an invariant imbedding method for computation of eigenlengths of singular two point boundary value problems in this article we have described an invariant imbedding method for calculating the smallest eigenlength of a singular tpbvp with the singularity at the origin the invariant imbedding yields a first order nonlinear equation called a riccati equation and also gives the initial conditions at the origin for this equation with the aid of theorem 8 in section 3 we numerically integrate the riccati equation to blowup which gives our computed eigenlength in closing we would like to comment on the numerical merits of the integration to blowup technique on the basis of the examples presented it appears that this technique combined with the available numerical integrators with variable step size is capable of producing accurate results the feature of a variable step size is essential as the value of z approaches the actual eigenlength however it is desirable to have a priori estimate or bounds of the eigenlength similar to those of boland and nelson 2 for the nonsingular case the singular system however presents difficulties due to the lack of sign conditions on the coefficient matrices in obtaining such bounds hopefully an investigation of the matrix r z will yield these results

approximation algorithms for three dimensional assignment problems with triangle inequalities consider the following classical formulation of the axial three dimensional assignment problem 3da see e g balas and saltzman 1989 given is a complete tripartite graph kn n n i j k i j i k j k where i j k are disjoint sets of size n and a cost c ijk for each triangle i j k i j k the problem 3da is to find a subset a of n triangles a i j k such that every element of i j k occurs in exactly one triangle of a and the total cost c a i j k ac ijk is minimized some recent references to this problem are balas and saltzman 1989 frieze 1974 frieze and yadegar 1981 hansen and kaufman 1973

high accuracy cubic spline finite difference approximation for the solution of one space dimensional non linear wave equations abstract in this paper we propose a new three level implicit nine point compact cubic spline finite difference formulation of order two in time and four in space directions based on cubic spline approximation in x direction and finite difference approximation in t direction for the numerical solution of one space dimensional second order non linear hyperbolic partial differential equations we describe the mathematical formulation procedure in details and also discuss how our formulation is able to handle wave equation in polar coordinates the proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable numerical results are provided to justify the usefulness of the proposed method

numerical study of fisher s reaction diffusion equation by the sinc collocation method fisher s equation which describes a balance between linear diffusion and nonlinear reaction or multiplication is studied numerically by the sinc collocation method the derivatives and integrals are replaced by the necessary matrices and a system of algebraic equations is obtained to approximate solution of the problem the error in the approximation of the solution is shown to converge at an exponential rate numerical examples are given to illustrate the accuracy and the implementation of the method the results show that any local initial disturbance can propagate with a constant limiting speed when time becomes sufficiently large both the limiting wave fronts and the limiting speed are independent of the initial values

compositional verification and 3 valued abstractions join forces abstract two of the most promising approaches to fighting the state explosion problem are abstraction and compositional verification in this work we join their forces to obtain a novel fully automatic compositional technique that can determine the truth value of the full calculus with respect to a given system given a system m m 1 m 2 we view each component m i as an abstraction m i of the global system the abstract component m i is defined using a 3 valued semantics so that whenever a calculus formula has a definite value true or false on m i the same value holds also for m thus can be checked on either m 1 or m 2 or both and if any of them returns a definite result then this result holds also for m if both checks result in an indefinite value the composition of the components needs to be considered however instead of constructing the composition of m 1 and m 2 our approach identifies and composes only the parts of the components in which their composition is necessary in order to conclude the truth value of indeed relevant it ignores the parts which can be handled separately the resulting model is often can potentially be significantly smaller than the full system furthermore we explain how our compositional approach can be combined with abstraction of the components in order to further reduce the size of the checked components the result is an incremental compositional abstraction refinement framework which resembles automatic assume guarantee reasoning

numerical solution of singular regular boundary value problems by pole detection with qd algorithm in this paper we present a method using qd algorithm combined with finite difference method for numerically solving singular boundary value problems for certain ordinary differential equation having singular coefficients we can also use qd algorithm combined with other numerical methods these problems arise when reducing partial differential equation to ordinary differential equation by physical symmetry some illustrative examples are given to demonstrate the efficiency and high accuracy of the proposed method and compare it to the numerical results made with other methods

a note on parametric spline function approximation abstract the application of a parametric spline function which depends on a parameter p 0 to two linear partial differential equations is discussed for p 0 the parametric spline reduces to the ordinary cubic spline

high accuracy non polynomial spline in compression method for one space dimensional quasi linear hyperbolic equations with significant first order space derivative term abstract in this paper we propose a new three level implicit nine point compact non polynomial spline in compression finite difference method of order two in time and four in space directions based on non polynomial spline approximation in x direction and central difference approximation in t direction for the numerical solution of one space dimensional second order quasi linear hyperbolic partial differential equations with first order space derivative term we describe the mathematical details of the method and also discuss how our method is able to handle wave equation in polar coordinates the proposed method when applied to a linear hyperbolic equation is shown to be unconditionally stable numerical results are provided to justify the usefulness of the proposed method

parameters spline methods for the solution of hyperbolic equations abstract in this paper by using a parameters cubic spline in space and compact finite difference in time direction we get a class of finite difference schemes for solving second order hyperbolic equations with mixed boundary conditions stability analysis of the methods have been carried out it has been shown that by suitable choosing the cubic spline parameters most of the previous known methods for homogeneous and non homogeneous cases can be derived from our methods we also obtain new high accuracy schemes of o h 2 2 h 2 and o h 4 2 h 4 numerical comparison with rashidinia s method j rashidinia et al spline methods for the solutions of hyperbolic equations appl math comput 190 2007 882 886 shows the superiority of our presented schemes

non polynomial spline solution of singularly perturbed boundary value problems a generalized scheme based on quartic non polynomial spline functions is proposed this scheme is designed for numerical solution of singularly perturbed two point boundary value problems arising in the study of science and engineering the scheme leads to a five diagonal linear system of equations convergence analysis of the method is briefly discussed two numerical examples each of constant and variable coefficients are given to show practical usefulness of the method

numerical solution of singular boundary value problems via chebyshev polynomial and b spline in this paper we present b spline method for numerically solving singular two point boundary value problems for certain ordinary differential equation having singular coefficients these problems arise when reducing partial differential equation to ordinary differential equation by physical symmetry to remove the singularity we first use chebyshev economizition in the vicinity of the singular point and the boundary condition at a point email protected in the vicinity of the singularity is derived the resulting regular bvp is then efficiently treated by employing b spline for finding the numerical solution some examples have been included and comparison of the numerical results made with other methods

tension spline method for solution of non linear fisher equation tension spline method is proposed for the non linear fisher equation with initial boundary values the three time level implicit method based on the non polynomial cubic tension spline is developed for the solution of the non linear reaction diffusion equation the method involves the parameters and its order can be increased from o k 2 k 2 h 2 h 2 to o k 2 k 2 h 2 h 4 by an appropriate choice of the parameters the stability of proposed method is analyzed finally numerical results are presented to demonstrate the accuracy and efficiency of this method

an algorithm for the three index assignment problem we describe a branch and bound algorithm for solving the axial three index assignment problem the main features of the algorithm include a lagrangian relaxation that incorporates a class of facet inequalities and is solved by a modified subgradient procedure to find good lower bounds a primal heuristic based on the principle of minimizing maximum regret plus a variable depth interchange phase for finding good upper bounds and a novel branching strategy that exploits problem structure to fix several variables at each node and reduce the size of the total enumeration tree computational experience is reported on problems with up to 78 equations and 17 576 variables the primal heuristics were tested on problems with up to 210 equations and 343 000 variables

hybrid spline difference method for the burgers equation abstract this study developed a high accuracy hybrid spline difference method and deduced a difference equation that is very similar to finite difference method from the concept of spline difference as validated by nonlinear burgers equation the concept of difference made the computational process as simple as the finite difference method and easy to be implemented the free parameters 1 12 and 1 6 were combined in the concept of spline hybrid in order to increase the accuracy of the first and second derivatives of space from o h 2 of finite difference method to the o h 4 the accuracy was improved and the numerical oscillation with the increase in parameter r e was improved greatly

a class of difference scheme for solving telegraph equation by new non polynomial spline methods abstract in this paper by using a new non polynomial parameters cubic spline in space direction and compact finite difference in time direction we get a class of new high accuracy scheme of o 4 h 2 and o 4 h 4 for solving telegraph equation if we suitably choose the cubic spline parameters meanwhile stability condition of the difference scheme has been carried out finally numerical examples are used to illustrate the efficiency of the new difference scheme

a new spline in compression approximation for one space dimensional quasilinear parabolic equations on a variable mesh in this paper we propose a new two level implicit method of order two in time and four in space directions based on spline in compression approximation for the numerical solution of one space dimensional quasi linear parabolic partial differential equation on a uniform mesh the derivation and the stability of the proposed method are discussed in details we have extended the method to non uniform mesh numerical results are given to illustrate the usefulness of the proposed method

a new unconditionally stable compact difference scheme of o τ2 h4 for the 1d linear hyperbolic equation abstract in this paper we propose a three level compact difference scheme of o 2 h 4 for the difference solution of one dimensional second order non homogeneous linear hyperbolic equation u tt 2 u t 2 u u xx f x t 0 x 1 t 0 where 0 stability analysis of the method has been carried out finally numerical examples are used to illustrate the efficiency of the new difference scheme

multi symplectic runge kutta collocation methods for hamiltonian wave equations a number of conservative pdes like various wave equations allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes we show that gauss legendre collocation in space and time leads to multi symplectic integrators i e to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for hamiltonian odes we also discuss the issue of conservation of energy and momentum since time discretization by a gauss legendre method is computational rather expensive we suggest several semi explicit multi symplectic methods based on gauss legendre collocation in space and explicit or linearly implicit symplectic discretizations in time

invariant imbedding applied to homogeneous two point boundary value problems with a singularity of the first kind it is shown that previous work of elder can be used to extend the version of invariant imbedding due to m r scott to homogeneous vector differential systems having a singularity of the first kind the boundary conditions considered consist of existence finite at the singularity and specified values for some subset of the dependent variables at a second point the important special case of a second order equation is discussed in some detail computational considerations are discussed and numerical examples are presented

numerical simulation on hyperbolic diffusion equations using modified cubic b spline differential quadrature methods in this article a modified cubic b spline differential quadrature method mcb dqm is proposed to solve a hyperbolic diffusion problem in which flow motion is affected by both convection and diffusion one dimensional hyperbolic non homogeneous heat wave and telegraph equations are also considered along with two dimensional hyperbolic diffusion problem the method reduces the hyperbolic problem into a system of nonlinear ordinary differential equations the system is then solved by the optimal four stage three order strong stability preserving time stepping runge kutta ssp rk43 scheme the reliability and efficiency of the method has been tested on seven examples the stability of the method is also discussed and found to be unconditionally stable

cubic spline fractal solutions of two point boundary value problems with a non homogeneous nowhere differentiable term abstract fractal interpolation functions fifs supplement and subsume all classical interpolants the major advantage by the use of fractal functions is that they can capture either the irregularity or the smoothness associated with a function this work proposes the use of cubic spline fifs through moments for the solutions of a two point boundary value problem bvp involving a complicated non smooth function in the non homogeneous second order differential equation in particular we have taken a second order linear bvp y x q x y x p x y x r x with the dirichlet s boundary conditions where p x and q x are smooth but r x may be a continuous nowhere differentiable function using the discretized version of the differential equation the moments are computed through a tridiagonal system obtained from the continuity conditions at the internal grids and endpoint conditions by the derivative function these moments are then used to construct the cubic fractal spline solution of the bvp where the non smooth nature of y can be captured by fractal methodology when the scaling factors associated with the fractal spline are taken as zero the fractal solution reduces to the classical cubic spline solution of the bvp we prove that the proposed method is convergent based on its truncation error analysis at grid points numerical examples are given to support the advantage of the fractal methodology

an unconditionally stable fourth order method for telegraph equation based on hermite interpolation there have been many approaches to solve telegraph equations numerically in literature however very few are two level difference schemes in solving telegraph equations with dirichlet boundary conditions in this paper based on cubic hermite interpolation a two level method is presented for the numerical solutions of one dimensional telegraph equations the accuracy of the present scheme is of order o k 4 h 4 it is proved that the scheme is unconditionally stable numerical results are given to illustrate the efficiency of our method

a new algorithm for a class of singular boundary value problems abstract in this paper we present a new algorithm to solve a class of singular boundary value problems in the reproducing kernel space w 2 3 0 1 the algorithm is efficiently applied to solving some model problems with the comparison between the numerical solutions and the exact solutions it is demonstrated by the numerical examples that this algorithm is of high precision

a lagrangian relaxation algorithm for multidimensional assignment problems arising from multitarget tracking the central problem in multitarget tracking is the data association problem of partitioning the observations into tracks in some optimal way so that an accurate estimate of the true tracks can be recovered this work considers what is perhaps the simplest multitarget tracking problem in a setting where the issues are easily delineated i e straight lines in two dimensional space time with an error component introduced into the observations a multidimensional assignment problem is formulated using gating techniques to introduce sparsity into the problem and filtering techniques to generate tracks which are then used to score each assignment of a collection of observations to a filtered track problem complexity is further reduced by decomposing the problem into disjoint components which can then be solved independently a recursive lagrangian relaxation algorithm is developed to obtain high quality suboptimal solutions in real time the algorithms are however applicable to a large class of sparse mul

spline methods for the solutions of hyperbolic equations abstract second order hyperbolic equations with mixed boundary conditions are solved by using a non polynomial cubic spline in space and finite difference in time direction we develop new classes of three level methods stability analysis of the methods have been carried out it has been shown that by suitably choosing the cubic spline parameters most of the previous known methods for homogeneous and non homogeneous cases can be derived from our methods we also obtain new high accuracy schemes of o k 2 h 2 and o k 2 h 4 numerical example is given to illustrate the applicability and efficiency of the new methods

quartic spline methods for solving one dimensional telegraphic equations in this paper by using a quartic spline in space and generalized trapezoidal formula in time direction one dimensional telegraphic equations are solved we present new classes of two level schemes the accuracy of these methods are o k 2 h 4 it has been shown that by suitably choosing parameter a high accuracy scheme of o k 3 h 4 can be derived from our methods numerical results demonstrate the superiority of the new scheme

spline methods for the solution of fourth order parabolic partial differential equations in this paper a fourth order non homogeneous parabolic partial differential equation that governs the behaviour of a vibrating beam is solved by using a new three level method based on parametric quintic spline in space and finite difference discretization in time stability analysis of the method has been carried out it has been shown that by suitably choosing the parameters most of the previous known methods for homogeneous and non homogeneous cases can be derived from our method we also obtain two new high accuracy schemes of o k 4 h 6 and o k 4 h 8 and two new schemes which are analogues of jain s formula for the non homogeneous case comparison of our results with those of some known methods show the superiority of the present approach

fourth order approximations at first time level linear stability analysis and the numerical solution of multidimensional second order nonlinear hyperbolic equations in polar coordinates in this article three level implicit difference schemes of o k4 k2h2 h4 where k 0 h 0 are grid sizes in time and space coordinates respectively are proposed for the numerical solution of one two and three space dimensional nonlinear wave equations in polar coordinates subject to appropriate initial and dirichlet boundary conditions we also discuss fourth order approximation at first time level for more general case we also obtain the stability range of the difference scheme when applied to a test equation utt urr aruu ar2u g r t a 1 and 2 numerical examples are provided to demonstrate the required order of convergence of the methods

variable mesh methods for the numerical solution of two point singular perturbation problems abstract in this paper variable mesh difference methods of third order are derived for the solution of the two point second order singular perturbation problems y f x y y e these methods are applied to a number of examples and their superiority over the existing methods is shown

cubic spline solution of singularly perturbed boundary value problems with significant first derivatives abstract we develop a numerical technique for a class of singularly perturbed two point singular boundary value problems on an uniform mesh using polynomial cubic spline the scheme derived in this paper is second order accurate the resulting linear system of equations has been solved by using a tri diagonal solver numerical results are provided to illustrate the proposed method and to compared with the methods in r k mohanty urvashi arora a family of non uniform mesh tension spline methods for singularly perturbed two point singular boundary value problems with significant first derivatives appl math comput 172 2006 531 544 m k kadalbajoo v k aggarwal fitted mesh b spline method for solving a class of singular singularly perturbed boundary value problems int j comput math 82 2005 67 76

high order variable mesh off step discretization for the solution of 1 d non linear hyperbolic equation abstract in this paper we propose a new high order three level implicit method based on off step discretization on a non uniform mesh for the solution of 1 d non linear hyperbolic partial differential equation of the form u tt u xx g x t u u x u t subject to appropriate initial and dirichlet boundary conditions we use only three evaluations of the function g and three grid points at each time level in a compact cell our method is directly applicable to the wave equation in polar coordinates and we do not require any special technique to handle singular coefficients of the differential equation the method is convergent for uniform mesh numerical results are provided to justify the usefulness of the proposed method

on the galerkin and collocation methods for two point boundary value problems using sinc bases this paper presents a study of the performance of the collocation and galerkin methods using sinc basis functions for solving linear and nonlinear second order two point boundary value problems the two methods have the linear systems solved by the q r method and have the nonlinear systems solved by newton s method this study shows that the collocation method performs better than the galerkin method for the cases considered

an unconditionally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficients we propose a three level implicit unconditionally stable difference scheme of o k 2 h 2 for the difference solution of second order linear hyperbolic equation u t email protected x t u email protected 2 x t u a x t u x x f x t 00 subject to appropriate initial and dirichlet boundary conditions where a x t 0 a x t b x t 0 the proposed formula is applicable to the problems having singularity at x 0 the resulting tri diagonal linear system of equations is solved by using gauss elimination method numerical examples are provided to illustrate the unconditionally stable character of the proposed method

an o h4 accurate cubic spline tage method for nonlinear singular two point boundary value problems in this paper we propose two parameter alternating group explicit tage method for the numerical solution of u aru ar 2u f r 0

stability interval for explicit difference schemes for multi dimensional second order hyperbolic equations with significant first order space derivative terms in this piece of work we introduce a new idea and obtain stability interval for explicit difference schemes of o k2 h2 for one two and three space dimensional second order hyperbolic equations utt a x t uxx x t ux 2 2 x t u utt a x y t uxx b x y t uyy x y t ux x y t uy 2 2 x y t u and utt a x y z t uxx b x y z t uyy c x y z t uzz x y z t ux x y z t uy x y z t uz 2 2 x y z t u 0 0 subject to appropriate initial and dirichlet boundary conditions where h 0 and k 0 are grid sizes in space and time coordinates respectively a new idea is also introduced to obtain explicit difference schemes of o k2 in order to obtain numerical solution of u at first time step in a different manner

a new fourth order discretization for singularly perturbed two dimensional non linear elliptic boundary value problems abstract in this article we derive a new difference method of o h4 so called arithmetic average discretization for the solution of two dimensional non linear singularly perturbed elliptic partial differential equation of the form e uxx uyy f x y u ux uy 0 0 is a small parameter we also derive new methods of o h4 for the estimates of u n which are quite often of interest in many physical problems in all cases we require only 9 grid points and a single computational cell the main advantage of the proposed methods is that the methods are directly applicable to singular problems we do not require any special technique or modification to solve singular problems numerical results are provided to demonstrate the usefulness of the methods discussed

a new high order space derivative discretization for 3d quasi linear hyperbolic partial differential equations abstract in this paper we propose a new high accuracy numerical method of o k2 k2h2 h4 for the solution of three dimensional quasi linear hyperbolic partial differential equations where k 0 and h 0 are mesh sizes in time and space directions respectively we mainly discretize the space derivative terms using fourth order approximation and time derivative term using second order approximation we describe the derivation procedure in details and also discuss how our formulation is able to handle the wave equation in polar coordinates the proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable the proposed method behaves like a fourth order method for a fixed value of k h 2 some examples and their numerical results are provided to justify the usefulness of the proposed method

spline function approximation for differential equations abstract a spline function which depends on a parameter p 0 has been constructed the spline function is of trigonometric and polynomial order one for p 0 it reduces to ordinary cubic spline the application of the parametric spline to the solution of the system of ordinary and partial differential equations is discussed the numerical results are given

cubic spline solution of two point boundary value problems with significant first derivatives a cubic spline function has been applied to the solution of the second order nonlinear two point boundary value problems with significant first derivatives the method obtained involves some parameters these parameters may be suitably chosen to obtain an oscillation free solution of the test equation y ky k 1 depending upon the choice of the parameters methods of order two three and four may be obtained the methods have been tested on both linear and nonlinear problems

on a class of singular boundary value problems with singular perturbation abstract in this paper we investigate a class of singular second order differential equations with singular perturbation subject to three point boundary value conditions whose solution exhibits a couple of boundary layers at two endpoints we first establish a lower upper solutions theorem by using the schauder fixed point theorem by the asymptotic expansions and the lower upper solutions theorem we obtain the existence asymptotic estimates and uniqueness for the proposed problem several examples are given for illustrating our results

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

a survey on parametric spline function approximation this survey paper contains a large amount of material and indeed can serve as an introduction to some of the ideas and methods for the solution of ordinary and partial differential equations starting from schoenberg s work quart appl math 4 1946 345 369 the parametric spline function which depends on a parameter w 0 is reduces to the ordinary cubic or quintic spline for w 0 a note on parametric spline function approximation which is special case of this work has been published in comp math applics 29 1995 67 73 this article deals with the odd order parametric spline relations

tage iterative algorithm and nonpolynomial spline basis for the solution of nonlinear singular second order ordinary differential equations abstract in the present paper we discuss three point difference method based on nonpolynomial spline basis for the second order ordinary differential equation difference schemes are derived for linear and nonlinear case and are used to solve via two parameter alternating group explicit iterative algorithm the schemes have a fourth and second order of uniform convergence for the choice of the parameters involved in the method computational results are presented comparing the two methods in terms of accuracy and execution times the results indicate the advantage of using parallel implementation of the new method

mixed spline function method for reaction subdiffusion equations in this paper we develop two classes of finite difference schemes for the reaction subdiffusion equations by using a mixed spline function in space direction forward and backward difference in time direction respectively it has been shown that some of the previous known difference schemes can be derived from our schemes if we suitably choose the spline parameters by fourier method we prove that one class of difference scheme is unconditionally stable and convergent the other is conditionally stable and convergent finally some numerical results are provided to demonstrate the effectiveness of the proposed difference schemes

numerical solution of singular boundary value problems by invariant imbedding abstract an invariant imbedding method is proposed for numerically solving singular boundary value problems with a regular singular point at one end of the interval the singular problem is first reduced to a regular boundary value problem by using series solution in the vicinity of the singular point to subtract out the singularity the regular boundary value problem is then solved by employing invariant imbedding method some numerical examples have been included to demonstrate the efficiency of the method

the solution of a time dependent problem by the b spline method abstract in this paper we apply the b spline method for solving the boundary value problems involving non homogeneous heat convection diffusion wave and telegraph equations the accuracy of the method for the equation is o t h 2 several examples are given to verify the reliability and efficiency of the method

a framework for compositional verification of multi valued systems via abstraction refinement we present a framework for fully automated compositional verification of calculus specifications over multi valued systems based on abstraction and refinement in a multi valued model of a system both the system transitions and the state labels are assigned values from a lattice we formalize our framework based on bilattices consisting of a truth lattice and an information lattice formulas are interpreted on the truth lattice the information lattice determines how definite the value is in terms of the concrete system being modeled our compositional approach views each component as an abstraction of the entire system and checks it separately only if all individual checks return indefinite values the parts of the components which are responsible for these values are composed and checked if the latter check is still indefinite a refinement of the multi valued system is needed refinement is aimed at increasing the information level of model details

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

the impact of smooth w grids in the numerical solution of singular perturbation two point boundary value problems this paper develops a semi analytic technique for generating smooth nonuniform grids for the numerical solution of singularly perturbed two point boundary value problems it is based on the usual idea of mapping a uniform grid to the desired nonuniform grid we introduce the w grid which depends on the perturbation parameter epsilon 1 for problems on 0 1 with a boundary layer at one end point the local mesh width h i x i 1 x i with 0 x 0 x 1 x n 1 is condensed at either 0 or 1 two simple 2nd order finite element and finite difference methods are combined with the new mesh and computational experiments demonstrate the advantages of the smooth w grid compared to the well known piecewise uniform shishkin mesh for small epsilon neither the finite difference method nor the finite element method produces satisfactory results on the shishkin mesh by contrast accuracy is vastly improved on the w grid which typically produces the nominal 2nd order behavior in l 2 for large as well as small values of n and over a wide range of values of epsilon we conclude that the smoothness of the mesh is of crucial importance to accuracy efficiency and robustness published by elsevier inc less

when not losing is better than winning abstraction and refinement for the full μ calculus this work presents a novel game based approach to abstraction refinement for the full m calculus interpreted over 3 valued semantics a novel notion of non losing strategy is introduced and exploited for refinement previous works on refinement in the context of 3 valued semantics require a direct algorithm for solving a 3 valued model checking game this was necessary in order to have the information needed for refinement available on one game board in contrast while still considering a 3 valued model checking game here we reduce the problem of solving the game to solving two 2 valued model checking parity games in case the result is indefinite don t know the corresponding non losing strategies when combined hold all the information needed for refinement this approach is beneficial since it can use any solver for 2 valued parity games thus it can take advantage of newly developed such algorithms with improved complexity

cubic spline for solving singular two point boundary value problems in this paper we present another version of cubic spline method for numerically solving singular two point boundary value problems for certain ordinary differential equation having singular coefficients to remove the singularity we first use series solution in the vicinity of the singular point and the boundary condition at a point x d in the vicinity of the singularity is derived the resulting regular bvp is then efficiently treated by employing cubic spline for finding the numerical solution some examples have been included and comparison of the numerical results made with other methods

analytical linear stability criteria for the leap frog dufort frankel method abstract the criterion of linear numerical stability of the combined leap frog dufort frankel scheme for advective diffusive problems in two dimensions is x x 2 y y 2 u 2 x v 2 y t 2 1 where x and y are the grid spacings in the x and y directions u and v the velocity components x and y the diffusion coefficients and t the time step although this stability requirement does not depend explicitly on the magnitude of the diffusivity only on the ratio of the diffusivity coefficients the presence of the diffusive terms renders the criterion more severe than the one obtained for purely advective problems u x v y t 1 only in one dimension are both criteria identical therefore at more than one dimension the unconditionally stable scheme dufort frankel combined with the conditionally stable scheme leap frog leads to the more restrictive stability condition the maximum allowable time step occurs when the grid spacings in both directions have the same ratio as the square roots of the diffusivity coefficients a new method is presented for discussing the stability of a finite difference scheme without actually solving for the modulus of the amplification factor this method is also extended to more general cases

compact difference schemes for solving telegraphic equations with neumann boundary conditions in this paper based on the generalized trapezoidal formula a family of unconditionally stable compact difference schemes including a parameter q email protected 0 1 are discussed for the numerical solution of one dimensional telegraphic equations with neumann boundary conditions in general the accuracy of these schemes is second order in time and third order in time and third in space interestingly there exist a method of the family which is third order in time we also consider extensions of the presented difference schemes to a nonlinear problem numerical results demonstrate the superiority of our new schemes

a new fast algorithm based on half step discretization for one space dimensional quasilinear hyperbolic equations abstract in this article we describe a new compact three level implicit method of order four in time and space based on half step discretization for one space dimensional quasilinear hyperbolic equation u tt a x t u u xx f x t u u x u t defined in the semi infinite solution region where a 0 we require only nine grid points for the unknown variable u x t and two extra half step points each for x and t variables the proposed method is directly applicable to wave equation with singular coefficients which is main attraction of our work we do not require extra grid points for computation we describe the derivation of the method in detail the proposed method when applied to damped wave equation is shown to be unconditionally stable many benchmark problems are solved to confirm the fourth order convergence of the proposed method

difference schemes based on splines in compression for the solution of conservation laws abstract implicit finite difference schemes are developed for the efficient numerical integration of one and two dimensional scalar hyperbolic equations and a system of conservation laws using the spline in compression function approximation the difference schemes for one space dimension are of second order accuracy dissipative of order four and unconditionally stable the algorithms for two space dimensions are second order accurate and non iterative the difference schemes have been applied to problems with smooth and discontinuous initial conditions

a family of non uniform mesh tension spline methods for singularly perturbed two point singular boundary value problems with significant first derivatives in this paper tension spline numerical methods are developed on a non uniform mesh for the efficient numerical integration of singularly perturbed two point singular boundary value problems the proposed methods are applicable to problems both in rectangular and polar coordinates error analysis of a method is analyzed numerical experiments are presented to demonstrate the utility and the efficiency of the proposed computational procedure

a variable mesh c splage method of accuracy o k2hl 1 khl hl3 for 1d nonlinear parabolic equations a new high accuracy two level implicit cubic spline formulation for the solution of nonlinear parabolic equation u x email protected x t u u x u t is presented the strategy of the cubic spline alternating group explicit c splage method is developed the proposed c splage method requires only 3 spatial grid points and applicable to singular problems the convergence analysis of the proposed c splage method is given the obtained c splage method is compared to the corresponding successive over relaxation sor method both in terms of stability and performance

numerical solutions of second order matrix models using cubic matrix splines many studies of mechanical systems in engineering are based on second order matrix models this work discusses the second order generalization of previous research on matrix differential equations dealing with the construction of approximate solutions for the initial problem y x f x y x y x using matrix cubic splines without any dimensional increase an estimation of the approximation error the corresponding algorithm of our approach and some illustrative examples are presented

on the use of high order difference methods for the system of one space second order nonlinear hyperbolic equations with variable coefficients abstract implicit difference schemes of o k 4 k 2 h 2 h 4 where k 0 h 0 are grid sizes in time and space coordinates respectively are developed for the efficient numerical integration of the system of one space second order nonlinear hyperbolic equations with variable coefficients subject to appropriate initial and dirichlet boundary conditions the proposed difference method for a scalar equation is applied for the wave equation in cylindrical and spherical symmetry the numerical examples are given to illustrate the fourth order convergence of the methods

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

numerical solution of singularly perturbed two point boundary value problems by spline in tension some difference schemes for singularly perturbed two point boundary value problems are derived using spline in tension these schemes are second order accurate numerical examples are given in support of the theoretical results

numerical solution of the sine gordon equation two new difference schemes are proposed for the sine gordon equation they have the advantage that there is a discrete energy which is conserved their stability and convergence is proved also we show some numerical computations using one of these schemes

geometric three dimensional assignment problems abstract we investigate two geometric special cases of the three dimensional assignment problem given are three sets b r and g blue red and green each containing n grid points in the euclidean plane we want to find a partition of b r g into n three0colored triangles such that a the total circumference of all triangles or b the total area of all triangles becomes minimum both versions of the problem are proved to be np hard

a class of variable mesh spline in compression methods for singularly perturbed two point singular boundary value problems we report numerical techniques for a class of singularly perturbed two point singular boundary value problems on a non uniform mesh using spline in compression the proposed methods are stable everywhere in the solution region including the vicinity of the singularity error analysis of a method is briefly discussed numerical results are provided to illustrate the proposed methods

a stable noniterative prediction correction domain decomposition method for hyperbolic problems in this paper a noniterative domain decomposition algorithm for solving hyperbolic partial differential equations is presented the algorithm includes a prediction to estimate the values at the interface and these values are corrected by a scheme that improves the accuracy it has been shown that the algorithm is unconditionally stable efficiency is analyzed in terms of speedup and operation ratio numerical experiments illustrate that the method is stable and efficient