a simple solution of the bratu problem a brief survey of the properties and different treatments of the one dimensional 1d and 2d bratu problems is presented different iterative treatments of the resulting nonlinear system of equations are discussed the finite difference treatment of the problem is considered nonstandard finite difference methods with a simple sinusoidal starting function having an appropriate amplitude are recommended bounds on the amplitude for yielding both lower and upper solutions are given

comments on a necessary and sufficient condition for the stability of nonnegative interval discrete systems absrac in the above note the authors gave necessary and sufficient conditions for the schur stability of nonnegative interval matrices we comment that these conditions may be more conveniently stated in terms of characterizations of m matrices and are easy consequences of some well known facts in nonnegative matrix theory the m matrix characterization further reveals that the conditions in the note fall into a special case of a known result for the general interval matrices in a recent note shafai et al gave necessary and sufficient conditions for the schur stability of nonnegative interval matrices specifically they considered the set of matrices

an efficient iterative method for solving bratu type equations abstract in this paper an efficient analytical iterative method is introduced for solving the initial value and boundary value bratu problems a quasilinearization method with an optimal parameter is derived and utilized to reduce the highly nonlinear bratu equations to a sequence of linear problems an optimal picard iteration method is used to obtain approximate analytical solution of linearized versions of bratu s equations the numerical results presented the effectiveness and high performance of the proposed method and comparisons are made with many other existing numerical solution techniques the comparisons reveal the efficiency and applicability of the present work

a modified perturbation solution to the one dimensional bratu problem abstract an approximate analytical solution to the one dimensional bratu boundary value problem is introduced in this paper the solution is based on some perturbation expansion methods the first step is taken from the linearized perturbation technique which was developed to solve initial value problem with a nonlinear term and no small parameter an artificial small parameter is embedded and the dependent variable can then be expanded in terms of this parameter however necessary modifications are introduced to implement some techniques to solve a boundary value problem in order to allow for different nonlinearities the current solution showed good convergence at any value of the bratu constant when compared with the exact lower branch of the solution to the one dimensional bratu problem also its recursive nature allows for more iterations and adding more correction terms to the final approximate solution which increases the accuracy

a pseudo arclength continuation method for nonlinear eigenvalue problems a variant of the classical pseudo arclength continuation method is proposed basically the method can be viewed as pseudo arclength continuation in r lambda space where r is a functional of the solution another difference is a three parameter predictor instead of the standard euler step this predictor as well as the newton corrector iteration are justified and some numerical results for reaction diffusion equations are presented the method provides a simple algebraic check for symmetry breaking bifurcation the most common type of secondary bifurcation in physical examples

stability of interval matrices in this paper new sufficient conditions for stability and complete instability of an interval matrix jv p q are presented they include the corresponding results derived by heinen 1984 and xu daoyi 1985 as particular cases secondly we present the concept of composite type for stability and simple criteria for this type

solving nonlinear equations by a new derivative free iterative method we develop a new simple iteration formula which does not require any derivatives of f x for solving a nonlinear equation f x 0 it is proved that the convergence order of the new method is quadratic furthermore the new method can approximate complex roots by several numerical examples we show that the presented method will give desirable approximation to the root without a particularly good initial approximation and be efficient for all cases regardless of the behavior of f x

design and multidimensional extension of iterative methods for solving nonlinear problems in this paper a three step iterative method with sixth order local convergence for approximating the solution of a nonlinear system is presented from ostrowski s scheme adding one step of newton with frozen derivative and by using a divided difference operator we construct an iterative scheme of order six for solving nonlinear systems the computational efficiency of the new method is compared with some known ones obtaining good conclusions numerical comparisons are made with other existing methods on standard nonlinear systems and the classical 1d bratu problem by transforming it in a nonlinear system by using finite differences from this numerical examples we confirm the theoretical results and show the performance of the presented scheme

adomian decomposition method for a reliable treatment of the bratu type equations in this paper we present a framework to determine exact solutions of bratu type equations the algorithm rests mainly on the adomian decomposition method the proposed scheme is illustrated by studying two boundary value problems and an initial value problem of bratu type the first type gives a solution that blows up at the middle of the domain whereas the other equations give bounded solutions

on integral representation of the generalized inverse at s 2 we present a general integral representation for the generalized inverse a t s 2 which extends earlier result on the moore penrose inverse weighted moore penrose inverse and drazin inverse

an algorithm for solving boundary value problems in this paper a numerical algorithm based on the decomposition technique is presented for solving a class of nonlinear boundary value problems the method is implemented for well known examples including troesch s and bratu s problems which have been extensively studied the scheme is shown to be highly accurate and only a few terms are required to obtain accurate computable solutions

eigenvalue and eigenvector approximate analysis for repeated eigenvalue problems the method developed for approximate analysis involves a reparamaterization of the multivariable structural eigenvalue problems in terms of a single positive valued parameter the resulting equations yield first order approximations of changes in both the eigenvalues and eigenvectors associated with the repeated eigenvalue problem

sufficient conditions on stability of interval matrices connections and new results the stability properties of interval matrices are discussed sufficient conditions are given for these properties based upon gershgorin s theorem and its extension and they are shown to improve previous results the derivation helps clarify relations between certain previously available results and shows they can be derived using gershgorin s theorem furthermore the problem of determining the stability of interval matrices is related to that of characterizing certain nonsingular m matrices

one point pseudospectral collocation for the one dimensional bratu equation abstract the one dimensional planar bratu problem is u xx exp u 0 subject to u 1 0 because there is an analytical solution this problem has been widely used to test numerical and perturbative schemes we show that over the entire lower branch and most of the upper branch the solution is well approximated by a parabola u x u 0 1 x 2 where u 0 is determined by collocation at a single point x the collocation equation can be solved explicitly in terms of the lambert w function as u 0 w 1 2 2 1 2 where both real valued branches of the w function yield good approximations to the two branches of the bratu function we carefully analyze the consequences of the choice of we also analyze the rate of convergence of a series of even chebyshev polynomials which extends the one point approximation to arbitrary accuracy the bratu function is so smooth that it is actually poor for comparing methods because even a bad inefficient algorithm is successful it is however a solution so smooth that a numerical scheme the collocation or pseudospectral method yields an explicit analytical approximation we also fill some gaps in theory of the bratu equation we prove that the general solution can be written in terms of a single parameter free x without knowledge of the explicit solution the analytical solution can only be evaluated by solving a transcendental eigenrelation whose solution is not known explicitly we give three overlapping perturbative approximations to the eigenrelation allowing the analytical solution to be easily evaluated throughout the entire parameter space

comparison between adomian s method and he s homotopy perturbation method in this paper it is revealed that modified form of he s homotopy perturbation method corresponds to adomian s decomposition method for certain nonlinear problems

comment on stability of interval matrices liao xiao xin 1987 gave necessary and sufficient conditions for the stability of a class of interval matrices it can easily be shown that these results follow directly from a theorem relating stable metzler matrices and quasidominant negative diagonal matrices by mckenzie 1966 which states see also siljak 1978 a metzler matrix a is stable if and only if it is quasidominant negative diagonal as the author considers metzler matrices theorems 1 2 and 3 are obvious also a theorem by fiedler and ptak 1962 concerning a pair of minkowski matrices see also siljak 1978 leads to the same results

an analytic approach to solve multiple solutions of a strongly nonlinear problem based on a new kind of analytic method namely the homotopy analysis method an analytic approach to solve multiple solutions of strongly nonlinear problems is described by using gelfand equation as an example its validity is verified by comparing the approximation series with the known exact solution and different from perturbation techniques this approach is independent upon any small large perturbation quantities so the basic ideas of this approach can be employed to search for multiple solutions of strongly nonlinear problems in science and engineering

sensitivities and linear stability analysis around a double zero eigenvalue a general multiparameter system admitting a double zero eigenvalue at a critical equilibrium point is considered a sensitivity analysis of the critical eigenvalues is performed to explore the neighborhood of the critical point in the parameter space because the coalescence of the eigenvalues implies that the jacobian matrix is defective or nilpotent well suited techniques of perturbation analysis must be employed to evaluate the eigenvalues and the eigenvector sensitivities different asymptotic methods are used based on perturbations both of the eigenvalue problem and the characteristic equation the analysis reveals the existence of a generic nonsingular case and of a nongeneric singular case however even in the generic case a codimension 1 subspace exists in the parameter space on which a singularity occurs by the use of the relevant asymptotic expansions linear stability diagrams arebuiltup and different bifurcation mechanisms divergence hopf doubledivergence doubledivergence hopf degeneratehopf arehighlighted the problem of e nding a uniqueexpression uniformly valid in the wholespaceis then addressed it isfound that a second degreealgebraic equation governsthebehaviorofthecriticaleigenvalues it also permits clarie cation of the geometrical meaning of the unfolding parameters which has been discussed in literature for the takens bogdanova bifurcation finally a mechanical system loaded by nonconservative forces and exhibiting a double zero bifurcation is studied as an example

chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one dimensional bratu equation we introduce two methods for simultaneously approximating both branches of a two branched function using chebyshev polynomials both schemes remove the pernicious convergence wrecking effects of the square root singularity at the limit point where the two branches meet the chebyshev shafer method gives the approximants as the solution to a quadratic equation the mapping algorithm makes a quadratic change of variable for both the only input information is the set of values of f x at a set of discrete points there is little to choose between the two schemes in accuracy but the single expansion mapping method is more flexible in that it can accommodate different ranges on the two branches the eigenrelation for the one dimensional bratu equation is an interesting example because the upper branch is also singular at another point besides the limit point this too can be removed by subtracting the asymptotic solution which is the lambert w function from the upper branch only when the domain of the variable is infinite the quadratic change of variable can still be applied by substituting rational chebyshev functions which are basis functions for an unbounded interval for chebyshev polynomials we illustrate this by approximating the real valued root of the brill quintic u 5 u l l which was first solved by hermite using elliptic modular functions more than a century ago

iterative methods for large continuation problems abstract the computation of solution paths for continuation problems requires the solution of a sequence of nonlinear systems of equations each nonlinear system can be solved by computing the solution of a succession of linear systems of equations determined by jacobian matrices associated with the nonlinear system of equations points on the solution path where the jacobian matrix is singular are referred to as singular points and require special handling they may be turning points or bifurcation points in order to detect singular points it is essential to monitor the eigenvalues of smallest magnitude of the jacobian matrices generated as the solution path is traversed we describe iterative methods for the computation of solution paths for continuation problems so large that factorization of the jacobian matrices is infeasible or impractical the iterative methods simultaneously solve linear systems of equations determined by the jacobian matrices and compute a few eigenvalue eigenvector pairs associated with the eigenvalues of smallest magnitude of each jacobian a bordering algorithm with a pseudo arclength parametrization is applied in the vicinity of turning points to overcome the singularity of the jacobian a bifurcation perturbation strategy is used to compute solution paths at bifurcation points our iterative methods are based on the block lanczos algorithm and are applicable to problems with large symmetric jacobian matrices

on stability of interval matrices new sufficient and sometimes necessary and sufficient conditions are obtained for schur and hurwitz stability of interval matrices by relying on the concept of connective stability and m matrices the necessity part is broadened to include interval matrices with mixed signs of the off diagonal elements provided the sign patterns follow that of the morishima matrix the obtained results are extended to cover convex combinations of interval matrices

mode tracking issues in structural optimization within the context of optimization of the structural dynamics properties of finite element models methodology is developed for the tracking of eigenpairs through changes in the structural eigenvalue problem the goal is to eliminate difficulties caused by mode switching i e frequency crossing out of several candidate methods two methods for mode tracking are successful the first method the higher order eigenpair perturbation algorithm is based on a perturbation expansion of the eigenproblem it iteratively computes changes in the eigenpairs due to parameter perturbations with the important feature of maintaining the correspondence between the baseline and perturbed eigenpairs the second method is a cross orthogonality check method which uses mass orthogonality to reestablish correspondence after a standard reanalysis modified eigenpair extraction routines lanczos subspace iteration inverse power were unsuccessful in tracking modes applications of mode tracking technology that are presented are frequency constrained optimization and optimization with mode shape constraints each application procedure is outlined and examples are given recommendations are made based on method efficiency and robustness in the example problems

performance analysis of preconditioners based on broyden method solving nonlinear equations is a problem often needed to be dealt with in the practical engineering application this paper discusses preconditioning methods based on block broyden method it first introduces the block broyden method and preconditioning technique then it presents four different preconditioners for the block broyden method and discusses the implementation process it also analyzes the time complexity of the four algorithms finally it solves the nonlinear systems arising from the bratu problem by the four preconditioning algorithms experimental results show that the preconditioning technique is effective for the block broyden method and that the preconditioner ssor has the fastest solving speed and the best performance among the four preconditioners therefore it can be used in the large scale problems arising from scientific and engineering computing

an efficient nonlinear multigrid scheme for 2d boundary value problems abstract in this article a two dimensional nonlinear boundary value problem which is strongly related to the well known gelfand bratu model is solved numerically the numerical results are obtained by employing three different numerical strategies namely finite difference based method a newton multigrid method and a nonlinear multigrid full approximation storage fas we are able to handle the difficulty of unstable convergence behaviour by using minres method as a relaxation smoother in multigrid approach with an appropriate sinusoidal approximation as an initial guess a comparison in terms of convergence accuracy and efficiency among the three numerical methods demonstrate an improvement for the values of 0 c numerical results illustrate the performance of the proposed numerical methods wherein fas mg method is shown to be the most efficient further we present the numerical bifurcation behaviour for two dimensional gelfand bratu models and find new multiplicity of solutions in the case of a quadratic and cubic approximation of the nonlinear exponential term numerical experiments confirm the convergence of the solutions for different values of and prove the effectiveness of the nonlinear fas mg scheme

the liouville bratu gelfand problem for radial operators abstract we determine precise existence and multiplicity results for radial solutions of the liouville bratu gelfand problem associated with a class of quasilinear radial operators which includes perturbations of k hessian and p laplace operators

an efficient method for solving bratu equations in this paper we present a numerical technique for solving bratu equation it is based on the laplace adomain decomposition method which produces an implicit equation in two variables we used the predictor corrector technique to trace the solution curve generated from this equation numerical results and conclusions will be presented

superconvergence and the computation of generalized turning points for b v p s abstract we consider projection methods in general and collocation at gauss points in particular for the numerical computation of generalized turning points for two point b v p s we discretize the original problem using collocation to obtain a finite dimensional problem we will employ a direct method for the characterization and computation of simple turning points proper extensions of this direct method will be used for the computation of bifurcation points and cubic turning points we comment also on the superconvergence which will be clear from the numerical examples

an iterative finite difference method for solving bratu s problem in this paper we propose a new iterative finite difference ifd scheme based on the newton raphson kantorovich approximation method in function space to solve the classical one dimensional bratu s problem this new numerical method produces accurate solutions with low computational cost the effectiveness and accuracy of the ifd method are confirmed through several numerical examples and compared to some existing numerical solvers

on the iterative solutions of some nonlinear eigenvalue problems a simple refinement procedure of iteration methods for nonlinear systems is presented it is based upon the introduction of a numerical measure of the separation between iterates which themselves depend on parameters to be determined by requirements on the magnitude of this separation as a simple but informative application of this idea the refined picard iteration of the null dirichlet problem for u f x u 0 is considered to illustrate the use of this technique the equation of bratu and the so called gel fand problem is observed a discussion follows

eigenvector derivatives with repeated eigenvalues in this paper an algorithm is derived for computing the derivatives of eigenvalues and eigenvectors for real symmetric matrices in the case of repeated eigenvalues where the matrices are functions of real parameters such as mass density or moment of inertia the algorithm is an extension of recent work by i u ojalvo the key step in this extended derivation is to differentiate the eigenvalue equation twice the algorithm preserves the symmetry and band structure of the matrices allowing efficient computer storage and solution techniques applications include sensitivity analysis and optimization of the normal modes of finite element modeled structures such as large space structures a cantilever beam finite element example is included

simplified calculation of eigenvector derivatives with repeated eigenvalues an efficient and practical procedure based on the generalized schmidt orthogonalization is presented to compute the eigenvector derivatives when repeated eigenvalues exist

new smoother to enhance multigrid based methods for bratu problem abstract in this paper we present more investigations of the numerical solution of the 2d bratu equation to obtain the second solution on the upper branch by multigrid classical smoothers such as gauss seidel and weighted jacobi have proven ineffective for obtaining the second solution due to the loss of diagonal dominance and the presence of indefinite jacobian system at some parameter values in this paper we modify the multigrid algorithms by adding and combining some krylov methods as smoothers to enhance the multigrid efficiency though the idea is not new but we could get new enhanced results compared to that presented by hackbusch w hackbusch comparison of different multi grid variants for nonlinear equations zamm z angew math mech 72 1992 148 151 and washio and oosterlee t washio c w oosterlee krylov subspace acceleration for nonlinear multigrid schemes electron trans numer anal 6 1997 271 290

letter to the editor on the integral solution of the one dimensional bratu problem a brief survey of the properties and different treatments of the one dimensional 1d planar bratu problem is presented the treatment using green s function is considered different iterative treatments of the resulting nonlinear system of equations are discussed starting the solution with simple sinusoidal function having appropriate amplitude is recommended then the resulting system is solved using a nonlinear conjugate gradient nl cg method the importance of the proper choice of the starting function as well as the stopping criterion of the iterative solution are stressed

calculation of eigenvector derivatives for structures with repeated eigenvalues in structural optimization and system identification eigenvector derivatives provide important information for updating design model parameters when the current parameter values yield repeated eigenvalues it has not been possible previously to calculate unique eigenvector derivatives recent work has provided a method for determining unique eigenvalue derivatives for this case but methods for calculating the eigenvector sensitivities have been incomplete in this work a complete method for calculation of repeated root eigenvector derivatives is shown for the real symmetric structural eigenproblem the derivation is completed by using information from the second derivative of the eigen problem and is limited to the case of distinct eigenvalue sensitivities as an example the repeated root eigenvector sensitivities are calculated for a simple three degree of freedom beam grillage comparisons of linear approximations using these derivatives to the calculated eigenvectors support the accuracy of the formulation

weighted residual method for obtaining positive solutions of two point nonlinear boundary value problems this paper applies the weighted residual method to solving two point nonlinear boundary value problems that have at least one positive solution numerical verification is given revealing the efficiency and convenience of the method proposed comparison with the work of others is also illustrated

nonstandard finite differences for a truncated bratu picard model abstract in this paper we consider theoretical and numerical properties of a nonlinear boundary value problem which is strongly related to the well known gelfand bratu model with parameter when approximating the nonlinear term in the model via a taylor expansion we are able to find new types of solutions and multiplicities depending on the final index n in the expansion the number of solutions may vary from 0 1 2 to in the latter case of infinitely many solutions we find both periodic and semi periodic solutions numerical experiments using a non standard finite difference nsfd approximation illustrate all these aspects we also show the difference in accuracy for different denominator functions in nsfd when applied to this model a full classification is given of all possible cases depending on the parameters n and

truncation method with point transformation for exact solution of liouville bratu gelfand equation abstract in this paper we construct the exact solution of liouville bratu gelfand lbg equation this equation has painleve property under a point transformation painleve analysis based on the truncated singular manifold method is satisfied backlund transformation is obtained and is applied to reduce the problem to a system of nonlinear differential equations this system is reduced to a factorized form due to the symmetry of the equation by solving this factorized form we obtain exact solution for lbg equation satisfying the boundary conditions the solution is plotted for different values of and is compared with previous numerical results according to our intensive search there is no exact solution of the problem the obtained solution shows that the transformation does not change the properties of the equation

new perturbation iteration solutions for bratu type equations perturbation iteration theory is systematically generated for both linear and nonlinear second order differential equations and applied to bratu type equations different perturbation iteration algorithms depending upon the number of taylor expansion terms are proposed using the iteration formulas derived using different perturbation iteration algorithms new solutions of bratu type equations are obtained solutions constructed using different perturbation iteration algorithms are contrasted with each other as well as with numerical solutions it is found that algorithms with more taylor series expansion terms yield more accurate results

efficient finite differences for parameter dependent singularities in two point boundary value problems we will consider the implementation of finite differences to the numerical computation of simple turning points cubic turning points simple bifurcation points and bifurcation points with corank 2 in b v ps for ordinary differential equations the singularity will be characterized by an equation or a set of equations which will be augmented with the original system to produce a regular problem block elimination will be employed for saving on the amount of work while richardson extrapolation will be used to produce accurate results numerical examples will be presented to show the efficiency of the above described algorithm

multilevel continuation techniques for nonlinear boundary value problems with parameter dependence a new technique is described to continue along solution branches of parameter dependent nonlinear boundary value problems in order to obtain accurate multigrid solutions at specified points in particular the implementation for a general class of second order equations in divergence form in the program pltmg is described and analyzed this method has proven to be very robust and efficient as is illustrated by several examples

block nystrom type integrator for bratu s equation abstract in this paper we use a block nystrom method bnm to obtain the numerical solution for one dimensional bratu s problem the convergence analysis of the method is discussed and it is shown that the theoretical order of the method is consistent with its numerical rate of convergence the accuracy benefit of the bnm is demonstrated by comparing it to several other known methods given in the literature it is demonstrated that the bnm can also be used to solve bratu s problem associated with initial conditions by simply adjusting the boundary conditions in the algorithm

a new approach to bratu s problem in this paper a laplace transform decomposition numerical algorithm is introduced for solving bratu s problem the numerical scheme is based on the application of laplace transform integral operator to the differential equation the nonlinear term is then decomposed and an iterative algorithm is constructed for the determination of the infinite series solution the technique is illustrated with two numerical examples and the results show that the method converges rapidly and approximates the exact solution very accurately using only few iterates of the recursive scheme

iterative differential quadrature solutions for bratu problem abstract a numerical scheme based on differential quadrature methods is introduced for solving bratu problem the problem is firstly reduced to an iterative one then both of differential quadrature method dqm and moving least squares differential quadrature method mlsdqm are applied to solve iteratively the nonlinear problem the proposed scheme successfully computes multiple solutions to bratu s problem the obtained results agree with the 1d and 2d closed forms further a parametric study is introduced to investigate the computational characteristics of the proposed scheme

large scale continuation and numerical bifurcation for partial differential equations in this paper the problem of computing bifurcation diagrams for large scale nonlinear parameter dependent steady state systems which arise following the spatial discretization of semilinear pdes is investigated a continuation algorithm which employs a preconditioned version of the recursive projection method rpm is presented the rpm is often expensive when it is used in conjunction with the numerical method of lines preconditioning the jacobian of the underlying fixed point operator results in an algorithm the preconditioned recursive projection method prpm which is capable of efficiently computing equilibrium solution diagrams of large stiff systems for many pde problems the prpm is a fast and effective means of detecting both steady state and hopf bifurcation along a branch of solutions a description of the performance of the prpm when applied to two numerical examples is given

nonlinear cg like iterative methods abstract a nonlinear conjugate gradient method has been introduced and analyzed by j w daniel this method applies to nonlinear operators with symmetric jacobians orthomin 1 is an iterative method which applies to nonsymmetric and definite linear systems in this article we generalize orthomin 1 to a method which applies directly to nonlinear operator equations each iteration of the new method requires the solution of a scalar nonlinear equation under conditions that the hessian is uniformly bounded away from zero and the jacobian is uniformly positive definite the new method is proved to converge to a globally unique solution error bounds and local convergence results are also obtained numerical experiments on solving nonlinear operator equations arising in the discretization of nonlinear elliptic partial differential equations are presented

dynamics of the solution of bratu s equation abstract in the present report we examine the dynamics exhibited by the solution of bratu s equation it represents a one dimensional map with control parameter for certain values of the parameter it exhibits successive bifurcations and shows chaotic regimes this behaviour was confirmed by calculating the corresponding lyapunov exponent power spectra and cobweb diagrams indicating similarities with other well known one dimensional maps