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

a fast method for solving the heat equation by layer potentials boundary integral formulations of the heat equation involve time convolutions in addition to surface potentials if m is the number of time steps and n is the number of degrees of freedom of the spatial discretization then the direct computation of a heat potential involves order n2m2 operations this article describes a fast method to compute three dimensional heat potentials which is based on chebyshev interpolation of the heat kernel in both space and time the computational complexity is order p4q2nm operations where p and q are the orders of the polynomial approximation in space and time

dynamical properties of s gap shifts and other shift spaces abstract we study the dynamical properties of certain shift spaces to help study these properties we introduce two new classes of shifts namely boundedly supermultiplicative bsm shifts and balanced shifts it turns out that any almost specified shift is both bsm and balanced and any balanced shift is bsm however as we will demonstrate there are examples of shifts which are bsm but not balanced we also study the measure theoretic properties of balanced shifts we show that a shift space admits a gibbs state if and only if it is balanced restricting ourselves to s gap shifts we relate certain dynamical properties of an s gap shift to combinatorial properties from expansions in non integer bases this identification allows us to use the machinery from expansions in non integer bases to give straightforward constructions of s gap shifts with certain desirable properties we show that for any q 0 1 there is an s gap shift which has the specification property and entropy q we also use this identification to address the question for a given q 0 1 how many s gap shifts exist with entropy q for certain exceptional values of q there is a unique s gap shift with this entropy

a fast galerkin method for parabolic space time boundary integral equations an efficient scheme for solving boundary integral equations of the heat equation based on the galerkin method is introduced the parabolic fast multipole method pfmm is applied to accelerate the evaluation of the thermal layer potentials in order to remain attractive for a wide range of applications a key issue is to ensure efficiency for a big variety of temporal to spatial mesh ratios within the parabolic galerkin fmm pgfmm it turns out that the temporal nearfield can become very costly to that end a modified fast gauss transform fgt is developed the complexity and convergence behavior of the method are analyzed and numerically investigated on a range of model problems the results demonstrate that the complexity is nearly optimal in the number of discretization parameters while the convergence rate of the galerkin method is preserved

series solution of nonlinear two point singularly perturbed boundary layer problems the present paper is concerned with the approximate analytic series solution of the nonlinear two point second order singularly perturbed boundary value problems in place of the traditional numerical perturbation or asymptotic methods a homotopy technique is employed it is shown that proper choices of an auxiliary linear operator and also an initial approximation during the implementation of the homotopy analysis method ham can yield uniformly valid and accurate solutions the fast convergence of the method is ensured by the optimal convergence control parameter obtained through the absolute residual error concept to demonstrate the favor of the ham over the traditional finite difference techniques several nonlinear problems have been solved and compared

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

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

on the finitary isomorphisms of markov shifts that have finite expected coding time a necessary condition is given for a finitary isomorphism between mixing markov shifts of equal entropy to have finite expected coding time

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

a fast sparse grid based space time boundary element method for the nonstationary heat equation this article presents a fast sparse grid based space time boundary element method for the solution of the nonstationary heat equation we make an indirect ansatz based on the thermal single layer potential which yields a first kind integral equation this integral equation is discretized by galerkin s method with respect to the sparse tensor product of the spatial and temporal ansatz spaces by employing the mathcal h matrix and toeplitz structure of the resulting discretized operators we arrive at an algorithm which computes the approximate solution in a complexity that essentially corresponds to that of the spatial discretization nevertheless the convergence rate is nearly the same as in case of a traditional discretization in full tensor product spaces

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

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

a fast algorithm for the evaluation of heat potentials a fast algorithm for the evaluation of heat potentials to appear in communications and pure and applied mathematics l greengard j strain may 1989 abstract numerical methods for solving the heat equation via potential theory have been hampered by the high cost of evaluating heat potentials when m points are used in the discretization of the boundary and n time steps are computed an amount of work of the order o n 2 m 2 has traditionally been required in this paper we present an algorithm which requires an amount of work of the order o n m and we observe speedups of five orders of magnitude for large scale problems thus the method makes it possible to solve the heat equation by potential theory in practical situations keywords heat equation potential theory fast algorithms integral equations introduction a classical approach to the solution of the heat equation u t u q t in a space time domain t t 0 t see fig 1 is through the use of heat potentials 3 7 given zero initial conditions one seeks a representation of u as a single layer heat potential z t z s x t k x x 0 t t 0 x 0 t 0 dx 0 dt 0 t 0 or a double layer heat potential d x t z t z t 0 k x x 0 t t 0 x 0 t 0 dx 0 dt 0 n 0 where k is a fundamental solution of the heat equation in some region containing q n 0 denotes the unit t outward normal to t t at x and is a surface density defined on t t 0 t yale university and courant institute of mathematical sciences new york university the work of this author was supported in part by the office of naval research under grant n00014 89 j 1527 in part by ibm under grant p00038437 and in part by a nsf mathematical sciences postdoctoral fellowship courant institute of mathematical sciences new york university the work of this author was supported by darpa afosr contract no f 49620 87 c 0065

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

discontinuous galerkin method based on non polynomial approximation spaces in this paper we develop discontinuous galerkin dg methods based on non polynomial approximation spaces for numerically solving time dependent hyperbolic and parabolic and steady state hyperbolic and elliptic partial differential equations pdes the algorithm is based on approximation spaces consisting of non polynomial elementary functions such as exponential functions trigonometric functions etc with the objective of obtaining better approximations for specific types of pdes and initial and boundary conditions it is shown that l2 stability and error estimates can be obtained when the approximation space is suitably selected it is also shown with numerical examples that a careful selection of the approximation space to fit individual pde and initial and boundary conditions often provides more accurate results than the dg methods based on the polynomial approximation spaces of the same order of accuracy

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

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

an algorithm for solving nonlinear singular perturbation problems with mechanization in this paper by using the theories and methods of mathematical analysis and computer algebra a reliable algorithm of pade approximation method for solving nonlinear singular perturbation problems was established a new maple procedure spp was established too the procedure spp give not only the approximate analytic solutions but also the numerical solutions of spp six examples are presented to illustrate the implementation of the program the method is very simple and effective for most of singular perturbation problems this would be useful for solving differential equations

fitted fourth order tridiagonal finite difference method for singular perturbation problems in this paper a fitted fourth order tridiagonal finite difference scheme is presented for solving singularly perturbed two point boundary value problems with the boundary layer at one end left or right point we have taken a fourth order tridiagonal finite difference scheme by m m chawla a fourth order tridiagonal finite difference method for general nonlinear two point boundary value problems with mixed boundary conditions j inst maths appl 21 1978 83 93 and introduced a fitting factor the fitting factor is obtained from the theory of singular perturbations thomas algorithm is used to solve the system to demonstrate the applicability of the present method we have solved five linear problems three with left end and two with right end boundary layers solutions of these problems using the present fitted method are compared with chawla s solutions from the results it is observed that the present method is stable and has better approximation to the exact solution

the method of layer potentials for the heat equation in lipschitz cylinders pour l equation de la chaleur dans des cylindres de lipschitz on etudie des problemes aux valeurs limites d initiales avec conditions de neumann et de dirichlet

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

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

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

time domain boundary integral equation modeling of heat transmission problems this paper investigates the numerical modeling of a time dependent heat transmission problem by the convolution quadrature boundary element method it introduces the latest theoretical development into the error analysis of the numerical scheme semigroup theory is applied to obtain stability in the spatially semidiscrete scheme functional calculus is employed to yield convergence in the fully discrete scheme we compare the results to a more traditional laplace domain approach

quadrature for parabolic galerkin bem with moving surfaces abstract the successful implementation of the galerkin boundary element method hinges on the accurate and effective quadrature of the influence coefficients for parabolic boundary integral operators quadrature must be performed in space and time where integrals have singularities when source and evaluation points coincide for problems where the surface is fixed the time integration can be performed analytically but for moving geometries numerical quadrature in space and time must be used for this case a set of transformations is derived that render the singular space time integrals into smooth integrals that can be treated with standard tensor product gauss quadrature rules this methodology can be applied to the heat equation and to transient stokes flow

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

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

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

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

collocation for singular perturbation problems i first order systems with constant coefficients the application of collocation methods for the numerical solution of singularly perturbed ordinary differential equations is investigated collocation at gauss radau and lobatto points is considered for both initial and boundary value problems for first order systems with constant coefficients particular attention is paid to symmetric schemes for boundary value problems these problems may have boundary layers at both interval ends br our analysis shows that certain collocation schemes in particular those based on gauss or lobatto points do perform very well on such problems provided that a fine mesh with steps proportional to the layers width is used in the layers only and a coarse mesh just fine enough to resolve the solution of the reduced problem is used in between ways to construct appropriate layer meshes are proposed of all methods considered the lobatto schemes appear to be the most promising class of methods as they essentially retain their usual superconvergence power for the smooth reduced solution whereas gauss legendre schemes do not br we also investigate the conditioning of the linear systems of equations arizing in the discretization of the boundary value problem for a row equilibrated version of the discretized system we obtain a pleasantly small bound on the maximum norm condition number which indicates that these systems can be solved safely by gaussian elimination with scaled partial pivoting

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

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

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

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

mechanization for solving spp by reducing order method in this paper by using the theories and methods of mathematical analysis and computer algebra a reliable algorithm of reduction of order approximation method for solving singular perturbation problems was established a new maple procedure redordproc was established too the procedure redordproc give not only the approximate analytic solutions but also the numerical solutions of the kinetics problems some examples are presented to illustrate the implementation of the program

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

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

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

a parallel space time boundary element method for the heat equation abstract in this paper we introduce a new parallel solver for the weakly singular space time boundary integral equation for the heat equation the space time boundary mesh is decomposed into a given number of submeshes pairs of the submeshes represent dense blocks in the system matrices which are distributed among computational nodes by an algorithm based on a cyclic decomposition of complete graphs ensuring load balance in addition we employ vectorization and threading in shared memory to ensure intra node efficiency we present scalability experiments on different cpu architectures to evaluate the performance of the proposed parallelization techniques all levels of parallelism allow us to tackle large problems and lead to an almost optimal speedup

an efficient galerkin boundary element method for the transient heat equation we present boundary integral representations of several initial boundary value problems related to the heat equation a galerkin discretization with piecewise constant functions in time and piecewise linear functions in space leads to optimal a priori error estimates provided that the meshwidths in space and time satisfy h t mathcal o h x 2 each time step involves the solution of a linear system whose spectral condition number is independent of the refinement under the same assumption on the mesh we show that if the parabolic multipole method is used to apply parabolic boundary integral operators the overall complexity of the scheme is log linear while preserving the convergence of the galerkin discretization method the theoretical estimates are confirmed numerically at the end of the paper