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

the adaptation of structured grids to numerical solutions for transonic flow methods are described for adapting a structured grid in response to a numerical solution so that grid nodes become clustered where solution activity is high the aim being to reduce solution truncation errors without increasing the number of grid nodes employed or modifying their connectivity after introducing the concept of equidistribution and discussing options for the measurement of solution activity the paper concentrates mainly on two alternative techniques for producing smooth regular grids which apply constraints on this equidistribution the first technique described is based on a spring analogy and is demonstrated here with examples of two and three dimensional inviscid flows and with two dimensional viscous flows the second technique employs a poisson grid generator with adaptive terms included in the control functions and is demonstrated with a two dimensional inviscid flow a third method is then introduced termed the lpe method which allows a compromise to be chosen between grids generated by solving laplace equations poisson equations and equidistribution equations since this method is still being developed results are currently limited and tentative

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

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

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

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

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

an edge based unstructured mesh discretisation in geospherical framework an arbitrary finite volume approach is developed for discretising partial differential equations governing fluid flows on the sphere unconventionally for unstructured mesh global models the governing equations are cast in the anholonomic geospherical framework established in computational meteorology the resulting discretisation retains proven properties of the geospherical formulation while it offers the flexibility of unstructured meshes in enabling irregular spatial resolution the latter allows for a global enhancement of the spatial resolution away from the polar regions as well as for a local mesh refinement a class of non oscillatory forward in time edge based solvers is developed and applied to numerical examples of three dimensional hydrostatic flows including shallow water benchmarks on a rotating sphere

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

support theorem for the fundamental solution to the schrodinger equation on certain compact symmetric spaces abstract in this paper we construct the fundamental solution to the schodinger equation on a compact symmetric space with even root multiplicities using shift operators of heckman and opdam next we prove that the support of the fundamental solution becomes a lower dimensional subset at a rational time whereas its support and its singular support coincide with the whole symmetric space at an irrational time moreover we also show that generalized gauss sums appear in the expression of the fundamental solution

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

adaptivity techniques for compressible inviscid flows abstract to enhance the resolution of numerically simulated flowfields grid adaptivity is vitally important further to afford complete flexibility in the spatial discretisation of the flow domain both structured unstructured and combinations of grid types are likely to be required hence the development of grid adaptivity techniques for general grids is important this paper discusses adaptivity achieved through point enrichment point derefinement and point movement applied on general grids a flexible data structure is described together with suitable strategies for adaptively refining a computational grid applications are presented of adaptive grid refinement for two dimensional compressible flow simulation

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

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

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

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

two dimensional navier stokes equations with adaptivity on structured meshes abstract adaptivity techniques employed on two dimensional structured multiblock meshes are examined they include mesh point enrichment mesh movement and their combination in all approaches flow variables are taken for the adaptivity criteria steady state solutions for inviscid and viscous laminar flows are obtained by the finite volume method based on the jameson runge kutta scheme some applications for turbulent flows are also provided all three adaptivity techniques significantly improve the accuracy with which flow features such as shock waves and viscous boundary layers are captured the employment of the same integration method allows for a reliable comparison between the proposed techniques although the study has been performed on structured meshes the application to unstructured and hybrid meshes is straightforward

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

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

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

scattering theory for the laplacian on symmetric spaces of noncompact type and its application to a conjecture of strichartz abstract in this paper we develop the scattering theory for the laplacian on symmetric spaces of noncompact type we study the asymptotic properties of the resolvent in the framework of the agmon hormander space our approach is based on a detailed analysis of the helgason fourier transform and generalized spherical functions on symmetric spaces of noncompact type as an application of our scattering theory we prove a conjecture by strichartz concerning a characterization of a family of generalized eigenfunctions of the laplacian

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

adaptive remeshing for compressible flow computations an adaptive mesh procedure for improving the quality of steady state solutions of the euler equations in two dimensions is described the procedure is implemented in conjunction with a finite element solution algorithm using linear triangular elements and an explicit time stepping scheme the mesh adaptation is accomplished by regeneration using information provided by the computed solution on the current mesh the meshes produced are characterised by the appearance of stretched elements in the vicinity of one dimensional flow features and can exhibit a large variation in element size this allows solutions of enhanced quality to be produced in a computationally efficient manner