factorization of saddle point matrices in dynamical systems optimization reusing pivots abstract in this paper we consider the application of direct methods for solving a sequence of saddle point systems our goal is to design a method that reuses information from one factorization and applies it to the next one in more detail when we compute the pivoted l d l t factorization we speed up computation by reusing already computed pivots and permutations we develop our method in the frame of dynamical systems optimization experiments show that the method improves efficiency over bunch parlett and bunch kaufman while delivering the same results

singular perturbation analysis of boundary value problems for differential difference equations vi small shifts with rapid oscillations this paper continues the authors study of boundary value problems for singularly perturbed linear second order differential difference equations with small shifts this study was initiated in the companion paper siam j appl math 54 1994 pp 249 272 in this paper the study is extended to problems that have solutions that exhibit rapid oscillations restrictions on the sizes of the shifts in terms of the small parameter are found such that generally the shifted terms cannot be replaced with truncated taylor series in particular it is shown that even when the shifts are small relative to the width of an oscillation they can affect the solution to leading order the conclusion is that oscillatory solutions are more sensitive to small delays than are layer solutions it is shown that a suitably modified version of the standard wkb method can be used to obtain leading order oscillatory solutions of these differential difference equations these preliminary studies of differential difference eq

numerical continuation of bifurcations of limit cycles in matlab rm cl matcont and matcont are matlab continuation packages for the interactive numerical study of a range of parameterized nonlinear dynamical systems in particular odes matcont is an interactive graphical package and rm cl matcont is a commandline version both packages allow us to compute curves of equilibria limit points hopf points limit cycles flip fold and torus bifurcation points of limit cycles we discuss computational details of the continuation of limit cycles and flip fold and torus bifurcations of limit cycles in matcont and rm cl matcont using orthogonal collocation instead of the more commonly used fully extended systems we use minimally extended systems we further describe the use of the matlab sparse matrix routines and the initialization and adaptation of the bordering vectors that are essential in minimally extended systems finally we compare the use of the minimally and the fully extended systems in the matlab environment

modified ode solver for vibration of tube in tube structures abstract for some complex boundary value problems such as ordinary differential equations odes with eigenvalue problems finding a closed form of solution is usually very difficult sometimes even impossible the main purpose of this paper is to present a numerical solution of eigenvalues for free vibration of tube in tube tall buildings by using ode solver modified in the paper in order that an existing ode solver may be compatible with solving the eigenvalue problem a solution strategy for the ode solver is modified that is the strategy organically puts the following techniques together standard transform for non linear problem inverse iteration orthogonalization and eigenvalue shift etc in comparison to the results calculated by other analytical methods and the finite element method shows that the present ode solver can be conveniently taken for accurate and reliable solutions of the eigenvalue problem of the vibration

branch switching in bifurcation problems for ordinary differential equations the problem of switching branches in boundary value problems of ordinary differential equations is considered three non local methods for calculating emanating solutions near a nontrivial bifurcation point are proposed these methods calculate one solution on an emanating branch without a priori exact knowledge of the bifurcation point other solutions on the branch can be obtained by global continuation the methods are convenient as they consist in solving boundary value problems by standard software the construction of an initial approximation of the emanating solution is outlined a characteristic feature of the proposed methods is that they can be easily automated the user can avoid nearly all preparatory work the methods are tested on several examples arising in different application areas

algorithms for almost block diagonal linear systems codes for solving systems of ordinary differential equations for use in the method of lines for partial differential equations pdes usually provide only a banded system solver in this context a frequently occurring structure is almost block diagonal abd solving abd systems by imposing banded structure introduces fill in and is inefficient though robust efficient abd software has been developed and used in packages for solving boundary value problems with separated boundary conditions for ordinary differential equations bvodes it has not been generally exploited in pde software the situation with bordered almost block diagonal system software for bvodes with nonseparated boundary conditions is less satisfactory this survey draws on material from a variety of sources particularly p amodio et al numer linear algebra appl 7 2000 pp 275 317 and b garrett and i gladwell j comput methods sci engrg 1 2001 pp 75 98 and the references therein

a survey on free boundary identification of the truncated boundary in numerical bvps on infinite intervals a free boundary formulation for the numerical solution of boundary value problems on infinite intervals was proposed recently in fazio siam j numer anal 33 1996 1473 we consider here a survey on recent developments related to the free boundary identification of the truncated boundary the goals of this survey are to recall the reasoning for a free boundary identification of the truncated boundary to report on a comparison of numerical results obtained for a classical test problem by three approaches available in the literature and to propose some possible ways to extend the free boundary approach to the numerical solution of problems defined on the whole real line

a finite element method of lines scheme for guided wave structures a hybrid scheme that incorporates the method of finite elements the method of lines variational techniques and conformal mapping is proposed for the analysis of guided wave structures this technique is devised to analyze complex waveguide structures with irregular composite geometries high numerical accuracy is maintained by treating key singularities by semianalytical procedures compared with the conventional method of lines with finite difference techniques the proposed scheme is able to solve many complex guided wave problems with fewer mesh lines without compromise on accuracy this gain is due to the added variable mesh directions and the flexibility in the method of finite elements in choosing appropriate functions for fitting the configuration of concern

a model category structure on the category of simplicial categories in this paper we put a cofibrantly generated model category struc ture on the category of small simplicial categories the weak equivalences are a simplicial analogue of the notion of equivalence of categories

a numerical study for multiple solutions of a singular boundary value problem arising from laminar flow in a porous pipe with moving wall abstract this paper is concerned with multiple solutions of a singular nonlinear boundary value problem bvp on the interval 0 1 which arises in a study of the laminar flow in a porous pipe with an expanding or contracting wall for the singular nonlinear bvp the correct boundary conditions are derived to guarantee that its linearization has a unique smooth solution then a numerical technique is proposed to find all possible multiple solutions for the suction driven pipe flow with the expanding wall e g 2 we find a new solution numerically and classify it as a type vi solution the computed results agree well with what can be obtained by the bifurcation package auto in addition we also construct asymptotic solutions for a few cases of parameters which agree well with numerical solutions these serve as validations of our numerical results thus we believe that the numerical technique designed in the paper is reliable and may be further applied to solve a variety of nonlinear equations that arise from other flow problems

vibration of composite beams with deformable shear connection mathematical modelling and numerical solution using general purpose software for two point boundary value problems abstract in this paper we develop from first principles a linear one dimensional mathematical model for the dynamic response of planar composite beams with deformable shear connection in which the inertias associated with the axial motion of the components and with the bending rotations of their cross sections are consistently taken into account to the best of our knowledge the paper by auclair et al j sound vib 366 2016 230 247 is the only work which heretofore considered these two effects however we point out and correct a flaw in the derivations of these authors to solve the eigenproblem defining the natural vibration frequencies of the composite beam and the corresponding vibration modes we advocate an untried approach the eigenproblem is converted into a standard inhomogeneous boundary value problem which can be solved using freely available state of the art mathematical software for benchmarking purposes the formulation of a conventional displacement based conforming finite element is also briefly addressed the effect of rotational and axial inertias on the natural vibration frequencies is found to be marginal and considerably less pronounced than the predictions of auclair and coworkers in disagreement with the findings of these authors it appears legitimate to ignore the above mentioned effect in a full dynamic analysis of composite beams at least within the practical range of span to depth ratios

numerical computation of periodic orbits that bifurcate from stationary solutions of ordinary differential equations the computation of periodic orbits of autonomous ordinary differential equations is considered a new method especially suited for computing limit cycles that bifurcate from stationary solutions is established this novel approach is very easy to implement the method requiring the programming of the right hand side only another topic of this paper is the demonstration of how a direct method should be applied in the computation of hopf bifurcation points these methods solve boundary value problems by means of standard software only the methods are tested on four examples arising in different application areas including chemistry and biology

computation of the phase response curve a direct numerical approach neurons are often modeled by dynamical systems parameterized systems of differential equations a typical behavioral pattern of neurons is periodic spiking this corresponds to the presence of stable limit cycles in the dynamical systems model the phase resetting and phase response curves prcs describe the reaction of the spiking neuron to an input pulse at each point of the cycle we develop a new method for computing these curves as a by product of the solution of the boundary value problem for the stable limit cycle the method is mathematically equivalent to the adjoint method but our implementation is computationally much faster and more robust than any existing method in fact it can compute prcs even where the limit cycle can hardly be found by time integration for example because it is close to another stable limit cycle in addition we obtain the discretized phase response curve in a form that is ideally suited for most applications we present several examples and provide the implementation in a freely available matlab code

modified deflation algorithm for the solution of singular problems ii nonlinear multipoint boundary value problems abstract various kinds of iterative methods have been proposed for the solution of nonlinear multipoint boundary value problems mpbvp s however it is necessary for these methods that the adjusting matrix which corresponds to the jacobian of nonlinear equations is nonsingular at the solution in this paper an algorithm for the singular solution of nonlinear mpbvp s which is an extension of the modified deflation algorithm for the singular root of nonlinear algebraic equations developed by the author is presented according to the present method the singular solution can ultimately be reduced to the usual simple solution and both convergency and accuracy can greatly be improved the effectiveness of the present method is shown by solving two illustrative examples

finite element methods for spherically symmetric elliptic equations in this paper we obtain error estimates for the numerical solution by the rayleigh ritz galerkin method with piecewise polynomial subspaces of singular two point boundary value problems which arise from spherically symmetric elliptic equations our results are stated in the natural sobolev norms which are l 2 norms with respect to a weight function the original elliptic problem may be set in a space of arbitrary dimension moreover we can enforce smoothness of the numerical solution at the origin

application of global methods in parallel shooting a new divide and conquer method is proposed it employs a coarse grid discretization like multiple shooting does but solves the local problems in a bvp way i e by some global method a sophisticated error control is developed to combine local and global convergence of the newton updating an implementation based on using the collocation code colnew is briefly discussed and a number of examples are given to illustrate the success of the method in particular for singular perturbation problems

a numerical method for solving interface problems arising in two point boundary value problems abstract a numerical approach iterating on the position of interface points is suggested for solving interface problems arising in two point boundary value problems given an interface problem it is decomposed into several standard local boundary value problems which are coupled at the interface points meanwhile a nonlinear equation involving the interface points are formulated from the interface conditions the advantages of this approach are that the boundary conditions for each local problem can easily be selected by considering the natural physical requirements and that each of the local problems can be solved independently by standard numerical bvp techniques

singular perturbation analysis of boundary value problems for differential difference equations v small shifts with layer behavior in this paper an investigation is initiated of boundary value problems for singularly perturbed linear second order differential difference equations with small shifts i e where the second order derivative is multiplied by a small parameter and the shift depends on the small parameter similar boundary value problems are associated with expected first exit times of the membrane potential in models for neurons in particular this paper focuses on problems with solutions that exhibit layer behavior at one or both of the boundaries the analyses of the layer equations using laplace transforms lead to novel results it is shown that the layer behavior can change its character and even be destroyed as the shifts increase but remain small in the companion paper siam j appi math 54 1994 pp 273283 similar boundary value problems with solutions that exhibit rapid oscillations are studied

on a class of rellich inequalities we prove the rellich and the improved rellich inequalities that involve the distance function from a hypersurface of codimension k under a certain geometric assumption in case the distance is taken from the boundary that assumption is the convexity of the domain we also discuss the best constant of these inequalities

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

a similarity approach to the numerical solution of free boundary problems the aim of this work is to point out that within a similarity approach some classes of free boundary value problems governed by ordinary differential equations can be transformed to initial value problems the interest in the numerical solution of free boundary problems arises because these are always nonlinear problems furthermore we show that free boundary problems arise also via a similarity analysis of moving boundary hyperbolic problems and they can be obtained as approximations of boundary value problems defined on infinite intervals most of the theoretical content of this survey is original it generalizes and unifies results already available in literature as far as applications of the proposed approach are concerned three problems of interest are considered and numerical results for each of them are reported

reformulation of boundary value problems into standard form boundary value problems in ode s arising in various applications are frequently not in the standard form required by the currently existing software however many problems can be converted to such a form thus enabling the practitioner to take advantage of the availability and reliability of this general purpose software here various conversion devices are surveyed