authentication method based on hamming code image authentication has come through a history of several years however up to the present time most mainstream image authentication schemes are still unable to detect burst bit errors moreover the capability of recovering tampered pixels in detail complex areas has not been very satisfactory either in this paper we offer to combine the hamming code technique torus automorphism and bit rotation technique to do tamper proofing according to our experimental results our new hybrid method can effectively eliminate burst bit errors and our recovered pixels in detail areas can actually gain very high clarity the results show that our scheme is quite a practical method which is quite able to detect and recover tampered areas

multigrid applied to singular perturbation problems the solution of the singular perturbation problem email protected u b x u f 0 0 u 0 u 0 u 1 u 1 by a multigrid algorithm is considered theoretical and experimental results for a number of different discretizations are presented the theoretical and observed rates agree with the results developed in an earlier work of kamowitz and parter in addition the rate of convergence of the algorithm when the coarse grid operator is the natural finite difference analogue of the fine grid operator is presented this is in contrast to the case in the previous work where the galerkin choice i h hl hi h h was used for the coarse grid operators

fitted mesh b spline collocation method for singularly perturbed differential difference equations with small delay abstract this paper deals with the singularly perturbed boundary value problem for a linear second order differential difference equation of the convection diffusion type with small delay parameter of o e whose solution has a boundary layer the fitted mesh technique is employed to generate a piecewise uniform mesh condensed in the neighborhood of the boundary layers b spline collocation method is used with fitted mesh parameter uniform convergence analysis of the method is discussed the method is shown to have almost second order parameter uniform convergence the effect of small delay on boundary layer has also been discussed several examples are considered to demonstrate the performance of the proposed scheme and how the size of the delay argument and the coefficient of the delay term affects the layer behavior of the solution

an inverse eigenvalue problem for a general convex domain the underlying problem is to deduce the precise shape of a membrane from the complete knowledge of the eigenvalues i for the lapace operator a cf a ax in the x1x2 plane let d g r be a simply connected bounded domain with a smooth boundary ad let us suppose that the a are the eigenvalues of the impedance problem au k o in d

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

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

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

on dichotomy and well conditioning in bvp we investigate the relationships between the stability bounds of the problem on the one hand and the growth behaviour of the fundamental solution on the other hand it is shown that if these stability bounds are moderate i e if the problem is well conditioned then the homogeneous solution space is dichotomic which means that it can be split into a subspace of nondecreasing and a complementary subspace of nonincreasing modes this is done by carefully examining the green s functions if these exhibit an exponential behaviour then the solution space is also exponentially dichotomic on the other hand we also show that exponential dichotomy implies moderate stability constants i e well conditioning from this it follows that both concepts are more or less equivalent

an inverse eigenvalue problem of the wave equation for a multi connected region in r2 together with three different types of boundary conditions the trace of the wave kernel m t email protected w 1exp ite 1 2 w where e w w 1 are the eigenvalues of the negative laplacian email protected email protected 2 k 1 email protected x k 2 in the x 1 x 2 plane is studied for a variety of bounded domains where

some improvements of the nonequidistant engquist osher scheme two nonequidistant finite difference schemes for quasilinear singular perturbation problems are given and their properties are discussed and compared one of them corresponds to the equidistant lorenz modification of the engquist osher scheme the other one switches in dependence on the cell reynolds number both schemes show quadratic l 1 accuracy but the second one gives better pointwise results

the wave equation approach to an inverse problem for a general multi connected domain in r2 with mixed boundary conditions the spectral function email protected t email protected j 1 exp email protected j 1 2 where m j j 1 are the eigenvalues of the negative laplacian email protected email protected u 1 email protected x u 2 in r 2 is studied for small t for a variety of domains where

singularly perturbed problems in partial differential equations a survey this survey paper contains a surprisingly large amount of material on singularly perturbed partial differential equations and indeed can serve as an introduction to some of the ideas and methods of the 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 methods developed by numerous researchers after 1980 2000 in this review we have covered singularly perturbed partial differential equations about odes the survey has already been done by us see m k kadalbajoo k c patidar appl math comput 130 2002 457 510

global uniformly convergent schemes for a singularly perturbed boundary value problem using patched base spline functions abstract for a self adjoint second order model singular perturbation problem global uniformly convergent schemes are constructed the basic idea consists in replacing the coefficients by piecewise polynomials and solving the resulting problems exactly

a fast adaptive numerical method for stiff two point boundary value problems we describe a robust adaptive algorithm for the solution of singularly perturbed two point boundary value problems many different phenomena can arise in such problems including boundary layers dense oscillations and complicated or ill conditioned internal transition regions working with an integral equation reformulation of the original differential equation we introduce a method for error analysis which can be used for mesh refinement even when the solution computed on the current mesh is underresolved based on this method we have constructed a black box code for stiff problems which automatically generates an adaptive mesh resolving all features of the solution the solver is direct and of arbitrarily high order accuracy and requires an amount of time proportional to the number of grid points

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

numerical solution of singular perturbation problems via deviating arguments we present a new approach to numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the interval the original problem is divided into outer and inner region problems a terminal boundary condition in the implicit form is derived then the outer region problem is solved as a two point boundary value problem tpbvp and an explicit terminal boundary condition is obtained in turn a new inner region problem is obtained and solved as a tpbvp using the explicit terminal boundary condition the proposed method is iterative on the terminal point of the inner region some numerical examples have been solved to demonstrate the applicability of the method

uniform pointwise convergence on shishkin type meshes for quasi linear convection diffusion problems a singularly perturbed quasi linear two point boundary value problem with an exponential boundary layer is considered the problem is discretized using a nonstandard upwinded first order difference scheme on generalized shishkin type meshes we give a uniform error estimate in a discrete l infty norm numerical experiments support the theoretical results

accurate discretization for singular perturbations the one dimensional case this paper develops a discretization method for one dimensional singular perturbation problems based on a petrov galerkin finite element or an equivalent finite volume scheme the method is unique in that its discretization error has a bound that is second order in the mesh size and uniform in the perturbation parameter it satisfies a local discrete conservation law and it exhibits a discrete maximum principle numerical results are included for comparison with the theory