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

uniform pointwise convergence for a singularly perturbed problem using arc length equidistribution a singularly perturbed two point boundary value problem with an exponential boundary layer is solved numerically by using an adaptive grid method the mesh is constructed adaptively by equidistributing a monitor function based on the arc length of the exact solution the error analysis for this approach was carried out by qiu et al j comput appl math 101 1999 1 25 in this work their error bound will be improved to the optimal order which is independent of the perturbation parameter the main ingredient used to obtain the improved result is the theory of the discrete green s function

valuation like maps and the congruence subgroup property let d be a finite dimensional division algebra and n a subgroup of finite index in d a valuation like map on n is a homomorphism n from n to a not necessarily abelian linearly ordered group satisfying n 1 n for some nonnegative such that n where n x n x we show that this implies the existence of a nontrivial valuation v of d with respect to which n is v adically open we then show that if n is normal in d and the diameter of the commuting graph of d n is 4 then n admits a valuation like map this has various implication in particular it restricts the structure of finite quotients of d the notion of a valuation like map is inspired by 27 and in fact is closely related to part u3 of the u hypothesis in 27

exponential methods for singularly perturbed ordinary differential difference equations abstract a variety of exponential methods based on piecewise analytical solutions of advection reaction diffusion operators is proposed for the numerical solution of linear ordinary differential difference equations with small delay these methods are shown to be exact in exact arithmetic for linear ordinary differential equations with constant coefficients and right hand side are non standard and include three point non local approximations for the coefficients right hand side and dependent variable also a variety of exponential methods based on the piecewise analytical solution of advection diffusion operators with non local approximations is presented it is shown that exponential methods based on the advection diffusion reaction operator provide more accurate results than those based on the analytical solution of the advection diffusion operator it is also shown that at least for the problems considered here non local approximations for either the coefficients or the right hand side of the differential equations do not result in an increase of accuracy over those based on local approximations

singular perturbation analysis of boundary value problems for differential difference equations vi small shifts with rapid oscillations this paper continues the authors study of boundary value problems for singularly perturbed linear second order differential difference equations with small shifts this study was initiated in the companion paper siam j appl math 54 1994 pp 249 272 in this paper the study is extended to problems that have solutions that exhibit rapid oscillations restrictions on the sizes of the shifts in terms of the small parameter are found such that generally the shifted terms cannot be replaced with truncated taylor series in particular it is shown that even when the shifts are small relative to the width of an oscillation they can affect the solution to leading order the conclusion is that oscillatory solutions are more sensitive to small delays than are layer solutions it is shown that a suitably modified version of the standard wkb method can be used to obtain leading order oscillatory solutions of these differential difference equations these preliminary studies of differential difference eq

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

singular perturbation analysis of boundary value problems for differential difference quations ii rapid oscillations and resonances this paper continues a study of a class of boundary value problems for linear second order differential difference equations in which the second order derivative is multiplied by a small parameter siam j appl math 42 1982 pp 502 531 the previous paper focused on problems involving boundary and interior layer phenomena here the problems studied have solutions exhibiting rapid oscillations the presence of the shift terms can induce large amplitudes multiphase behavior and resonance phenomena n particular we study two types of resonance phenomena namely global and local resonance a combination of exact solutions singular perturbation methods and numerical computations are used in these studies in a companion paper siam j appl math 45 1985 pp 708 734 we study problems with solutions which exhibit turning point behavior

robust solution of singularly perturbed problems using multigrid methods we consider the problem of solving linear systems of equations that arise in the numerical solution of singularly perturbed ordinary and partial differential equations of reaction diffusion type standard discretization techniques are not suitable for such problems and so specially tailored methods are required usually involving adapted or fitted meshes that resolve important features such as boundary and or interior layers in this study we consider classical finite difference schemes on the layer adapted meshes of shishkin and bakhvalov we show that standard direct solvers exhibit poor scaling behavior with respect to the perturbation parameter when solving the resulting linear systems we propose and prove optimality of a new block structured preconditioning approach that is robust for small values of the perturbation parameter and compares favorably with standard robust multigrid preconditioners for these linear systems we also derive stopping criteria which ensure that the potential accuracy

strong convergence of an iterative algorithm for pseudocontractive mapping in banach spaces abstract let c be a nonempty closed convex subset of a real uniformly smooth banach space x t c c a pseudocontractive mapping let n n and n be three real sequences in 0 1 satisfying appropriate conditions then for fixed u c and arbitrary x 0 c the sequence x n generated by x n n u n x n 1 n t x n n 1 converges strongly to a fixed point of t

on the congruence subgroup problem for anisotropic groups of inner type a_n an let g be an absolutely almost simple simply connected algebraic group of inner form of type a n defined and anisotropic over a number field k let v denote a complete set of inequivalent valuations of k and v f resp infty the subset of non archimedean resp archimedean valuations in v this work deals with the congruence subgroup problem for groups g of inner form of type a n with respect to a particular class of valuations s subset v which we term as d amenable fix a realization g subset mathrm gl n as a k subgroup and set lambda s g k cap gl n mathcal o s where mathcal o s is the ring of s integers in k then we show that if s is d amenable and gamma is any subgroup of finite index in lambda s there exists a finite set s gamma and positive integers r v mid v in s gamma such that gamma contains gamma in lambda s mid gamma in 1 mathfrak p v r v text for v in s gamma the d amenability property holds for any s with v smallsetminus s finite we also show that the set of primes in v in an arithmetic progression is d amenable for any d so that we recover the result of prasad and rapinchuk proc steklov inst math 292 1 216 246 2016

a parameter uniform grid equidistribution method for singularly perturbed degenerate parabolic convection diffusion problems abstract we consider a class of singularly perturbed degenerate parabolic convection diffusion problems on a rectangular domain a numerical method is constructed using the implicit euler scheme on a uniform mesh in the time direction and the upwind finite difference scheme on a layer adaptive non uniform mesh in the spatial direction the layer adaptive non uniform mesh in the spatial direction is generated through the equidistribution of a suitably chosen monitor function we perform error analysis through the truncation error and barrier function approach and prove that the method is uniformly convergent with first order in both time and space numerical results are given in support of theoretical findings

graded mesh difference schemes for singularly perturbed two point boundary value problems etude de l approximation numerique des operateurs differentiels sans points de retournement par des schemas aux differences finies compacts analyse de la stabilite

maximum norm a posteriori error estimate for a 2d singularly perturbed semilinear reaction diffusion problem a singularly perturbed semilinear reaction diffusion equation posed in the unit square is discretized on arbitrary nonuniform tensor product meshes we establish a second order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter no mesh aspect ratio assumption is made numerical results are presented that support our theoretical estimate

strong convergence theorems for finitely many nonexpansive mappings and applications abstract let e be a uniformly convex banach space which satisfies opial s condition or whose norm is frechet differentiable recently takahashi and shimoji w takahashi k shimoji convergence theorems for nonexpansive mappings and feasibility problems math comput modelling 32 2000 1463 1471 introduced an iterative scheme given by finitely many nonexpansive mappings in e and proved weak convergence theorems which are connected with the problem of image recovery in this paper we introduce a new iterative scheme which includes their iterative scheme as a special case under the assumption that e is a reflexive banach space whose norm is uniformly g teaux differentiable and which has a weakly continuous duality mapping we prove strong convergence theorems which are connected with the problem of image recovery using the established results we consider the problem of finding a common fixed point of finitely many nonexpansive mappings

maximum norm a posteriori error estimates for a one dimensional convection diffusion problem a singularly perturbed quasi linear two point boundary value problem with an exponential boundary layer is discretized on arbitrary nonuniform meshes using first and second order difference schemes including upwind schemes we give first and second order maximum norm a posteriori error estimates that are based on difference derivatives of the numerical solution and hold true uniformly in the small parameter numerical experiments support the theoretical results

an asymptotic theory of large deviations for markov jump processes we present new asymptotic methods for the analysis of markov jump processes the methods based on the wkb and other singular perturbation techniques are applied directly to the kolmogorov equations and not to approximate equations that come e g from diffusion approximations for time homogeneous processes we construct approximations to the stationary density function and the mean first passage time from a given domain examples involving a random walk and a problem in queueing theory are presented to illustrate our methods for a class of time inhomogeneous processes we construct long time approximations to the transition probability density function and the probability of large deviations from a stable state the law of large numbers is obtained as a special case

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

analysis of difference approximations to a singularly perturbed two point boundary value problem on an adaptively generated grid over the last few years there has been a significant growth in the use of adaptive grid methods for the numerical solution of differential equations with steep solutions little has been done however on the error analysis of adaptive methods in this paper we present an analysis for an upwind finite difference solution of a singular perturbation problem on a grid that is generated adaptively by equidistributing a monitor function based on the exact solution it is shown that the discrete solutions converge uniformly with respect to the perturbation parameter epsilon this epsilon uniform convergence is illustrated by numerical computations

maximum norm a posteriori error estimates for a singularly perturbed differential difference equation with small delay we consider an adaptive grid approach to solve a singularly perturbed differential difference equation with small delay the mesh is constructed adaptively by equidistributing a arc length monitor function we derive a first order maximum norm a posteriori estimate for the full discretization scheme a first order rate of convergence independent of the perturbation parameter and the small delay parameter is established a singularly perturbed differential difference equation with small delay is discretized on an adaptive grid which is formed by equidistributing arc length monitor function we first derive first order maximum norm a posteriori estimates for the full discretization scheme of these problems then a first order rate of convergence independent of the perturbation parameter and the small delay parameter is established numerical results are provided that support our theoretical estimates

strong convergence of an iterative algorithm for nonself multimaps in banach spaces abstract let e be a uniformly convex banach space having a uniformly g teaux differentiable norm d a nonempty closed convex subset of e and t d k e a nonself multimap such that f t 0 and p t is nonexpansive where f t is the fixed point set of t k e is the family of nonempty compact subsets of e and p t x u x t x x u x d x t x suppose that d is a nonexpansive retract of e and that for each v d and t 0 1 the contraction s t defined by s t x t p t x 1 t v has a fixed point x t d let n n and n be three real sequences in 0 1 satisfying approximate conditions then for fixed u d and arbitrary x 0 d the sequence x n generated by x n n u n x n 1 n p t x n n 0 converges strongly to a fixed point of t

asymptotic analysis of a state dependent m g 1 queueing system we present new asymptotic methods for the analysis of queueing systems these methods are applied to a state dependent m g 1 queue we formulate problems for and compute approximations to i the stationary density of the unfinished work ii the mean length of time until the end of a busy period iii the mean length of a busy period and iv the mean time until the unfinished work reaches or exceeds a specified capacity the methods are applied to the full kolmogorov equations scaled so that the arrival rate is rapid and the mean service is small thus we do not truncate equations as in diffusion approximations for state independent m g 1 queues our results are shown to agree with the known exact solutions we include comparisons both analytic and numerical between our results and those obtained from diffusion approximations

upwind and midpoint upwind difference methods for time dependent differential difference equations with layer behavior abstract in this paper we have considered an important class of time dependent singularly perturbed convection diffusion problems with retarded terms which often arise in computational neuroscience we used taylors series to approximate the retarded terms and the resulting time dependent singularly perturbed differential equation is approximated using parameters uniform numerical methods based on euler implicit upwind and midpoint upwind finite difference schemes we discretize the continuous problem using implicit euler scheme in the time direction with a constant step size and the resulting system of equations is approximated using upwind and midpoint upwind difference schemes on a piecewise uniform mesh we will prove the uniform convergence of these two schemes numerical experiments support the convergence results

a posteriori error analysis for finite element solution of one dimensional elliptic differential equations using equidistributing meshes the paper is concerned with the adaptive linear finite element solution of linear one dimensional elliptic differential equations using equidistributing meshes a strategy is developed for defining such meshes based on a residual based a posteriori error estimate the mesh and the finite element solution are determined by the coupled system of the finite element equation and the equidistribution relation i e the mesh equation an iterative algorithm is proposed for solving this system for the mesh and the finite element solution the existence of an equidistributing mesh is proven for a given sufficiently large number of the points with help of a result on the continuous dependence of the finite element solution on the mesh which is also established in the current work error bounds for the finite element solution are obtained for the equidistributing and quasi equidistributing meshes they show that adaptive meshes can lead to more accurate solutions than a uniform mesh and it is unnecessary to compute the equidistribution relation accurately for the equidistributing meshes the departure from the equidistributing meshes has only a mild effect on the finite element error numerical examples are given to illustrate the convergence of the iterative algorithm and the theoretical findings

implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings in this paper we propose a new implicit iteration scheme with perturbed mapping for approximation of common fixed points of a finite family of nonexpansive mappings we establish some convergence theorems for this implicit iteration scheme in particular necessary and sufficient conditions for strong convergence of this implicit iteration scheme were obtained

robust adaptive computation of a one dimensional q tensor model of nematic liquid crystals this paper illustrates the use of an adaptive finite element method to solve a non linear singularly perturbed boundary value problem which arises from a one dimensional q tensor model of liquid crystals the adaptive non uniform mesh is generated by equidistribution of a strictly positive monitor function which is a linear combination of a constant floor and a power of the first derivative of the numerical solution by an appropriate selection of the monitor function parameters we show that the computed numerical solution converges at an optimal rate with respect to the mesh density and that the solution accuracy is robust to the size of the singular perturbation parameter

the efficient generation of simple two dimensional adaptive grids we present an efficient adaptive procedure for solving steady partial differential equations based on the simple two dimensional quadrilateral grid generator of huang and sloan siam j sci comput 15 1994 pp 776 797 the mesh equations are derived using an equidistribution principle and are discretized in the physical domain this allows the application of the adaptive grid generator to finite element and finite volume methods to efficiently solve the nonlinear algebraic equations describing the adapted grid we propose an alternating line gauss seidel relaxation procedure the beneficial effect of adaptive grids is demonstrated by coupling the grid generator to finite volume and finite difference approximations of advection and advection diffusion test problems for some examples the proposed method is shown to be 50 times more efficient than previously used solution procedures

strong convergence theorems for multivalued nonexpansive nonself mappings in banach spaces abstract let e be a uniformly convex banach space with a uniformly g teaux differentiable norm c a nonempty closed convex subset of e and t c k e a nonexpansive mapping for u c and t 0 1 let x t be a fixed point of a contraction g t c k e defined by g t x t t x 1 t u x c it is proved that if c is a nonexpansive retract of e x t is bounded and t z z for any fixed point z of t then the strong lim t 1 x t exists and belongs to the fixed point set of t furthermore we study the strong convergence of x t with the weak inwardness condition on t in a reflexive banach space with a uniformly g teaux differentiable norm

numerical analysis of singularly perturbed delay differential equations with layer behavior in this paper we present a numerical method to solve boundary value problems for singularly perturbed differential difference equations with negative shift in recent papers the term negative shift has been used for delay siam j appl math 54 1994 249 siam j appl math 42 1982 502 siam j appl math 45 1985 687 siam j appl math 45 1985 708 siam j appl math 54 1994 273 similar bvps are associated with expected first exit time problem of the membrane potential in models for neuron and in variational problem in control theory the stability and convergence analysis of the method is discussed also the effect of small shift on the boundary layer solution in both the cases i e boundary layer on the left side as well as the right side is discussed by considering numerical experiments several numerical examples are solved using the presented method compared the computed result with exact solution and plotted the graphs of the solution of the problems

qualitatively stable finite difference schemes for advection reaction equations a systematic procedure is proposed and implemented for the design of nonstandard finite difference methods as reliable numerical simulations that preserve significant properties inherent to the solutions of advection reaction equations in the case of hyperbolic fixed points a renormalization of the denominators of the discrete derivatives is performed for the numerical solutions to display the linear stability properties of the exact solutions non hyperbolic fixed points are described with the help of two new monotonic properties the construction of schemes which preserve these properties being done by nonlocal approximation of nonlinear terms in the reaction terms

particular solutions of forced generalized airy equations differential equations with delayed forcing terms arise naturally in the explicit solution of differential delay equations 11 in our present studies of singular perturbation analyses of boundary value problems for secondorder differential difference equations with a turning point see 2 for earlier results we have had to deal with forcing terms consisting of shifted airy and lommel functions and their derivatives one example of the type of equation which arises is the inhomogeneous airy equation see the appendix for a derivation y xy c ai x a c bi x a c l x a d ai x a d bi x a d3 l x a 1 where a is a constant 0 a co ci di i 1 2 3 are constants ai bi are airy functions and l is a lommel function see 31 satisfying l xl 1 2

reproducing kernel method for solving singularly perturbed differential difference equations with boundary layer behavior in hilbert space abstract we consider singularly perturbed differential difference equation which contains negative shift in the differentiated term with boundary layer behavior rkhsm reproducing kernel hilbert space method without gram schmidt orthogonalization process is considered in the present paper we decompose the domain of the problem into two subintervals one of them has not the boundary layer and the other one has the side of the interval in which the boundary layer exists is important if the boundary layer of this problem exists on the left side of interval the rkhsm will provide a proper approximation of solution otherwise for the implement of rkhsm we need to change the variable of the singularly perturbed problem to shift the boundary layer region

a non standard numerical scheme for a generalized gause type predator prey model abstract a non standard finite difference scheme is constructed to simulate a predator prey model of gause type with a functional response using fixed point analysis it is shown that the scheme preserves the physical properties of the model and gives results that are qualitatively equivalent to the real dynamics of the model it is also shown that the scheme undergoes a supercritical hopf bifurcation for a specific value of the bifurcation parameter k0 this leads to the existence of a stable limit cycle created by the scheme when the bifurcation parameter passes through k0 as predicted by the continuous model the scheme is used to simulate the model with the functional responses of holling types ii and iii the simulation results generated by the non standard finite difference scheme are compared with those obtained from the standard methods such as forward euler and runge kutta methods these comparisons show that the standard methods give erroneous results that disagree with the theoretical predictions of the model however it is proved that the proposed non standard finite difference scheme is consistent with the asymptotic dynamics of the model numerical simulations are presented to support these facts

a conservative uniformly accurate difference method for a singular perturbation problem in conservation form a conservative three point finite difference method is presented for the numerical solution of the singular perturbation problem varepsilon u xx b x u x f x 0 x 1 u 0 and u 1 given b x ne 0 certain a priori estimates are established for this problem and are then used to obtain uniform error estimates for the difference scheme some illustrative numerical results are also given

numerical construction of optimal adaptive grids in two spatial dimensions abstract this article concerns a new procedure to generate a solution adaptive grid for convection dominated problems in two spatial dimensions based on finite element approximations the procedure extends a one dimensional equidistribution principle which minimizes the interpolation error in appropriate norms the idea in extending such a technique to two spatial dimensions is to select two directions which can reflect the physics of the problems and then the one dimensional equidistribution principle is applied to the chosen directions the final grids generated are connected through a sweep line based unstructured grid technique model problems considered are the two dimensional convection diffusion problems where boundary and interior layers occur numerical results of model problems illustrating the efficiency of the proposed scheme are presented comparisons of the solutions with an existing scheme will also be provided

comparison of some recent numerical methods for initial value problems for stiff ordinary differential equations we consider the combustion equation as one of the candidates from the class of stiff ordinary differential equations a solution over a length of time that is inversely proportional to d 0 where d 0 is a small disturbance of the pre ignition state is sought this problem has a transient at the midpoint of the integration interval the solution changes from being non stiff to stiff and afterwards becomes non stiff again we provide its asymptotic and numerical solution obtained via a variety of methods comparisons are made for the numerical results which we obtain with the matlab ode solvers ode45 ode15s and ode23s and some nonstandard finite difference methods results corresponding to standard finite difference method are also presented furthermore the discussion on these approaches along with the others provides several open problems for new and young researchers

singular perturbation analysis of boundary value problems for differential difference equations v small shifts with layer behavior in this paper an investigation is initiated of boundary value problems for singularly perturbed linear second order differential difference equations with small shifts i e where the second order derivative is multiplied by a small parameter and the shift depends on the small parameter similar boundary value problems are associated with expected first exit times of the membrane potential in models for neurons in particular this paper focuses on problems with solutions that exhibit layer behavior at one or both of the boundaries the analyses of the layer equations using laplace transforms lead to novel results it is shown that the layer behavior can change its character and even be destroyed as the shifts increase but remain small in the companion paper siam j appi math 54 1994 pp 273283 similar boundary value problems with solutions that exhibit rapid oscillations are studied

numerical analysis of singularly perturbed delay differential turning point problem abstract in this paper we describe a numerical method based on fitted operator finite difference scheme for the boundary value problems for singularly perturbed delay differential equations with turning point and mixed shifts similar boundary value problems are encountered while simulating several real life processes for instance first exit time problem in the modelling of neuronal variability a rigorous analysis is carried out to obtain priori estimates on the solution and its derivatives for the considered problem in the development of numerical methods for constructing an approximation to the solution of the problem a special type of mesh is generated to tackle the delay term along with the turning point then to develop robust numerical scheme and deal with the singularity because of the small parameter multiplying the highest order derivative term an exponential fitting parameter is used several numerical examples are presented to support the theory developed in the paper

optimal error estimate using mesh equidistribution technique for singularly perturbed system of reaction diffusion boundary value problems in this article we study the problem of determining an appropriate grading of meshes for a system of coupled singularly perturbed reaction diffusion problems having diffusion parameters with different magnitudes the central difference scheme is used to discretize the problem on adaptively generated mesh where the mesh equation is derived using an equidistribution principle an a priori monitor function is obtained from the error estimate a suitable a posteriori analogue of this monitor function is also derived for the mesh construction which will lead to an optimal second order parameter uniform convergence we present the results of numerical experiments for linear and semilinear reaction diffusion systems to support the effectiveness of our preferred monitor function obtained from theoretical analysis

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

a robust adaptive method for a quasi linear one dimensional convection diffusion problem a quasi linear conservative convection diffusion two point boundary value problem is considered to solve it numerically an upwind finite difference scheme is applied the mesh used has a fixed number n 1 of nodes and is initially uniform but its nodes are moved adaptively using a simple algorithm of de boor based on equidistribution of the arc length of the current computed piecewise linear solution it is proved for the first time that a mesh exists that equidistributes the arc length along the polygonal solution curve and that the corresponding computed solution is first order accurate uniformly in varepsilon where varepsilon is the diffusion coefficient in the case when the boundary value problem is linear if n is sufficiently large independently of varepsilon it is shown that after o rm ln 1 varepsilon rm ln n iterations of the algorithm the piecewise linear interpolant of the computed solution achieves first order accuracy in the l infty 0 1 norm uniformly in varepsilon numerical experiments are presented that support our theoretical results

hybrid method for numerical solution of singularly perturbed delay differential equations this paper is concerned with the numerical technique for a singularly perturbed second order differential difference equation of the convection diffusion type with small delay parameter d whose solution has a single boundary layer we analyse three difference operators l k n k 1 2 3 a simple upwind scheme midpoint upwind scheme and a hybrid scheme respectively on a shishkin mesh to approximate the solution of the problem the hybrid algorithm uses central difference in the boundary layer region and midpoint upwind scheme outside the boundary layer we have concluded that the hybrid scheme gives better accuracy the paper concludes with a few numerical results exhibiting the performance of these three schemes

on mann iteration in hilbert spaces abstract let k be a compact convex subset of a real hilbert space h t k k a hemicontractive map let n be a real sequence in 0 1 satisfying appropriate conditions then for arbitrary x 0 k the sequence x n defined iteratively by x n n x n 1 1 n t x n n 1 converges strongly to a fixed point of t

numerical treatment of a mathematical model arising from a model of neuronal variability in this paper we describe a numerical approach based on finite difference method to solve a mathematical model arising from a model of neuronal variability the mathematical modelling of the determination of the expected time for generation of action potentials in nerve cells by random synaptic inputs in dendrites includes a general boundary value problem for singularly perturbed differential difference equation with small shifts in the numerical treatment for such type of boundary value problems first we use taylor approximation to tackle the terms containing small shifts which converts it to a boundary value problem for singularly perturbed differential equation a rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives up to third order then a parameter uniform difference scheme is constructed to solve the boundary value problem so obtained a parameter uniform error estimate for the numerical scheme so constructed is established though the convergence of the difference scheme is almost linear but its beauty is that it converges independently of the singular perturbation parameter i e the numerical scheme converges for each value of the singular perturbation parameter however small it may be but remains positive several test examples are solved to demonstrate the efficiency of the numerical scheme presented in the paper and to show the effect of the small shift on the solution behavior

e uniformly convergent non standard finite difference methods for singularly perturbed differential difference equations with small delay abstract non standard finite difference methods nsfdms now a days are playing an important role in solving the real life problems governed by odes and or by pdes many differential models of sciences and engineerings for which the existing methodologies do not give reliable results these nsfdms are solving them competitively to this end in this paper we consider second order linear singularly perturbed differential difference equations using the second of the five non standard modeling rules of mickens r e mickens nonstandard finite difference models of differential equations world scientific singapore 1994 the new finite difference methods are obtained for the particular cases of these problems this rule suggests us to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures most of the significant properties of the governing differential equation s both theoretically and numerically we show that these nsfdms are e uniformly convergent

computational method based on reproducing kernel for solving singularly perturbed differential difference equations with a delay abstract in this paper reproducing kernel hilbert space method rkhsm based on collocation scheme is used to solve singularly perturbed second order differential difference equations we implement rkhsm without gram schmidt orthogonalization process for singularly perturbed differential difference equation with boundary layer behavior and also oscillatory behavior with small delay rkhsm in this study is based on the division of the problem domain into two subintervals one with boundary layer and the other one without such a boundary layer several numerical examples are studied to demonstrate the accuracy of the present method results of the present scheme indicate that new algorithm has the following advantages small computational work fast convergence and high precision

a simple adaptive grid method in two dimensions this paper gives an interpretation of the concept of equidistribution in the context of adaptive grid generation for multidimensional problems it is shown that the equidistribution principle cannot be satisfied throughout the domain of the problem and based on this recognition a local equidistribution principle is developed a discrete formulation is described for grid generation in two space dimensions and a smoothing mechanism is presented for improving mesh quality the adaptive grid method that is constructed contains three grid quality parameters numerical examples illustrate adaptive grid generation using a prescribed monitor function and grid generation for numerical solution of partial differential equations results show that the method produces high quality grids and that it is fairly insensitive to the choice of parameters

optimal adaptive grids of least squares finite element methods in two spatial dimensions this article concerns a procedure to generate optimal adaptive grids for convection dominated problems in two spatial dimensions based on least squares finite element approximations the procedure extends a one dimensional equidistribution principle which minimizes the interpolation error in some norms the idea is to select two directions which can reflect the physics of the problems and then apply the one dimensional equidistribution principle to the chosen directions model problems considered are the two dimensional convection diffusion problems where boundary and interior layers occur numerical results of model problems illustrating the efficiency of the proposed scheme are presented in addition to avoid skewed mesh in the optimal grids generated by the algorithm an unstructured local mesh smoothing will be considered in the least squares approximations comparisons with the gakerkin finite element method will also be provided

uniformly convergent non standard finite difference methods for singularly perturbed differential difference equations with delay and advance a new class of fitted operator finite difference methods are constructed via non standard finite difference methods nsfdm s for the numerical solution of singularly perturbed differential difference equations having both delay and advance arguments the main idea behind the construction of our method s is to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures significant properties of the governing differential equation and thus provides the reliable numerical results unlike other fofdms constructed in standard ways the methods that we present in this paper are fairly simple to construct and thus enrich the class of fitted operator methods by adding these new methods these methods are shown to be e uniformly convergent with order two which is the highest possible order of convergence obtained via any fitted operator method for the problems under consideration this paper further clarifies several doubts e g why a particular scheme is not suitable for the whole range of values of the associated parameters and what could be the possible remedies finally we provide some numerical examples which illustrate the theoretical findings copyright 2005 john wiley sons ltd

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

comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problems this paper deals with the adaptive mesh generation for singularly perturbed nonlinear parameterized problems with a comparative research study on them we propose an a posteriori error estimate for singularly perturbed parameterized problems by moving mesh methods with fixed number of mesh points the well known a priori meshes are compared with the proposed one the comparison results show that the proposed numerical method is highly effective for the generation of layer adapted a posteriori meshes a numerical experiment of the error behavior on different meshes is carried out to highlight the comparison of the approximated solutions

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

finite quotients of the multiplicative group of a finite dimensional division algebra are solvable we prove that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable let d be a finite dimensional division algebra having center k and let n d be a normal subgroup of finite index suppose d n is not solvable then we may assume that h d n is a minimal nonsolvable group mns group for short i e a nonsolvable group all of whose proper quotients are solvable our proof now has two main ingredients one ingredient is to show that the commuting graph of a finite mns group satisfies a certain property which we denote property 3 12 this property includes the requirement that the diameter of the commuting graph should be 3 but is in fact stronger another ingredient is to show that if the commuting graph of d n has the property 312 then n is open with respect to a nontrivial height one valuation of d assuming without loss as we may that k is finitely generated after establishing the openness of n when d n is an mns group we apply the nonexistence theorem whose proof uses induction on the transcendence degree of k over its prime subfield to eliminate h as a possible quotient of d thereby obtaining a contradiction and proving our main result

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

stability and accuracy of adapted finite element methods for singularly perturbed problems the stability and accuracy of a standard finite element method fem and a new streamline diffusion finite element method sdfem are studied in this paper for a one dimensional singularly perturbed connvection diffusion problem discretized on arbitrary grids both schemes are proven to produce stable and accurate approximations provided that the underlying grid is properly adapted to capture the singularity often in the form of boundary layers of the solution surprisingly the accuracy of the standard fem is shown to depend crucially on the uniformity of the grid away from the singularity in other words the accuracy of the adapted approximation is very sensitive to the perturbation of grid points in the region where the solution is smooth but in contrast it is robust with respect to perturbation of properly adapted grid inside the boundary layer motivated by this discovery a new sdfem is developed based on a special choice of the stabilization bubble function the new method is shown to have an optimal maximum norm stability and approximation property in the sense that u u n infty leq c inf v n in v n u v n infty where un is the sdfem approximation in linear finite element space vn of the exact solution u finally several optimal convergence results for the standard fem and the new sdfem are obtained and an open question about the optimal choice of the monitor function for the moving grid method is answered

a posteriori error estimation in maximum norm for a strongly coupled system of two singularly perturbed convection diffusion problems abstract a system of two singularly perturbed convection diffusion problems with strong coupling is studied this system is discretized by an upwind finite difference scheme on an arbitrary non uniform mesh an a posteriori error estimation in the maximum norm is derived based on the a posteriori error estimation a monitor function is proposed to design an adaptive grid algorithm especially to implement the adaptive grid approach we not only give a mesh generation algorithm but also give a sub algorithm which is called an arc length equidistribution algorithm numerical results are given to confirm our theoretical result

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

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

on the stability of mesh equidistribution strategies for time dependent partial differential equations abstract we study the stability of several mesh equidistribution schemes for time dependent partial differential equations in one space dimension the schemes move a finite difference or finite element mesh so that a given quantity is uniform over the domain we consider mesh moving methods that are based on solving a system of ordinary differential equations for the mesh velocities and show that some of these methods are unstable with respect to an equidistributing mesh when the partial differential system is dissipative using linear perturbation techniques we are able to develop simple criteria for determining the stability of a particular method and show how to construct stable differential systems for the mesh velocities several examples illustrating stable and unstable mesh motions are presented

a fitted numerical method for a system of partial delay differential equations we consider a system of two coupled partial delay differential equations pddes describing the dynamics of two cooperative species the original system is reduced to a system of ordinary delay differential equations ddes obtained by applying the method of lines then we construct a fitted operator finite difference method fofdm to solve this resulting system the model considered in this paper is very sensitive to small changes in the parameters associated in with the model depending on the values of these parameters the solution can be stable periodic and or aperiodic such behavior of the solution is exploited via the proposed fofdm this fofdm is analyzed for convergence and it is seen that this method is unconditionally stable and has the accuracy of o k h 2 where k and h denote time and space step sizes respectively some numerical results confirming theoretical observations are also presented these results are comparable with those obtained in the literature

improved reproducing kernel method for singularly perturbed differential difference equations with boundary layer behavior this paper is devoted to the numerical treatment of singularly perturbed differential difference equations with small delay whose solutions exhibiting boundary layer the reproducing kernel method presented in the previous work is not valid for singularly perturbed differential difference equations with small delay in this work we will improve the reproducing kernel method in order to obtain accurate approximation to the solutions of considered singularly perturbed differential difference equations two numerical examples are provided to show the performance of the present scheme

a numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations abstract a boundary value problem for second order singularly perturbed delay differential equation is considered when the delay argument is sufficiently small to tackle the delay term the researchers m k kadalbajoo k k sharma numerical analysis of singularly perturbed delay differential equations with layer behavior appl math comput 157 2004 11 28 r e o malley jr singular perturbation methods for ordinary differential equations springer verlag new york 1991 used taylor s series expansion and presented an asymptotic as well as numerical approach to solve such type boundary value problem but the existing methods in the literature fail in the case when the delay argument is bigger one because in this case the use of taylor s series expansion for the term containing delay may lead to a bad approximation in this paper to short out this problem we present a numerical scheme for solving such type of boundary value problems which works nicely in both the cases i e when delay argument is bigger one as well as smaller one to handle the delay argument we construct a special type of mesh so that the term containing delay lies on nodal points after discretization the proposed method is analyzed for stability and convergence to demonstrate the efficiency of the method and how the size of the delay argument and the coefficient of the delay term affects the layer behavior of the solution several test examples are considered

numerical study of singularly perturbed differential difference equation arising in the modeling of neuronal variability the objective of this paper is to present a numerical study of a class of boundary value problems of singularly perturbed differential difference equations spdde which arise in computational neuroscience in particular in the modeling of neuronal variability the mathematical modeling of the determination of the expected time for the generation of action potential in the nerve cells by random synaptic inputs in dendrites includes a general boundary value problem for singularly perturbed differential difference equation with shifts the problem considered in this paper exhibit turning point behavior which add to the complexity in the construction of numerical approximation to the solution of the problem as well as in obtaining theoretical estimates on the solution exponentially fitted finite difference scheme based on il in allen southwell fitting is used on a specially designed mesh some numerical examples are given to validate convergence and computational efficiency of the proposed numerical scheme effect of the shifts on the layer structure is illuminated for the considered examples

analysis of a system of singularly perturbed convection diffusion equations with strong coupling a system of singularly perturbed convection diffusion equations with strong coupling via the convection term is studied the system is discretized by an upwinded first order finite difference scheme for which both an a priori and an a posteriori error analysis in the maximum norm are conducted based on the a priori error bound we show that there exists a mesh with n subintervals that gives optimal error bounds of mathcal o left n 1 right which are robust with respect to the diffusion parameters the a posteriori error bounds are used to design an adaptive method numerical experiments complement the theoretical findings

a study of monitor functions for two dimensional adaptive mesh generation in this paper we study the problem of two dimensional adaptive mesh generation using a variational approach and specifically the effect that the monitor function has on the resulting mesh behavior the basic theoretical tools employed are green s function for elliptic problems and the eigendecomposition of symmetric positive definite matrices based upon this study a general strategy is suggested for how to choose the monitor function and numerical results are presented for illustrative purposes the three dimensional case is also briefly discussed it is noted that the strategy used here can be applied to other elliptic mesh generation techniques as well

reformulation of boundary value problems into standard form boundary value problems in ode s arising in various applications are frequently not in the standard form required by the currently existing software however many problems can be converted to such a form thus enabling the practitioner to take advantage of the availability and reliability of this general purpose software here various conversion devices are surveyed

singular perturbation analysis of boundary value problems for differential difference equations iii turning point problems this paper continues a study of a class of boundary value problems for linear second order differential difference equations in which the second order derivative is multiplied by a small parameter siam j appl math 42 1982 pp 502 531 45 1985 pp 687 707 the previous papers focused on problems involving boundary and interior layer phenomena rapid oscillations and resonance behavior the problems studied here have solutions which exhibit turning point behavior i e transition regions between rapid oscillations and exponential behavior the presence of the shift terms can induce large amplitudes and multiphase behavior over parts of the interval a combination of exact solutions singular perturbation methods and numerical computations are used in these studies

numerical methods on shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy abstract a numerical study is made for a class of boundary value problems of second order differential equations in which the highest order derivative is multiplied by a small parameter and both the differentiated convection and undifferentiated reaction terms are with negative shift we analyze three difference operators l k h k 1 2 3 a simple upwind scheme midpoint upwind scheme and a hybrid scheme respectively on a shishkin mesh to approximate the solution of the problem theoretical error bounds are established the accuracy by one of the grid adaptation strategies namely grid redistribution is examined by solving the considered problem using hybrid method for some values of and n as a result the accuracy gets improved by this grid adaptation strategy with almost the same computational cost a few numerical results exhibiting the performance of these three schemes are presented

the margulis platonov conjecture for sl1 d and 2 generation of finite simple groups we give a new proof of the margulis platonov conjecture for the groups sl 1 d which is shorter than the previous proofs and in which the use of the classification of finite simple groups is limited to the fact that every such group is generated by two elements the argument is based on further development of the methods of 18