solution of the large matrix equations which occur in response theory abstract the eigenvalue equation and the sets of linear equations that occur in linear and nonlinear response function calculations have a specific paired structure we have developed iterative algorithms which utilize this structure to efficiently solve the equations the algorithms have been designed so they do not require the matrices to be explicitly available which makes it possible to perform accurate calculations with a dimension of maybe 106 numerical tests show convergence behavior superior to previously suggested algorithms

explicit streamline diffusion finite element methods for the compressible euler equations in conservation variables the paper concerns the streamline diffusion finite element method applied to one and two dimensional gas flow described by the inviscid euler equations in conservation variables we point out that the streamline diffusion method is a natural finite element analogue to upstream type finite difference volume schemes and in fact constitutes a general framework for a large class of them we study explicit implementations of the method and derive different choices of stabilizing streamline diffusion matrices in particular we propose a consistent fully multidimensional version a brief review of the theoretical background to the method is presented and some numerical results in two dimensions are given

positive solutions of nth order impulsive boundary value problems with integral boundary conditions in banach spaces abstract by using the fixed point theory for strict set contraction operator this paper investigates the existence of positive solutions for a class of n th order impulsive boundary value problems with integral boundary conditions in banach spaces moreover we offer some interesting discussion of the associated boundary value problems since the technique to deal with n th order bvp is completely different from that of previous papers the expression of the associated integral equation is simpler than that of relevant papers and upper and lower bounds for these positive solutions also are given our work is new

boundary value problems for nth order lipschitz equations on considere l unicite des solutions de problemes aux valeurs limites pour une equation scalaire d ordre n y n f t y y y n 1 ou f est continue sur a b r n et satisfait une condition de lipschitz

initial value methods for discrete boundary value problems motivated by the work of angel and kalaba s and golberg 8 on twopoint boundary value problems for difference equations in this paper we shall study several initial value methods for multipoint boundary value problems for linear and nonlinear difference systems we generalize method of adjoints goodman and lance 9 method of complementary functions tifford 17 d an method of particular solutions miele 111 proposed for two point continuous problems to linear multipoint problems for difference equations next we show that method of adjoint and method of complementary functions can be used in an iterative way to solve nonlinear problems to justify the results obtained here several examples are also given the methods discussed here are similar to our earlier work l 3 on continuous problems we will take up the initial adjusting method of ojika kasue and welsh 12 15 18 for the discrete case later throughout the paper we shall assume the existence uniqueness and stability properties of solutions of the problems discussed and refer to 4 for some of these questions

existence of solutions for third order boundary value problems on a time scale abstract let t be a closed subset of r with inf t and sup t for the third order nonlinear equation u 3 f t u u u t t where f t r3 r is continous we assume solutions of initial value problems are unique and extend to t we consider questions of the uniqueness of solutions implying the existence of solutions for conjugate boundary problems on t

quantitative error control for finite element methods for one dimensional compressible flow we prove a posteriori l 2 l 2 and l infty h 1 residual based error estimates the estimates contain certain strong stability factors or weights related to the solution of a linearized dual problem associated with the euler equations we compute the stability factors and weights by solving the dual problem numerically and show that quantitative error control is possible for one dimensional 1 d compressible flow

iterative methods for solving right focal point boundary value problems abstract we provide necessary and sufficient conditions for the existence of a solution of n th order differential equation satisfying right focal point boundary conditions for the linear problems we propose forward forward and backward backward one pass practical shooting methods which convert the boundary value problems to its equivalent initial value problems for the nonlinear problems iterative methods are discussed and a priori conditions for the convergence are obtained several examples are also illustrated

optimality for boundary value problems for lipschitz equations on considere les solutions de problemes aux valeurs limites pour une equation scalaire d ordre n x n f t x x x n 1 ou f est continue sur a b r n et satisfait une condition de lipschitz on determine les sous intervalles de longueur optimale de a b en fonctions des coefficients de lipschitz sur lesquels des solutions sont uniques

multiple solutions and eigenvalues for third order right focal boundary value problems abstract we are concerned with the right focal boundary value problem x t f t x t t 1 t t 3 x t 1 x t 2 x t 3 0 and the associated eigenvalue problem x t a t f x t with the same boundary conditions under various assumptions on f a and we establish intervals of the parameter which yield the existence of a positive solution of the eigenvalue problem by placing certain restrictions on the nonlinearity we prove the existence of at least one at least two at least three and infinitely many positive solutions of the boundary value problem by applying some known fixed point theorems as well as some recent generalizations of these fixed point theorems

a new reproducing kernel method for linear nonlocal boundary value problems in the previous works the authors developed the reproducing kernel method rkm for nonlocal boundary value problems a key of the method is the construction of the reproducing kernel rk satisfying the homogenous boundary conditions bcs of the considered problems however it is very difficult to obtain the rk of a reproducing kernel space satisfying nonlocal bcs or nonlinear bcs even if the rk is found its representation is also very complicated compared with the rk without any bcs in this paper we will present a new rkm for linear nonlocal boundary value problems the method can avoid reducing the inhomogeneous bcs to homogeneous bcs and constructing rk satisfying homogeneous nonlocal linear bcs numerical examples are provided to show the effectiveness of the new method

aspects of conservation in finite element flow computations in this paper we consider different aspects of conservation in the finite element method in particular we address the problem of conservation using numerical integration related to mass lumping simplified projections quasilinear formulations and non conservative variables numerical results concerning simplified projection algorithms are given

uniqueness and existence for bounded boundary value problems the existence and uniqueness of solutions for the boundary value problems with general linear point evaluation boundary conditions is established we assume that f is bounded and that there is uniqueness on a homogeneous problem and on the linear variational problems

boundary value problems for higher order integro differential equations where r t x t x t x j t and a is a continuous operator map c l al a c ar a the function f is continuous in all of its arguments the problem l l 1 2 includes several particular cases for example the boundary value problems for the differential equations if a 0 considered in 7 8 and the references therein for the integro differential equations of the volterra type if

existence and uniqueness of solutions for nth order nonlinear two point boundary value problems in this paper we present some general results of existence and uniqueness of solutions for nth order nonlinear two point boundary value problems by using shooting method together with maximum principle and kneser hukahara continuum theorem as applications we give certain concrete sufficient conditions for the existence and uniqueness of solutions of nonlinear two point boundary value problems of nth order lipschitz equations

a finite element formulation of compressible flows using various sets of independent variables abstract this paper presents a finite element method for the simulation of compressible flows the navier stokes and euler equations are solved in the conservation form using various sets of independent variables a variational formulation is developed based upon a variant of the petrov galerkin method and uses a shock capturing operator an adaptive algorithm based on a particular residual norm is proposed several numerical examples are presented to demonstrate the performances of each set of variables in solving compressible high speed flows

disconjugacy disfocality and differentiation with respect to boundary conditions in particular under certain assumptions concerning the disconjugacy or disfocality of 1 we will consider the differentiability of solutions of 1 with respect to boundary conditions both boundary values and boundary points in so doing we will prove an analogue of a theorem of peano s which appears in hartman s text 1 theorem 3 1 p 951 with the consideration therein being with the differentiation of solutions of 1 with respect to initial conditions in this setting given a solution y x of 1 we are also interested in solutions of the linear nth order differential equation

a second order nonlinear difference equation oscillation and asymptotic behavior discrete analogues are investigated for well known results on oscillation growth and asymptotic behavior of solutions of y q t y 0 for q t 0 and for q t 0 the analogue of atkinson s oscillation criterion is shown to be true for 2yn 1 qn yn 0 but the analogue for atkinson s nonoscillation criterion is shown to be false

liapunov type inequalities for third order half linear equations and applications to boundary value problems abstract we derive liapunov type inequalities for third order half linear differential equations these inequalities utilize integrals of both q t and q t rather than those of q t as in most papers in the literature for higher order liapunov type inequalities furthermore by combining these inequalities with the uniqueness implies existence theorems by several authors we establish the uniqueness and hence existence uniqueness for several classes of boundary value problems for third order linear equations we believe that this is the first time for liapunov type inequalities to be used to deal with boundary value problems and expect that this approach can be further applied to study general higher order boundary value problems

a finite element based level set method for structural optimization a finite element based level set method is implemented for structural optimization the streamline diffusion finite element method is used for solving both the level set equation and the reinitialization equation the lumped scheme is addressed and the accuracy is compared with the conventional finite difference based level set method a dirichlet boundary condition is enforced during the reinitialization to prevent the boundary from drifting numerical examples of minimum mean compliance design illustrate the reliability of the proposed optimization method copyright 2009 john wiley sons ltd

existence of solutions of right focal point boundary value problems for ordinary differential equations y f x y y y 1 1 with respect to l two conditions which we will assume are a f x y 7 yz y is continuous on a b x r b solutions of initial value problems for 1 are unique motivated by the idea of right disfocality of a linear nth order differential operator given by muldowney 11 we shall be concerned with uniqueness of solutions of 1 which satisfy boundary conditions of the form y l x yi 1 i n where x1 x q a x for related results in the linear case where x1 x x x see for example nehari 2 peterson 3 and elias 4 5 1 to be more definite we define what is meant by a right m 1 m focal point boundary value problem for 1 on an interval i

existence and uniqueness of solutions of k 2 point nonlocal boundary value problems for ordinary differential equations abstract sufficient conditions are given under which the existence of solutions of 4 point nonlocal boundary value problems for n th order nonlinear ordinary differential equations yields the existence of unique solutions of k 2 point boundary value problems for 1 k n 1 the results are motivated by a henderson and jackson paper involving relationships between 2 point and n point conjugate boundary value problems

convergence to steady state solutions for stabilized finite element simulations of compressible flows abstract the main objective of the paper is to compare the numerical performance of several pseudo time integration algorithms used for driving finite element solutions of compressible flow equations towards the steady state the algorithms are implementations of four strategies explicit linear implicit multistage linear implicit and nonlinear all except the explicit scheme employ a particular implementation of gmres to solve systems of linear equations the nonlinear algorithm uses at each time step two iterations of the newton method with an approximate jacobian the finite element formulation is a version of the stabilized fem with h adaptivity the comparison is done for three flow problems in three flow regimes subsonic transonic and supersonic final conclusions concerning relative performance of the methods are drawn

optimal interval lengths for nonlocal boundary value problems associated with third order lipschitz equations abstract for the third order differential equation y f x y y y where f x y 1 y 2 y 3 is lipschitz continuous in terms of y i i 1 2 3 we obtain optimal bounds on the length of intervals on which there exist unique solutions of certain nonlocal three and four point boundary value problems these bounds are obtained through an application of the pontryagin maximum principle from the theory of optimal control

a conservative fractional step method to solve non isentropic euler equations abstract a conservative fractional step method to solve the non isentropic euler set of equations on unstructured meshes is presented herein the present two step method which is based on approximate riemann solvers is compared with roe s scheme the actual numerical rate of convergence is examined the c f l like conditions which must be fulfilled to obtain discrete positive values of density and pressure variables are then discussed it is also shown that the extension to the multi component case enables to fulfil the maximum principle for the mass fraction

eigenvalue problems for second order nonlinear dynamic equations on time scales abstract this paper is concerned with the existence and nonexistence of positive solutions of the second order nonlinear dynamic equation u t a t f u t 0 t 0 1 satisfying either the conjugate boundary conditions u 0 u 1 0 or the right focal boundary conditions u 0 u 1 0 where a and f are positive we show that there exists a 0 such that the above boundary value problem has at least two one and no positive solutions for 0 and respectively furthermore by using the semiorder method on cones of the banach space we establish an existence and uniqueness criterion for positive solution of the problem in particular such a positive solution u t of the problem depends continuously on the parameter i e u t is nondecreasing in lim 0 u 0 and lim u

differentiation of solutions of boundary value problems with respect to nonlinear boundary conditions we consider solutions of boundary value problems for the ordinary differential equation y n x y y y n 1 which satisfy gi y xi yi 1 i n where x1 xn and yi r 1 i n the implicit function theorem is used to establish results in which solutions of the boundary value problems are differentiated with respect to the boundary conditions

uniqueness implies existence on time scales abstract the uniqueness of solutions of initial value boundary value problems along with some uniqueness conditions on two point boundary value problems imply the existence of solutions for the same boundary value problems for the second order nonlinear dynamic equation y f t y y on a time scale this existence is established using shooting methods

discrete polynomial interpolation green s functions maximum principles error bounds and boundary value problems abstract we construct discrete interpolating polynomials provide explicit representations of discrete green s functions give several identities and inequalities for these green s functions use the explicit forms of the interpolating polynomials and that of green s functions to establish several maximum principles further we obtain error bounds in discrete polynomial interpolation and use them to study existence and uniqueness of the discrete boundary value problems these bounds are also used to provide sufficient conditions for the convergence of the picard s method the approximate picard s method quasilinearization and the approximate quasilinearization the monotone convergence of the picard s iterative method is also analysed

on the right focal point boundary value problems for integro differential equations abstract for the nth order nonlinear integro differential equations subject to right focal point boundary conditions we provide necessary and sufficient conditions for the existence uniqueness and convergence of an approximate iterative method

riccati type transformations for second order linear difference equations abstract oscillation and comparison theorems for a linear homogeneous second order difference equation are proved by employing various equivalent non linear equations obtained by means of transformations analogous to the riccati transformation for ordinary differential equations

existence and uniqueness of solutions of k point boundary value problems for ordinary differential equations for boundary value problems for linear differential equations in which the boundary conditions are linear the uniqueness of solutions implies the existence of solutions this is not the case in general for nonlinear differential equations but in the past several years a number of results of the uniqueness implies existence type for boundary value problems for nonlinear equations have been proven for example 1 2 4 5 9 111 in this paper we prove some additional results of this type we will be concerned with boundary value problems

criteria for right disfocality of linear difference equations abstract right disfocality and criteria for right disfocality of linear difference equations are considered the criteria require the positivity of appropriate minors of a wronskian determinant and are analogues to criteria for the disconjugacy of linear ordinary differential equations and linear difference equations

convergence of a residual based artificial viscosity finite element method we present a residual based artificial viscosity finite element method to solve conservation laws the galerkin approximation is stabilized by only residual based artificial viscosity without any least squares supg or streamline diffusion terms we prove convergence of the method applied to a scalar conservation law in two space dimensions toward an unique entropy solution for implicit time stepping schemes

efficient methods for implicit geometrical representation of complex material microstructures summary we present two methods for initializing distance functions on adaptively refined finite element meshes to represent complex material microstructures from segmented x ray tomographic data implicit microstructure representation combined with the extended fem allows modelers to represent complex material microstructures with consistent mesh quality and accuracy in the first method a level set evolution equation is formulated and solved by the galerkin method on an adaptively refined mesh we show that the convergence and stability of this method is optimal for the order of elements used in the second approach we initialize the distance field by the fast marching method on a uniform grid and then project the solution onto the finite element mesh by least squares we show that this latter approach is superior in speed and accuracy as an example problem both methods are demonstrated in the initialization of distance fields for two inclusion phases within a al 7075 alloy copyright 2014 john wiley sons ltd

uniqueness of solutions of right focal problems for third order differential equations on considere l unicite des solutions quand elles existent de certains problemes aux valeurs limites pour l equation differentielle d ordre 3 y f x y y y on suppose que f a b r 3 r est continue

a look ahead lanczos algorithm for unsymmetric matrices abstract the two sided lanczos algorithm is known to suffer from serious breakdowns these occur when the associated moment matrix does not permit triangular factorization we modify the algorithm slightly so that it corresponds to using a 2 x 2 pivot in triangular factorization whenever a 1 x 1 pivot would be dangerous the incidence of breakdown is greatly reduced the price paid is that the tridiagonal matrix produced by the algorithm now has bumps whenever a 2 x 2 pivot is used author

boundary data smoothness for solutions of nonlocal boundary value problems for second order differential equations abstract under certain conditions solutions of the boundary value problem y f x y y y x 1 y 1 and y x 2 i 1 m r i y i y 2 are differentiated with respect to boundary conditions where a x 1 1 m x 2 b r 1 r m r and y 1 y 2 r

a classification of the solutions of a difference equation according to their behavior at infinity on considere l equation aux differences lineaires d ordre n ly t p t y t 0 ou p t est de signe defini sur a a a 1 a est un entier

a convergence analysis for nonsymmetric lanczos algorithms a convergence analysis for the nonsymmetric lanczos algorithm is presented by using a tridiagonal structure of the algorithm some identities concerning ritz values and ritz vectors are established and used to derive approximation bounds in particular the analysis implies the classical results for the symmetric lanczos algorithm

uniqueness existence and optimality for fourth order lipschitz equations for each t y y y y4 t zi z2 z3 z e a 6 x r4 in particular we will characterize optimal length subintervals of a b in terms of the lipschitz coefficients k i 1 2 3 4 on which certain two three and four point boundary value problems for 1 have unique solutions the techniques we employ here involve applications of the pontryagin maximum principle 14 p 3141 in conjunction with uniqueness implies existence results for solutions of boundary value problems for 1 these techniques are motivated by works of melentsova 15 and melentsova and mil shtein 16 171 and most notably by the two papers by jackson ll 121 furthermore the results contained herein can be considered as extensions of a recent paper by henderson 6 dealing with boundary value problems for third order equations in relating the results of this paper to previous works we will formulate the boundary value problems in terms of the nth order differential equation

focal boundary value problems for nonlinear difference equations ii on considere l unicite des solutions impliquant l existence des solutions de problemes aux valeurs limites de type focal pour l equation aux differences non lineaire d ordre n u t n f t u t u t n 1 ou f a r n r est continue et l equation u n 1 f t u 1 u n peut etre resolue pour u 1 comme fonction continue des variables t u 2 u u 1

the iterative solutions of 2nth order nonlinear multi point boundary value problems abstract the aim of this paper is to investigate the existence of iterative solutions for a class of 2 n th order nonlinear multi point boundary value problems the multi point boundary condition under consideration includes various commonly discussed boundary conditions such as three or four point boundary condition n 2 point boundary condition and 2 n m point boundary condition the existence problem is based on the method of upper and lower solutions and its associated monotone iterative technique a monotone iteration is developed so that the iterative sequence converges monotonically to a maximal solution or a minimal solution depending on whether the initial iteration is an upper solution or a lower solution two examples are presented to illustrate the results

uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for nth order differential equations abstract for the nth order differential equation y n f x y y y n 1 we consider uniqueness implies existence results for solutions satisfying certain nonlocal k 2 point boundary conditions 1 k n 1 uniqueness of solutions when k n 1 is intimately related to uniqueness of solutions when 1 k n 2 these relationships are investigated as well

a novel nonsymmetric k lanczos algorithm for the generalized nonsymmetric k eigenvalue problems in this article we present a novel algorithm named nonsymmetric k lanczos algorithm for computing a few extreme eigenvalues of the generalized eigenvalue problem mx al where the matrices m and l have the so called k structures we demonstrate a k tridiagonalization procedure preserves the k structures an error bound for the extreme k ritz value obtained from this new algorithm is presented when compared with the class nonsymmetric lanczos approach this method has the same order of computational complexity and can be viewed as a special 2 x 2 block nonsymmetric lanczos algorithm numerical experiments with randomly generated k matrices show that our algorithm converges faster and more accurate than the nonsymmetric lanczos algorithm 0 ekevier science inc 1997

a local space time discontinuous finite element method we present a locally enriched space time finite element method for solving hyperbolic problems with discontinuities the space time formulation is coupled to a standard semidiscrete finite element method the discontinuities are captured by a space time version of the extended finite element method which treats arbitrary moving discontinuities since the discontinuities are local the enriched space time method is only needed around the discontinuities which provides significant computational savings the coupling is implemented through a weak enforcement of the continuity of the flux between the space time and semidiscrete domains in a manner similar to discontinuous galerkin methods the method is illustrated through one dimensional problems it displays remarkable accuracy in capturing moving discontinuities both in their amplitude and speed

the extended finite element method xfem for solidification problems an enriched finite element method for the multi dimensional stefan problems is presented in this method the standard finite element basis is enriched with a discontinuity in the derivative of the temperature normal to the interface the approximation can then represent the phase interface and the associated discontinuity in the temperature gradient within an element the phase interface can be evolved without re meshing or the use of artificial heat capacity techniques the interface is described by a level set function that is updated by a stabilized finite element scheme several examples are solved by the proposed method to demonstrate the accuracy and robustness of the method copyright 2001 john wiley sons ltd

a kqz algorithm for solving linear response eigenvalue equations abstract we develop a new algorithm kqz for solving the generalized eigenvalue problem mx lx which comes from the linear response lr eigenvalue equation in contrast to the qz algorithm our algorithm preserves the block structure of the pencil m l during the computations and only uses k orthogonal transformations we accelerate the convergence by using a quadruple implicit shift technique based on the implicit kq theorem the kqz algorithm saves about half the computational cost and storage of the qz algorithm

an assumed gradient finite element method for the level set equation the level set equation is a non linear advection equation and standard finite element and finite difference strategies typically employ spatial stabilization techniques to suppress spurious oscillations in the numerical solution we recast the level set equation in a simpler form by assuming that the level set function remains a signed distance to the front interface being captured as with the original level set equation the use of an extensional velocity helps maintain this signed distance function for some interface evolution problems this approach reduces the original level set equation to an ordinary differential equation that is almost trivial to solve further we find that sufficient accuracy is available through a standard galerkin formulation without any stabilization or discontinuity capturing terms several numerical experiments are conducted to assess the ability of the proposed assumed gradient level set method to capture the correct solution particularly in the presence of discontinuities in the extensional velocity or level set gradient we examine the convergence properties of the method and its performance in problems where the simplified level set equation takes the form of a hamilton jacobi equation with convex non convex hamiltonian importantly discretizations based on structured and unstructured finite element meshes of bilinear quadrilateral and linear triangular elements are shown to perform equally well copyright 2005 john wiley sons ltd

a symmetric structure preserving γqr algorithm for linear response eigenvalue problems abstract in this paper we present an efficient qr algorithm for solving the linear response eigenvalue problem h x x where h is symmetric with respect to 0 diag i n i n based on newly introduced orthogonal transformations the qr algorithm preserves the symmetric structure of h throughout the whole process and thus guarantees the computed eigenvalues to appear pairwise as they should with the help of a newly established implicit orthogonality theorem we incorporate the implicit multi shift technique to accelerate the convergence of the qr algorithm numerical experiments are given to show the effectiveness of the algorithm

numerical schemes for the hamilton jacobi and level set equations on triangulated domains borrowing from techniques developed for conservation law equations numerical schemes which discretize the hamilton jacobi h j level set and eikonal equations on triangulated domains are presented the first scheme is a provably monotone discretization for the h j equations unfortunately the basic scheme lacks lipschitz continuity of the numerical hamiltonian by employing a virtual edge flipping technique local lipschitz continuity of the numerical flux is restored on acute triangulations next schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous hamiltonians these schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local lipschitz continuity of the numerical hamiltonian finally a class of petrov galerkin approximations is considered these schemes are stabilized via a least squares bilinear form the petrov galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy discretization of the level set equation also requires the numerical approximation of a mean curvature term a simple mass lumped galerkin approximation is presented in section 6 and analyzed using maximum principle analysis the use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples these numerical examples include discretizations of convex and nonconvex forms of the h j equation the eikonal equation and the level set equation

on nonlinear discrete boundary value problems mo x 1 x n 0 1 4 where n is a fixed positive integer 8 represent n by n constant matrices a t is an n by n matrix for each te 0 l h t belongs to ir for each nonnegative integer t u is in r 1 is a real parameter f and g are continuously differentiable f r i ir and g r i r for systems 1 1 1 2 we prove that if the linear system x t l a l x t h t 1 5 subject to

right focal point boundary value problems for ordinary differential equations and variational equations where x g x such problems are called right focal point boundary value problems in the last few years a number of papers have appeared dealing with such boundary value problems most notable in the case when 1 is linear are papers by elias 11 muldowney 7 8 and peterson 19 io and for nonlinear considerations the papers by henderson 3 4 jackson 5 and peterson 11 121 in those papers several analogues of known results for conjugate type boundary value problems are considered for eq l we will have need of the hypotheses