sharp error bounds for the derivatives of lidstone spline interpolation ii abstract in this paper we shall derive explicit error estimates in l norm between a given function f x pc n a b 4 n 6 and its quintic lidstone spline interpolate the results obtained are then used to establish precise error bounds for the approximated and biquintic lidstone spline interpolates we also include applications to integral equations and boundary value problems as well as sufficient numerical examples which dwell upon the sharpness of the obtained results

the spectrum of eigenparameter dependent discrete sturm liouville equations let us consider the boundary value problem bvp for the discrete sturm liouville equation 0 1 a n 1y n 1 b ny n a ny n email protected n email protected n 0 2 c email protected email protected y 1 b email protected email protected y 0 0 where a n and b n email protected n are complex sequences c i b email protected c i 0 1 and l is a eigenparameter discussing the point spectrum we prove that the bvp 0 1 0 2 has a finite number of eigenvalues and spectral singularities with a finite multiplicities if email protected n exp en d 1 a n b n 0 and email protected email protected 1

numerical methods for the solution of special and general sixth order boundary value problems with applications to benard layer eigenvalue problems a family of numerical methods is developed for the solution of special nonlinear sixth order boundary value problems methods with second fourth sixth and eighth order convergence are contained in the family the problem is also solved by writing the sixth order differential equation as a system of three second order differential equations a family of second and fourth order convergent methods is then used to obtain the solution a second order convergent method is discussed for the numerical solution of general nonlinear sixth order boundary value problems this method with modifications where necessary is applied to the sixth order eigenvalue problems associated with the onset of instability in a benard layer numerical results are compared with asymptotic estimates appearing in the literature

the pseudospectral method for solving differential eigenvalue problems abstract a simple pseudospectral method is presented for the numerical solution of linear differential eigenvalue problems the method does not produce the spurious eigenvalues which generally occur when such problems are solved by the spectral tau method the pseudospectral method is presented using two model problems and the presentation contains a useful algorithm for the computation of the spectral differentiation matrices at general collocation points numerical results are offered for a set of benchmark problems including the orr sommerfeld equation for stability of plane poiseuille flow between parallel plates the results indicate that the new pseudospectral method is comparable in accuracy to the tau method

piecewise complementary lidstone interpolation and error inequalities the purpose of this paper is to develop piecewise complementary lidstone interpolation in one and two variables and establish explicit error bounds for the derivatives in l and l 2 norms

the chebyshev collocation path following method for solving sixth order sturm liouville problems in this paper we implement a chebyshev collocation method to approximate the eigenvalues of nonsingular sixth order sturm liouville problem this method transforms the sturm liouville problem to a sparse singular linear system which is solved by the path following technique numerical results demonstrate the accuracy and efficiency of the present algorithm

quadratic eigenparameter dependent discrete sturm liouville equations with spectral singularities abstract let us consider the boundary value problem bvp for the discrete sturm liouville equation 0 1 a n 1 y n 1 b n y n a n y n 1 y n n n 0 2 0 1 2 2 y 1 0 1 2 2 y 0 0 where a n and b n n n are complex sequences i i c i 0 1 2 and is a eigenparameter discussing the point spectrum we prove that the bvp 0 1 0 2 has a finite number of eigenvalues and spectral singularities with a finite multiplicities if sup n n exp e n 1 a n b n for some e 0 and 1 2 1

a note on the numerical solution of high order differential equations numerical solution of high order differential equations with multi boundary conditions is discussed in this paper motivated by the discrete singular convolution algorithm the use of fictitious points as additional unknowns is proposed in the implementation of locally supported lagrange polynomials the proposed method can be regarded as a local adaptive differential quadrature method two examples an eigenvalue problem and a boundary value problem which are governed by a sixth order differential equation and an eighth order differential equation respectively are employed to illustrate the proposed method

on the convergence of finite difference methods for a class of two point boundary value problems with periodic boundary conditions abstract in this paper we present a detailed convergence analysis to establish second order convergence for a three point two parameter family of finite difference methods based on one evaluation of f for the class of two point boundary value problems tpbvps y f x y o o x 1 y o y 1 y o y 1 under suitable conditions on f y and for y c 4 0 1 some numerical examples are provided to illustrate the results

dsc analysis of free edged beams by an iteratively matched boundary method this communication introduces a novel scheme for the treatment of free edge supports in the analysis of beams by using the discrete singular convolution dsc algorithm accommodating free edges has been a challenging issue in the dsc analysis of beams plates and shells an iteratively matched boundary imb method is proposed to overcome the difficulty numerical experiments are carried out to demonstrate that the proposed imb method works very well in dealing with arbitrary combinations of beam edge supports

eigenparameter dependent discrete dirac equations with spectral singularities let us consider the boundary value problem bvp for the discrete dirac equations 0 1 a n 1y n 1 2 b ny n 2 p ny n 1 email protected n 1 a n 1y n 1 1 b ny n 1 q ny n 2 email protected n 2 email protected n 0 2 c email protected email protected y 1 2 b email protected email protected y 0 1 0 where a n b n p n and q n email protected n are complex sequences c i b email protected c i 0 1 and l is a eigenparameter discussing the eigenvalues and the spectral singularities we prove that the bvp 0 1 0 2 has a finite number of eigenvalues and spectral singularities with a finite multiplicities if n 1 exp en d 1 a n 1 b n p n q n 0 and 12 d 1

parametric septic splines approach for the solution of linear sixth order two point boundary value problems parametric splines which reduces to seven degree polynomial splines have been used to develop a class of numerical methods for the solution of linear sixth order two point boundary value problems convergence analysis of the methods is discussed through standard procedure it is shown that the present methods give approximations which are better than those produced by other splines and domain decomposition methods three numerical examples are given to illustrate the practical usefulness of the new approach

numerical solution of nonlinear differential equations with algebraic constraints i convergence results for backward differentiation formulas in this paper we investigate the behavior of numerical ode methods for the solution of systems of differential equations coupled with algebraic constraints systems of this form arise frequently in the modelling of problems from physics and engineering we study some particular examples from electrical networks fluid dynamics and constrained mechanical systems we show that backward differentiation formulas converge with the expected order of accuracy for these systems

homotopy perturbation method for solving sixth order boundary value problems in this paper we apply the homotopy perturbation method for solving the sixth order boundary value problems by reformulating them as an equivalent system of integral equations this equivalent formulation is obtained by using a suitable transformation the analytical results of the integral equations have been obtained in terms of convergent series with easily computable components several examples are given to illustrate the efficiency and implementation of the homotopy perturbation method comparisons are made to confirm the reliability of the homotopy perturbation method we have also considered an example where the homotopy perturbation method is not reliable

explicit error bounds for the derivatives of piecewise lidstone interpolation abstract the purpose of this paper is to provide explicit error bounds for the derivatives of piecewise lidstone interpolation in l and l2 norms the results obtained are best possible sharp and supplement those established in earlier papers

difference equations of second order with spectral singularities abstract in this paper using the uniqueness theorem of analytic functions we investigated the eigenvalues and the spectral singularities of the difference equation a n 1 y n 1 b n y n a n y n 1 y n n z 0 1 2 where t a n n z b n n z are complex sequences and is a spectral parameter

spline solutions of linear twelfth order boundary value problems abstract linear twelfth order boundary value problems special case are solved using polynomial splines of degree 12 the spline function values at the midknots of the interpolation interval and the corresponding values of the even order derivatives are related through consistency relations the algorithm developed approximates the solutions and their higher order derivatives of differential equations two numerical illustrations are given to show the practical usefulness of the algorithm developed it is observed that this algorithm is second order convergent

numerical solution of sixth order boundary value problems with sixth degree b spline functions abstract siddiqi and twizell using sextic splines to approximate solution of sixth order linear boundary value problems in the present paper a sixth degree b spline functions is used to construct an approximate solution for sixth order linear and nonlinear boundary value problems in fact we will show that for every e 0 there exist an approximate solution ve such that the least square error of this ve is less than e 0 and also ve satisfies the exact boundary conditions

collocation methods for solving linear differential algebraic boundary value problems we consider boundary value problems for linear differential algebraic equations with variable coefficients with no restriction on the index a well known regularisation procedure yields an equivalent index one problem with d differential and a n d algebraic equations collocation methods based on the regularised bvp approximate the solution x by a continuous piecewise polynomial of degree k and deliver in particular consistent approximations at mesh points by using the radau schemes under weak assumptions the collocation problems are uniquely and stably solvable and if the unique solution x is sufficiently smooth convergence of order min k 1 2k 1 and superconvergence at mesh points of order 2k 1 is shown finally some numerical experiments illustrating these results are presented

one step collocation methods for differential algebraic systems of index 1 abstract in this paper we propose one step collocation methods for linear differential algebraic equation dae systems of index 1 the proposed methods give a continuous approximate solution on the integration interval t0 tn we study the uniform convergence properties of the proposed collocation methods some of the methods show an order reduction phenomenon at the nodes similar to that observed for runge kutta methods petzold 1986 since dae systems may be considered as a kind of very stiff differential systems the collocation methods for dae systems have to verify strong stability properties for example collocation methods at radau points are particularly suitable for solving dae systems however the numerical results will also show that gauss legendre collocation methods may be applied successfully to dae systems also if these are only a stable methods

jost solution and the spectral properties of the matrix valued difference operators abstract in this paper we find polynomial type jost solution of the selfadjoint matrix valued difference equation of second order then we investigate analytical properties and asymptotic behaviour of the jost solution using the weyl compact perturbation theorem we prove that the selfadjoint operator l generated by the matrix valued difference expression of second order has the continuous spectrum filling the segment 2 2 we also study the eigenvalues of l and prove that it has a finite number of simple real eigenvalues

an eigenfunction expansion for a quadratic pencil of a schrodinger operator with spectral singularities in this paper we consider the operatorlgenerated inl2 r by the differential expressionl y y q x 2 p x 2 y x r 0 and the boundary conditiony 0 0 wherepandqare complex valued functions andpis continuously differentiable onr we derive a two fold spectral expansion ofl in the sense of keldysh 1951 soviet math dokl 77 11 14 1971 russian math survey26 15 44 engl transl in terms of the principal functions under the conditionslimx p x 0 supx r eex q x p x 0 taking into account the spectral singularities also we investigate the convergence of the spectral expansion

differential quadrature solutions of eighth order boundary value differential equations special cases of linear eighth order boundary value problems have been solved using polynomial splines however divergent results were obtained at points adjacent to boundary points this paper presents an accurate and general approach to solve this class of problems utilizing the generalized differential quadrature rule gdqr proposed recently by the authors explicit weighting coefficients are formulated to implement the gdqr for eighth order differential equations a mathematically important by product of this paper is that a new kind of hermite interpolation functions is derived explicitly for the first time linear and non linear illustrations are given to show the practical usefulness of the approach developed using frechet derivatives non linear eighth order problems are also solved for the first time numerical results obtained using even only seven sampling points are of excellent accuracy and convergence in an entire domain the present gdqr has shown clear advantages over the existing methods and demonstrated itself as a general stable and accurate numerical method to solve high order differential equations

a numerical scheme for multi point special boundary value problems and application to fluid flow through porous layers abstract in this paper we propose a numerical scheme based on finite differences for the numerical solution of nonlinear multi point boundary value problems over adjacent domains in each subdomain the solution is governed by a different equation the solutions are required to be smooth across the interface nodes the approach is based on using finite difference approximation of the derivatives at the interface nodes smoothness across the interface nodes is imposed to produce an algebraic system of nonlinear equations a modified multi dimensional newton s method is proposed for solving the nonlinear system the accuracy of the proposed scheme is validated by examples whose exact solutions are known the proposed scheme is applied to solve for the velocity profile of fluid flow through multilayer porous media

spectrum and spectral singularities of a quadratic pencil of a schrodinger operator with a general boundary condition abstract in this article we investigate the spectrum and the spectral singularities of the quadratic pencil of schrodinger operatorlgenerated inl2 r by the differential expression l y y q x 2 p x 2 y x r 0 and the boundary condition 0 k x y x dx y 0 y 0 0 wherep q and k are complex valued functions p is continuously differentiable onr k l2 r and c with 0 discussing the spectrum we prove that l has a finite number of eigenvalues and spectral singularities with finite multiplicities if the conditions lim x p x 0 sup x r e e x q x p x k x 0 later we investigate the properties of the principal functions corresponding to the spectral singularities moreover some results about the spectrum oflare applied to non selfadjoint sturm liouville and klein gordons wave operators

application of least square method to arbitrary order problems with separated boundary conditions in this paper differential equations of arbitrary order with separated boundary conditions are converted into an optimal control problem then a convergent approximate solution is constructed such that the exact boundary conditions are satisfied in fact it will be shown that for every e 0 there exists an approximate solution v e of b spline functions such that the corresponding least square error is less than e 0 and also v e satisfies the exact boundary conditions some examples are given and the optimal errors are obtained for the sake of comparison

matched interface and boundary mib for the implementation of boundary conditions in high order central finite differences high order central finite difference schemes encounter great difficulties in implementing complex boundary conditions this paper introduces the matched interface and boundary mib method as a novel boundary scheme to treat various general boundary conditions in arbitrarily high order central finite difference schemes to attain arbitrarily high order the mib method accurately extends the solution beyond the boundary by repeatedly enforcing only the original set of boundary conditions the proposed approach is extensively validated via boundary value problems initial boundary value problems eigenvalue problems and high order differential equations successful implementations are given to not only dirichlet neumann and robin boundary conditions but also more general ones such as multiple boundary conditions in high order differential equations and time dependent boundary conditions in evolution equations detailed stability analysis of the mib method is carried out the mib method is shown to be able to deliver high order accuracy while maintaining the same or similar stability conditions of the standard high order central difference approximations the application of the proposed mib method to the boundary treatment of other non standard high order methods is also considered

a note on the spectral singularities of non selfadjoint matrix valued difference operators let l denote the non selfadjoint operator generated in 2 n e by the matrix difference expression y n a n 1y n 1 b ny n a ny n 1 n n and the boundary condition y 0 0 in this paper we investigate the jost solution the continuous spectrum the eigenvalues and the spectral singularities of l

new algorithms for solving high even order differential equations using third and fourth chebyshev galerkin methods this paper is concerned with spectral galerkin algorithms for solving high even order two point boundary value problems in one dimension subject to homogeneous and nonhomogeneous boundary conditions the proposed algorithms are extended to solve two dimensional high even order differential equations the key to the efficiency of these algorithms is to construct compact combinations of chebyshev polynomials of the third and fourth kinds as basis functions the algorithms lead to linear systems with specially structured matrices that can be efficiently inverted numerical examples are included to demonstrate the validity and applicability of the proposed algorithms and some comparisons with some other methods are made

on the stability of a classical second order method for solving a class of two point boundary value problems abstract we consider the stability of a classical second order method for the class of two point boundary value problems y 4 f x y 0 y 0 a1 y 1 b1 y 0 a2 y 1 b2 a1 a2 b1 and b2 being constants it is shown that for sufficiently small h this well known method is stable for all f y satisfying 4

a comparison between the sinc galerkin and the modified decomposition methods for solving two point boundary value problems one of the new techniques used in solving boundary value problems involving ordinary differential equations is the sinc galerkin method this method has been shown to be a powerful numerical tool for finding fast and accurate solutions a less known technique that has been around for almost two decades is the decomposition method in this paper we solve boundary value problems of higher order using these two methods and then compare the results it is shown that the sinc galerkin method in many instances gives better results

approximate solutions to boundary value problems of higher order by the modified decomposition method abstract in this paper we present an efficient numerical algorithm for approximate solutions of higher order boundary value problems with two point boundary conditions a modified form of the adomian decomposition method will be implemented to construct such solutions the approach provides the solution in the form of a rapidly convergent series the analysis is accompanied by numerical examples the obtained results demonstrate reliability and efficiency of the proposed scheme

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

non polynomial splines approach to the solution of sixth order boundary value problems non polynomial splines which are equivalent to seven degree polynomial splines are used to develop a class of numerical methods for computing approximations to the solution of sixth order boundary value problems with two point boundary conditions second fourth and sixth order convergence is obtained by using standard procedure it is shown that the present methods give approximations which are better than those produced by other spline and domain decomposition methods numerical examples are given to illustrate practical usefulness of the new approach

order results for implicit runge kutta methods applied to differential algebraic systems in this paper we study the order stability and convergence properties of implicit runge kutta methods applied to a relatively simple class of nonlinear differential algebraic systems these methods often do not attain the same order of accuracy for differential algebraic systems as they do for purely differential systems we derive a set of order conditions which the method coefficients should satisfy in addition to the usual order conditions to ensure a given order of accuracy and we present results on the stability and convergence properties of these methods

vibrations of non uniform rings studied by means of the differential quadrature method abstract the vibrational behavior of ring like structures is of interest in several technological situations from automotive and aircraft tires to electric motors and transformers this study deals with the solution of the governing sixth order differential equation by using the differential quadrature method the fundamental frequency of in plane inextensional vibration is thus determined and compared with the value predicted by the optimized rayleigh ritz method

the numerical solution of sixth order boundary value problems by the modified decomposition method a fast and accurate algorithm is developed for the solution of sixth order boundary value problems bvps with two point boundary conditions a modified form of the adomian decomposition method is applied to construct the numerical solution for such problems the scheme is tested on one linear and two nonlinear problems the results demonstrate reliability and efficiency of the algorithm developed

spline solutions of linear eighth order boundary value problems abstract linear eighth order boundary value problems special case are solved using polynomial splines of degree six the spline function values at the midknots of the interpolation interval and the corresponding values of the even order derivatives are related through consistency relations the algorithm developed approximates the solution and their higher order derivatives of differential equations four numerical illustrations are given to show the practical usefulness of the algorithm developed it is observed that this algorithm is second order convergent

automatic spectral collocation for integral integro differential and integrally reformulated differential equations automatic chebyshev spectral collocation methods for fredholm and volterra integral and integro differential equations have been implemented as part of the chebfun software system this system enables a symbolic syntax to be applied to numerical objects in order to pose and solve problems without explicit references to discretization the same objects can be used in matrix free iterative methods in linear algebra in order to avoid very large dense matrices or allow application to problems with nonsmooth coefficients as a further application of the ability to implement operator equations a method of greengard 1 for the recasting of differential equations as integral equations is generalized to mth order boundary value and generalized eigenvalue problems in the integral form large condition numbers associated with differentiation matrices in high order problems are avoided the ability to implement the recasting process generally follows from implementation of the operator expressions in chebfun the integral method also can be extended to first order systems although chebfun syntax does not currently allow easy implementation in this case

discrete singular convolution for the sine gordon equation abstract this paper explores the utility of a discrete singular convolution dsc algorithm for the integration of the sine gordon equation the initial values are chosen close to a homoclinic manifold for which previous methods have encountered significant numerical difficulties such as numerically induced spatial and temporal chaos a number of new initial values are considered including a case where the initial value is exactly on the homoclinic orbit the present algorithm performs extremely well in terms of accuracy efficiency simplicity stability and reliability

numerical methods for special nonlinear boundary value problems of order 2 m abstract general algorithms based on transforming the problem into a system of m second order problems and on solving the problem as a 2m th order problem are proposed for obtaining the solution of the special nonlinear boundary value problem y 2m x f x y x0 x x1 y 2i x0 a2i y 2i x1 b2i i 0 1 m 1 and m 1 the former is to be preferred to the latter as it does not suffer from rounding errors and also it is easy and inexpensive to implement the latter is found to be vulnerable to word length problems when it is used to solve high order boundary value problems beyond using finite difference methods an alternative method based on the assumption that the desired solution can be adequately represented by a finite polynomial is also outlined for solving the special nonlinear boundary value problem comparisons with the proposed equivalent low order form show that the results obtained are less accurate and apparently at more computational expense a simple example is carried out on the special nonlinear eighth order boundary value problem to illustrate the results given by the methods moreover numerical results for the special nonlinear tenth order boundary value problem using the equivalent low order method are also reported

eigenvalues of lidstone boundary value problems we consider the following boundary value problem 1 ny 2 n lf t y n 1 t 0 1 y 2 i 0 y 2 i 1 0 0 0 the values of l are characterized so that the boundary value problem has a positive solution in addition we derive explicit intervals of l such that for any l in the interval existence of a positive solution of the boundary value problem is guaranteed several examples are also included to dwell upon the importance of the results obtained

variational iteration method for twelfth order boundary value problems in this paper he s variational iteration method is applied to solve twelfth order boundary value problems the numerical results obtained with minimum amount of computation are compared with the exact solutions to show the efficiency of the method the results show that the variational iteration method is of high accuracy more convenient and efficient for solving high order boundary value problems

symmetric collocation methods for linear differential algebraic boundary value problems we present symmetric collocation methods for linear differential algebraic boundary value problems without restrictions on the index or the structure of the differential algebraic equation in particular we do not require a separation into differential and algebraic solution components instead we use the splitting into differential and algebraic equations which arises naturally by index reduction techniques and apply gaus type for the differential part and lobatto type for the algebraic part collocation schemes to obtain a symmetric method which guarantees consistent approximations at the mesh points under standard assumptions we show solvability and stability of the discrete problem and determine its order of convergence moreover we show superconvergence when using the combination of gaus and lobatto schemes and discuss the application of interpolation to reduce the number of function evaluations finally we present some numerical comparisons to show the reliability and efficiency of the new methods

three dimensional vibration analysis of curved and twisted beams with irregular shapes of cross sections by sub parametric quadrature element method abstract this paper presents a novel three dimensional 3d sub parametric quadrature element sp qe method for solving the coupled dynamic behavior of curved and pre twisted beamlike structures with irregular shapes of cross section the technique is an extension of the existing quadrature element method qem with regular shapes by mapping the irregular solid into a regular cube detailed formulations are worked out beams with rectangular circular elliptical and airfoil cross sections various curvature and pre twist rates and different boundary conditions are investigated either serendipity elements or lagrange elements are considered in the mapped regular domain convergence studies are carried out to show the computational performance of the proposed elements results are compared either with the existing 3d spectral tchebychev 3d st solutions or with the finite element data it is shown that the proposed method can yield accurate solutions with small number of degrees of freedom consistent or lumped mass matrix affects little on the accuracy of solutions therefore the element with lumped mass matrix can be efficiently used in dynamic analysis of solids with regular and irregular shapes

quintic spline solution of linear sixth order boundary value problems abstract there are few techniques available to numerically solve sixth order boundary value problems with two point boundary conditions in this paper quintic spline method is developed for the numerical solutions of linear special case sixth order boundary value problems two examples are considered for the numerical illustration of the method developed the method is also compared with those developed by el gamel et al mohamed el gamel john r cannon ahmed i zayed sinc galerkin method for solving sixth order bvps math comput 73 247 2003 and wazwaz a wazwaz the numerical solution of sixth order boundary value problems by the modified decomposition method appl math comput 118 2001 311 325 the comparison shows that the quintic spline method is more efficient and effective tool and yields better results

finite difference methods for twelfth order boundary value problems the effect of rotation acting on a layer of fluid heated from below is similar to the effect of a magnetic field acting under the same conditions they both inhibit the onset of instability and they both elongate the cells which appear at marginal stability however it must not be assumed that acting together rotation and magnetic field reinforce each other in fact they have conflicting tendencies when acting together the effect of rotation and magnetic field together leads to a twelfh order eigenvalue problem chandrasekhar 1961 which will be addressed in a future paper experience in solving high order boundary value problems has shown that considerable insight may be obtained by solving the special problem first moreover beside the mathematical interest the computational aspects of the special problem need to be considered to this end finite difference methods of order two and global extrapolation on two grids are proposed to solve the following special nonlinear twelfth order boundary value problem w xii x f x w a x b w 2i a a2i w 2i b b2i i 0 1 5 a simple example is carried out to illustrate the results given by the methods

spectral properties of non selfadjoint difference operators abstract let l denote the operator generated in l2 z by the difference expression ly n an 1yn 1 bnyn anyn 1 n z 0 1 2 where an n z and bn n z are complex sequences in this paper we investigated the spectrum the spectral singularities and the properties of the principal vectors corresponding to the spectral singularities of l we also studied similar problems for the discrete dirac operator m generated in l2 z c 2 by the system of difference expression y n 1 y n 2 y n y 2 n p n y 1 n y 1 n 1 q n y 2 n n z where y y 1 n y 2 n n z is the forward difference operator i e y i n y i n 1 y i n i 1 2 and pn n z qn n z are complex sequences

a perturbed collocation method for boundary value problems in differential algebraic equations this paper introduces a perturbed collocation method for linear boundary value problems bvp in differential algebraic equations dae with index one in contrast with the usually direct approximation of the collocation method for ordinary differential equations ode to bvps in daes the proposed method is stable and preserves superconvergence numerical experiments confirm the results

the generalized differential quadrature rule for fourth order differential equations the generalized differential quadrature rule gdqr proposed here is aimed at solving high order differential equations the improved approach is completely exempted from the use of the existing point technique by applying multiple conditions in a rigorous manner the gdqr is used here to static and dynamic analyses of bernoulli euler beams and classical rectangular plates numerical error analysis caused by the method itself is carried out in the beam analysis independent variables for the plate are first defined the explicit weighting coefficients are derived for a fourth order differential equation with two conditions at two different points it is quite evident that the gdqr expressions and weighting coefficients for two dimensional problems are not a direct application of those for one dimensional problems the gdqr are implemented through a number of examples good results are obtained in this work copyright 2001 john wiley sons ltd

ode methods for the solution of differential algebraic systems in this paper we study the numerical solution of the differential algebraic systems f t y y 0 many of these systems can be solved conveniently and economically using a range of ode methods others can be solved only by a small subset of ode methods and still others present insurmountable difficulty for all current ode methods we examine the first two groups of problems and indicate which methods we believe to be best for them then we explore the properties of the third group which cause the methods to fail we describe a reduction technique which allows systems to be reduced to ones that can be solved it also provides a tool for the analytical study of the structure of systems

on the spurious solutions in the high order finite difference methods for eigenvalue problems in this paper the origins of spurious solutions occurring in the high order finite difference methods are studied based on a uniform mesh spurious modes are found in the high order one sided finite difference discretizations of many eigenvalue problems spurious modes are classified as spectral pollution and non spectral pollution the latter can be partially avoided by mesh refinement while the former persists when the mesh is refined through numerical studies of some prototype eigenvalue problems such as those of the helmholtz and beam equations we show that perfect central differentiation schemes do not produce any spurious modes nevertheless high order central difference schemes encounter difficulties in implementing complex boundary conditions we further show that the central difference schemes will produce spurious modes as well if asymmetric approximation is involved in boundary treatments in general central difference schemes are less likely to produce spurious modes than one sided difference schemes 2007 elsevier b v all rights reserved

quadratic pencil of schrodinger operators with spectral singularities discrete spectrum and principal functions abstract in this article we investigated the spectrum of the quadratic pencil of schrodinger operatorsl generated inl2 r by the equation formula and the boundary condition formula whereu vare complex valued functions anduis absolutely continuous in each finite subinterval of r discussing the spectrum we proved thatl has a finite number of eigenvalues and spectral singularities with finite multiplicities if the conditions formula and formula hold it is shown that the principal functions corresponding to eigenvalues ofl are in the spacel2 r and the principal functions corresponding to spectral singularities are in another hilbert space which containsl2 r some results about the spectrum ofl have also been applied to radial klein gordon and one dimensional schrodinger equations

on numerical differential algebraic problems with application to semiconductor device simulation this paper considers questions regarding conditioning of and numerical methods for certain differential algebraic equations subject to initial and boundary conditions the approach taken is that of separating differential and algebraic solution components at least theoretically this yields conditioning results for differential algebraic boundary value problems in terms of pure differential problems for which existing theory is well developed the process is carried out for problems with global index 1 or 2 for semi explicit boundary value problems of index 1 where solution components are separated a convergence theorem is given for a special class of collocation methods for general index 1 problems advantages and disadvantages of certain symmetric difference schemes are discussed for initial value problems with index 2 the use of bdf schemes is examined with a summary of conditions for their successful and stable utilization finally the present considerations and analysis are applied to

sinc galerkin method for solving linear sixth order boundary value problems mmthere are few techniques available to numerically solve sixth order boundary value problems with two point boundary conditions in this paper we show that the sinc galerkin method is a very effective tool in numerically solving such problems the method is then tested on examples with homogeneous and nonhomogeneous boundary conditions and a comparison with the modified decomposition method is made it is shown that the sinc galerkin method yields better results