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

application of the multiscale finite element method to flow in heterogeneous porous media 1 the multiscale finite element method is applied in this paper to simulate groundwater flow in heterogeneous porous media the method can efficiently capture the large scale behavior of the solution without resolving all the small scale features by constructing the multiscale finite element base functions that are adaptive to the local property of the differential operator both multiscale finite element method and conventional finite element method are applied to five 2 d and two 3 d groundwater flow problems including a 2 d steady flow problem with continuous coefficients a 2 d steady flow with gradual change in coefficients a 2 d transient flow with gradual change in coefficients a 2 d steady flow problem with an abrupt change in coefficients a 2 d transient flow problem with an abrupt change in coefficients and a 3 d steady and a 3 d transient flow problem with gradual change in coefficients in the horizontal direction and with abrupt change in the vertical direction respectively the applications demonstrate the main advantages of the multiscale finite element method i e significantly reducing computational efforts and improving the accuracy of the solutions

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

universal analytic properties of noise introducing the j matrix formalism we propose a new method in the spectral analysis of noisy time series data for damped oscillators from the jacobi three terms recursive relation for the denominators of the pade approximations built on the well known z transform of an infinite time series we build a hilbert space operator a j operator where each bound state inside the unit circle in the complex plane is simply associated with one damped oscillator while the essential spectrum of the j operator which lies on the unit circle itself is shown to represent the noise signal and noise are thus clearly separated in the complex plane for a finite time series of length 2n the j operator is replaced by a finite order j matrix jn having n eigenvalues which are time reversal covariant different classes of input noise such as blank white and uniform gaussian and pink are discussed in detail the j matrix formalism allowing us to efficiently calculate hundreds of poles of the z transform evidence of a universal behavior in the final statistical distribution of the associated poles and zeros of the z transform is shown in particular the poles and zeros tend when the length of the time series goes to infinity to a uniform angular distribution on the unit circle therefore at finite order the roots of unity in the complex plane appear to be noise attractors we show that the z transform presents the exceptional feature of allowing lossless undersampling and how to make use of this property a few basic examples are given to suggest the power of the proposed method

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

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

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

a uniformly accurate finite element method for a singularly perturbed one dimensional reaction diffusion application d une methode a elements finis avec des elements de base exponentiels a un probleme autoadjoint a perturbation singuliere et a deux points limites generation d un schema aux differences tridiagonal uniformement precis du second ordre

robust finite time estimation of biased sinusoidal signals a novel finite time convergent estimation technique is proposed for identifying the amplitude frequency and phase of a biased sinusoidal signal resorting to volterra integral operators with suitably designed kernels the measured signal is processed yielding a set of auxiliary signals in which the influence of the unknown initial conditions is removed a second order sliding mode based adaptation lawfed by the aforementioned auxiliary signalsis designed for finite time estimation of the frequency amplitude and phase the worst case behavior of the proposed algorithm in presence of the bounded additive disturbances is fully characterized by input to state stability arguments the effectiveness of the estimation technique is evaluated and compared with other existing tools via extensive numerical simulations

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

necessary convergence conditions for upwind schemes in the two dimensional case this paper considers nine point difference schemes for a two dimensional boundary value singular perturbation problem without turning points and parabolic boundary layers necessary conditions are given for the uniform convergence in the sense of the maximum norm of a scheme using these conditions several widely used schemes are analysed it is shown that some common schemes are not uniformly convergent in and that in some cases we are able to compute uniquely free parameters in the scheme some remarks on the treatment of a problem with a parabolic boundary layer are given

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

a necessary and sufficient criteria for the existence of the least squares estimate for a 3 parametric exponential function given the data p i t i y i i 1 m m 3 we give necessary and sufficient conditions which guarantee the existence of the least squares estimate for a 3 parametric exponential function to this end we suggest a choice of the initial approximation and give some numerical examples

exact maximum likelihood parameter estimation of superimposed exponential signals in noise a unified framework for the exact maximum likelihood estimation of the parameters of superimposed exponential signals in noise encompassing both the time series and the array problems is presented an exact expression for the ml criterion is derived in terms of the linear prediction polynomial of the signal and an iterative algorithm for the maximization of this criterion is presented the algorithm is equally applicable in the case of signal coherence in the array problem simulation shows the estimator to be capable of providing more accurate frequency estimates than currently existing techniques the algorithm is similar to those independently derived by kumaresan et al in addition to its practical value the present formulation is used to interpret previous methods such as prony s pisarenko s and modifications thereof

the p version of the finite element method for a singularly perturbed boundary value problem the p version of the finite element method is applied to solve the singularly perturbed two point boundary value problem with or without turning point with the special choice of mesh points global error estimates are derived in some cases the exponential rate of convergence is obtained some numerical results are given to show the performance of the proposed method

a finite element method with a large mesh width for a stiff two point boundary value problem abstract we describe a finite element method for computation of numerical approximations of the solution of the second order singularly perturbed two point boundary value problem on 1 1 u pu f u 1 u 1 0 0 d dx on a quasi uniform mesh we construct exponentially fitted trial spaces which consist of piece wise polynomials and of exponentials which fit locally to the singular solution of the equation or its adjoint we discretise the galerkin form for the boundary problem using such exponentially fitted trial spaces we derive rigorous bounds for the error of discretisation with respect to the energy norm and we obtain superconvergence at the mesh points the error depending on the mesh width and the degree of the piece wise polynomials

existence of optimal solution for exponential model by least squares in this paper we prove the existence theorem for the best least squares approximation of the optimal parameters for the exponential model function we give sufficient conditions which guarantee the existence of such optimal parameters using these results and methods we are able to localize a sufficiently narrow area where one can choose a good initial approximation

prony pisarenko and the matrix pencil a unified presentation in the past the prony and pisarenko algorithms have been widely used in the pole zero identification of linear systems as well as in the problem of extracting harmonics embedded in noise these techniques have also been applied successfully to the direction of arrival doa problem it is shown that these algorithms are equivalent to a recently proposed doa estimation technique called the matrix pencil approach

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

diagonalization and spectral decomposition of factor block circulant matrices abstract we introduce factor circulant matrices matrices with the structure of circulants but with the entries below the diagonal multiplied by the same factor the diagonalization of a circulant matrix and spectral decomposition are conveniently generalized to block matrices with the structure of factor circulants differential equations involving factor circulants are considered

the harmonic archipelago as a universal locally free group abstract the harmonic archipelago is an important example of a two dimensional topological space whose fundamental group displays unusual properties this group can also be described as a quotient of two different products a representation somewhat akin to z z but nonabelian it is known to be locally free here it is shown that every countable locally free group embeds into this fundamental group

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

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

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

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

maximum likelihood estimation of structural wave components from noisy data in this paper a general framework is developed for determining the underlying parameters of general signal models through the application of maximum likelihood estimation theory for functions whose variables separate this method extends previous work in sinusoidal and exponential estimation to include models with other functional bases such as exponential functions with nonconstant amplitudes and bessel functions nonuniform spatial sampling is also possible with this technique the maximum likelihood method is applied to the identification of wave components along one dimensional structural elements results are given which demonstrate the viability and accuracy of the technique estimating exponential and bessel function model parameters from noisy simulation data

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

estimating components of univariate gaussian mixtures using prony s method a new technique for estimating the component parameters of a mixture of univariate gaussian distributions using the method of moments is presented the method of moments basically involves equating the sample moments to the corresponding mixture moments expressed in terms of component parameters and solving these equations for the unknown parameters these moment equations however are nonlinear in the unknown parameters and heretofore an analytic solution of these equations has been obtained only for two component mixtures 2 numerical solutions also tend to be unreliable for more than two components due to the large number of nonlinear equations and parameters to be solved for in this correspondence under the condition that the component distributions have equal variances or equal means the nonlinear moment equations are transformed into a set of linear equations using prony s method the solution of these equations for the unknown parameters is analytically feasible and numerically reliable for mixtures with several components numerous examples using the proposed technique for two three and four component mixtures are presented

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

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

a review of the parameter estimation problem of fitting positive exponential sums to empirical data exponential sum models are used frequently in heat diffusion diffusion of chemical compounds time series in medicine economics physical sciences and technology thus it is important to find good methods for the estimation of parameters in exponential sums in this paper we review and discuss results from the last forty years of research there are many different ways of estimating parameters in exponential sums and model a fit criterion which gives a valid result from the fit we find that a good choice is a weighted two norm objective function with weights based on the maximum likelihood ml criterion if the number of exponential terms is unknown statistical methods based on an information criterion or cross validation can be used to determine the optimal number it is suitable to use hybrid gauss newton gn and quasi newton algorithms to find the unknown parameters in the constrained weighted nonlinear least squares nlls problem formulated using an maximal likelihood ml objective function the problem is highly ill conditioned and it is crucial to find good starting values for the parameters to find the initial parameter values a modified prony method or a method based upon rewriting partial sums as geometrical sums is proposed

incremental stresses in loaded orthotropic circular membrane tubes ii numerical solution abstract in part i of this article 1995 int j solids structures 32 1907 1925 a simplified theory for the incremental state of stress in an orthotropic circular membrane tube was presented in this second part a numerical method is devised to solve the resulting elliptic sixth order system of equations for tubes which contain no slits or holes in this method fourier decomposition is used in the circumferential direction and finite element discretization is used in the longitudinal direction special finite elements with six degrees of freedom are employed using this numerical method the solutions of several specific problems of membrane tubes are obtained presented and discussed these include two problems concerning the decay of boundary disturbances the theoretical treatment of such decay is also discussed

employing symmetry constraints for improved frequency estimation by eigenanalysis methods the problem of extracting sinusoid signals from noisy observations made at equally spaced times is considered eigenanalysis methods such as pisarenko s method and the extended prony method find the eigenvector with minimum eigenvalue of a suitably chosen matrix and then obtain the complex sinusoids as the roots of the polynomial that has the components of the eigenvector as coefficients for the sinusoids to be undamped it is necessary that the roots lie on the unit circle and hence that the eigenvector be conjugate symmetric it is shown how this symmetry constraint can be incorporated into eigenanalysis estimation methods in a routine way the practical importance of the constraint is investigated by cramer rao variance bound calculations and by simulation the following conclusions are made 1 the symmetry constraint is straightforward to implement and reduces the amount of computation required 2 the relative reduction in variance of the constrained over the unconstrained frequency estimators

statistical analysis of pisarenko s method for sinusoidal frequency estimation statistical analysis is performed for pisarenko s method of sinusoidal frequency estimation by using the periodogram technique under the assumption that the data consist of several sinusoids plus white noise we give an asymptotic expression of the error variance of the sinusoidal frequency estimator also a modified method is proposed this is computationally efficient but the statistical property is not improved we also present simulation results and compare them to the theoretical ones lastly we point out that the pisarenko s method can be implemented as an on line bias compensated least squares algorithm thus avoiding the eigenvalue and eigenvector calculations

estimating the parameters of undamped exponential signals a nonlinear eigenvalue method is proposed for estimating the parameters of undamped exponential signals when the parameters are complex valued such data arise in several areas of application including communications radio location of objects seismic signal processing and computer assisted medical diagnosis osborne proposed a method to estimate the parameters of exponential models when the parameters are real valued the method is generalized to the complex parameters case it is shown to perform better than existing methods due to tufts and kumaresan and bai krishnaiah and zhao in the sense of having lower mean squared errors a simulation study showed that the observed mean squared errors are close to the cramer rao lower bound for frequency estimates

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

perfect blind restoration of images blurred by multiple filters theory and efficient algorithms we address the problem of restoring an image from its noisy convolutions with two or more unknown finite impulse response fir filters we develop theoretical results about the existence and uniqueness of solutions and show that under some generically true assumptions both the filters and the image can be determined exactly in the absence of noise and stably estimated in its presence we present efficient algorithms to estimate the blur functions and their sizes these algorithms are of two types subspace based and likelihood based and are extensions of techniques proposed for the solution of the multichannel blind deconvolution problem in one dimension we present memory and computation efficient techniques to handle the very large matrices arising in the two dimensional 2 d case once the blur functions are determined they are used in a multichannel deconvolution step to reconstruct the unknown image the theoretical and practical implications of edge effects and weakly exciting images are examined finally the algorithms are demonstrated on synthetic and real data

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

a modified prony algorithm for exponential function fitting a modification of the classical technique of prony for fitting sums of exponential functions to data is considered the method maximizes the likelihood for the problem unlike the usual implementation of prony s method which is not even consistent for transient signals proves to be remarkably effective in practice and is supported by an asymptotic stability result novel features include a discussion of the problem parametrization and its implications for consistency the asymptotic convergence proofs are made possible by an expression for the algorithm in terms of circulant divided difference operators

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

a uniformly accurate finite element method for a singular perturbation problem in conservative form a petrov galerkin finite element method with exponential basis elements is applied to a nonselfadjoint singularly perturbed two point boundary value problem in conservative form it is shown to be uniformly first order accurate in l infty and is uniformly second order accurate at the nodes i e the error constants are independent of the mesh size and the perturbation parameter

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

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

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

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

adaptivity and error estimation for the finite element method applied to convection diffusion problems a detailed analysis is performed for a finite element method applied to the general one dimensional convection diffusion problem piecewise polynomials are used for the trial space the test space is formed by locally projecting l spline basis functions onto upwinded polynomials the error is measured in the l p mesh dependent norm the method is shown to be quasi optimal provided that the input data is piecewise smooth a reasonable assumption in practice a posteriors error estimates are derived having the property that the effectivity index theta error estimate true error converges to one as the maximum mesh size goes to zero these error estimates are composed of locally computable error indicators providing for an adaptive mesh refinement strategy numerical results show that theta is nearly one even on coarse meshes and optimal rates of convergence are attained by the adaptive procedure the robustness of the algorithm is tested on a nonlinear turning point problem modeling flow th

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

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

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

an asymptotic finite element method for improvement of solutions of boundary layer problems a scheme that uses singular perturbation theory to improve the performance of existing finite element methods is presented the proposed scheme improves the error bounds of the standard galerkin finite element scheme by a factor of o en 1 where e is the small parameter andn is the order of the asymptotic approximation numerical results for linear second order o d e s are given and are compared with several other schemes

circulant matrices and differential delay equations on etudie les solutions periodiques non triviales de familles parametrisees d equations differentielles a retard

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

imaging of multiple linear cracks using impedance data this paper develops a fast simple algorithm for locating one or more perfectly insulating pair wise disjoint linear cracks in a homogeneous two dimensional electrical conductor using flux potential boundary measurements we also explore the issue of what types boundary inputs yield the most stable images

non local and semi local optimal weighting functions for symmetric problems involving a small parameter the method of optimal weighting functions for symmetric problems is described in a general form it is based on a petrov galerkin formulation in which the best approximation property and other mathematical features are achieved for a chosen norm different from the original energy norm of the problem the nonlocality of the weighting functions is shown to have only a minor effect on the efficiency of the method although a localization scheme is also suggested the method is applied to a one and two dimensional singular perturbation problems as well as to a cylindrical shell problem

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

the number of homomorphisms from the hawaiian earring group abstract we show a dichotomy for groups of cardinality less than continuum the number of homomorphisms from the hawaiian earring group to such a group g is either the cardinality of g in case g is noncommutatively slender or the number is 2 2 0 in case g is not noncommutatively slender an example of a noncommutatively slender group with nontrivial divisible element is also exhibited

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

limits of performance of quantitative polymerase chain reaction systems estimation of the dna copy number in a given biological sample is an important problem in genomics quantitative polymerase chain reaction qpcr systems detect the target dna molecules by amplifying their number through a series of thermal cycles and measuring the amount of created amplicons in each cycle ideally the number of target molecules doubles at the end of each cycle however in practice due to biochemical noise the efficiency of the qpcr reaction defined as the fraction of the target molecules which are successfully copied during a cycle is always less than 1 in this paper we formulate the problem of the joint maximum likelihood estimation of the qpcr efficiency and the initial dna copy number then we analytically determine the limits of performance of qpcr by deriving the cramer rao lower bound on the mean square estimation error as indicated by simulation studies the performance of the proposed estimator is superior compared to competing statistical approaches the proposed approach is validated using experimental data

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