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

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

x simple image eigencones of tropical matrices abstract we investigate max algebraic tropical one sided systems a x b where b is an eigenvector and x lies in an interval x a matrix a is said to have x simple image eigencone associated with an eigenvalue if any eigenvector x associated with and belonging to the interval x is the unique solution of the system a y x in x we characterize matrices with x simple image eigencone geometrically and combinatorially and for some special cases derive criteria that can be efficiently checked in practice

recognizing weakly stable matrices a max plus matrix a is called weakly stable if the sequence orbit x a otimes x a 2 otimes x ldots does not reach an eigenvector of a for any x unless x is an eigenvector this is in contrast to previously studied strongly stable robust matrices for which the orbit reaches an eigenvector with any nontrivial starting vector max plus matrices are used to describe multiprocessor interactive systems for which reachability of a steady regime is equivalent to reachability of an eigenvector by a matrix orbit we prove that an irreducible matrix is weakly stable if and only if its critical graph is a hamiltonian cycle in the associated graph we extend this condition to reducible matrices these criteria can be checked in polynomial time

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

numerical solution of evolution equations by the haar wavelet method an efficient numerical method for solution of nonlinear evolution equations based on the haar wavelets approach is proposed the method is tested in the case of burgers and sine gordon equations numerical results obtained by computer simulation are compared with other available solutions these calculations demonstrate that the accuracy of the haar wavelet solutions is quite high even in the case of a small number of grid points

an eight node curvilinear differential quadrature formulation for reissner mindlin plates a first endeavour to exploit the differential quadrature dq method as a simple accurate and efficient numerical technique for the bending of quadrilateral reissner mindlin plates with curvilinear boundaries is made the curvilinear quadrilateral is mapped onto a square domain by using the geometric coordinate transformation this geometric mapping technique is employed to transform the governing differential equations and boundary conditions of the problem from the physical domain onto the computational domain the dq procedures are then applied to discretise the transformed set of differential equations and boundary conditions into a set of linear algebraic equations the numerical solutions to the problem are obtained by solving this set of linear algebraic equations examples illustrating the accuracy and convergence of the dq method for reissner mindlin plates with curvilinear boundaries are presented the great development potential of the dq method as an alternative to other approximate techniques for solving problems related to engineering sciences and structural mechanics is exploited in the present study

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

an efficient multi step iterative method for computing the numerical solution of systems of nonlinear equations associated with odes an efficient multi step iterative method for computing the numerical solution of systems of nonlinear equations associated with odes

max algebraic attraction cones of nonnegative irreducible matrices abstract it is known that the max algebraic powers ar of a nonnegative irreducible matrix are ultimately periodic this leads to the concept of attraction cone attr a t by which we mean the solution set of a two sided system t a ar x ar t x where r is any integer after the periodicity transient t a and a is the maximum cycle geometric mean of a a question which this paper answers is how to describe attr a t by a concise system of equations without knowing t a this study requires knowledge of certain structures and symmetries of periodic max algebraic powers which are also described we also consider extremals of attraction cones in a special case and address the complexity of computing the coefficients of the system which describes attraction cone

bending and buckling analysis of antisymmetric laminates using the moving least square differential quadrature method in this paper the moving least square differential quadrature mlsdq method is employed to bending and buckling analysis of antisymmetric thick laminates based on the first order shear deformation theory the displacement and rotation components of the plate are independently assumed with the centered moving least square mls approximation within each domain of influence the weighting coefficients are associated with the nodal partial derivatives which are calculated through a fast computation procedure together with the mls nodal shape functions numerical examples are illustrated to study the accuracy stability and convergence of the mlsdq method effects of support size order of the complete basis functions and node irregularity on the numerical accuracy are investigated typical computational results of displacements stresses and critical buckling loads of various laminated plates are presented and compared with the available exact and finite element solutions

the legendre wavelet method for solving initial value problems of bratu type the aim of this work is to study the legendre wavelets for the solution of initial value problems of bratu type which is widely applicable in fuel ignition of the combustion theory and heat transfer the properties of legendre wavelets together with the gaussian integration method are used to reduce the problem to the solution of nonlinear algebraic equations also a reliable approach for convergence of the legendre wavelet method when applied to a class of nonlinear volterra equations is discussed and an error estimation for the proposed method is also introduced illustrative examples have been discussed to demonstrate the validity and applicability of the technique and the results have been compared with the exact solution we finally show the high accuracy and efficiency of the proposed method

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

interval systems of max separable linear equations for a given matrix interval a a a and a given vector interval b b b the notation a x b if max represents an interval system of linear max separable equations we define and characterize several types of solutions and solvability of interval systems of max separable linear equations

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

generators extremals and bases of max cones abstract max cones are max algebraic analogs of convex cones in the present paper we develop a theory of generating sets and extremals of max cones in r n this theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors we give new proofs of existing results suitably generalizing restating and refining them of these it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements we include results on properties of open and closed cones on properties of totally dependent sets and on computational bounds for the problem of finding the essentially unique basis of a finitely generated cone

min max theory for networks of constant geodesic curvature abstract we prove that on a closed surface for any c 0 our min max theory for prescribing mean curvature produces a solution given by a curve of constant geodesic curvature c which is almost embedded except for finitely many points at which the solution is a stationary junction with integer density moreover each smooth segment has multiplicity one the key is a classification of blowups which is new even for c 0

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

control solvability of interval systems of max separable linear equations abstract for a given matrix interval a a a and a given vector interval b b b the notation a x b if max represents an interval system of linear max separable equations several types of solvability of interval systems are known we define control solvability weak control solvability and weak universal solvability and give necessary and sufficient condition for an interval system to be solvable according to each definition

a meshless method for solving two dimensional variable order time fractional advectiondiffusion equation several physical phenomena such as transformation of pollutants energy particles and many others can be described by the well known convectiondiffusion equation which is a combination of the diffusion and advection equations in this paper this equation is generalized with the concept of variable order fractional derivatives the generalized equation is called variable order time fractional advectiondiffusion equation v otfade an accurate and robust meshless method based on the moving least squares mls approximation and the finite difference scheme is proposed for its numerical solution on two dimensional 2 d arbitrary domains in the time domain the finite difference technique with a weighted scheme and in the space domain the mls approximation are employed to obtain appropriate semi discrete solutions since the newly developed method is a meshless approach it does not require any background mesh structure to obtain semi discrete solutions of the problem under consideration and the numerical solutions are constructed entirely based on a set of scattered nodes the proposed method is validated in solving three different examples including two benchmark problems and an applied problem of pollutant distribution in the atmosphere in all such cases the obtained results show that the proposed method is very accurate and robust moreover a remarkable property so called positive scheme for the proposed method is observed in solving concentration transport phenomena the variable order time fractional advectiondiffusion equation in 2 d is defined an accurate and robust meshless method is proposed for its numerical solutions a variable order fractional modeling of air pollution is expressed and investigated it is shown that the proposed method is a positive scheme the proposed method can be extended for other types of variable order fractional problems

natural frequencies of a functionally graded cracked beam using the differential quadrature method the present work is concerned with the free vibration analysis of an elastically supported cracked beam the beam is made of a functionally graded material and rested on a winkler pasternak foundation the line spring model is employed to formulate the problem the method of differential quadrature is applied to solve it the obtained results agreed with the previous similar ones further a parametric study is introduced to investigate the effects of the geometric and elastic characteristics of the problem on the values of natural frequencies and mode shape functions

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

a new method for solving singular initial value problems in the second order ordinary differential equations singular initial value problems linear and nonlinear homogeneous and nonhomogeneous are investigated by using adomian decomposition method the solutions are constructed in the form of a convergent series a new general formula is established the approach is illustrated with few examples

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

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

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

an efficient wave based prediction technique for plate bending vibrations the finite element method is a commonly used prediction method for dynamic simulations of plate bending problems a major disadvantage of this method is its practical frequency limitation as its application involves prohibitively large computational efforts at higher frequencies a newly developed wave based prediction technique pushes up the frequency limit through an enhanced computational efficiency this paper discusses the principles of the wave based method for convex plate problems the beneficial convergence rate of the wave based method as compared with the finite element method is verified for various validation examples which include the most commonly encountered boundary conditions

static analysis of mindlin plates the differential quadrature element method dqem abstract the differential quadrature method dqm is a new numerical method for fast solving linear and nonlinear differential equations based on global basis functions in the present study a discrete approximate method based on the dqm the differential quadrature element method dqem is developed for the axisymmetric static analysis of moderately thick circular and annular plates described by the mindlin shear deformable theory the basic idea of the dqem is 1 to divide the whole variable domain into several sub domains elements 2 to form discretized element governing equations by applying the dqm to each element and 3 to assemble all the discretized element governing equations into the overall characteristic equations with the consideration of displacement and stress compatibility conditions between adjacent elements the annular and circular mindlin plate elements of differential quadrature dq are established the convergence characteristics of the proposed method are carefully investigated from the view points of element refinement and element grid refinement and some general regulations and suggestions on element division and element grid selection are provided a number of numerical examples are calculated and the results are compared with the corresponding exact solutions which exhibits high accuracy simplicity and applicability of the present method

systems of linear interval equations this paper presents theory and methods for computing the exact bounds on the solution of a system of n linear equations in n variables whose coefficients and right hand sides vary in some real intervals finite and iterative methods are given based on results from linear complementarity theory also regularity conditions regularity testing and computing the exact inverse of an interval matrix are dealt with

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

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

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

an efficient mixed methodology for free vibration and buckling analysis of orthotropic rectangular plates abstract in this paper a simple and efficient mixed ritz differential quadrature dq method is presented for free vibration and buckling analysis of orthotropic rectangular plates the mixed scheme combines the simplicity of the ritz method and high accuracy and efficiency of the dq method the accuracy of the proposed method is demonstrated by comparing the calculated results with those available in the literature it is shown that highly accurate results can be obtained using a small number of ritz terms and dq sample points the proposed method is suitable for the problem considered due to its simplicity and potential for further development

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

existence of infinitely many minimal hypersurfaces in positive ricci curvature in the early 1980s s t yau conjectured that any compact riemannian three manifold admits an infinite number of closed immersed minimal surfaces we use min max theory for the area functional to prove this conjecture in the positive ricci curvature setting more precisely we show that every compact riemannian manifold with positive ricci curvature and dimension at most seven contains infinitely many smooth closed embedded minimal hypersurfaces in the last section we mention some open problems related with the geometry of these minimal hypersurfaces

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

explicit form and efficient computation of mls shape functions and their derivatives this work presents a general and efficient way of computing both diffuse and full derivatives of shape functions for meshless methods based on moving least squares approximation mls and interpolation it is an extension of the recently introduced consistency approach based on lagrange multipliers which provides a general framework for constrained mls along with robust algorithms for the computation of shape functions and their diffuse derivatives the particularity of the proposed algorithms is that they do not involve matrix inversion or linear system solving the previous approach is limited to diffuse derivatives of the shape functions and not their full derivatives which are usually much more expensive to obtain in the present paper we propose to efficiently compute the full derivatives by a new algorithm based on the formal differentiation of the previous one in this way we obtain a unified low cost consistent methodology for evaluating the shape functions and both their diffuse and full derivatives in the second part of the paper we introduce explicit forms of mls shape functions in 1d 2d and 3d for an arbitrary number of nodes these forms are especially useful for comparing finite element and mls approximations finally we present a general architecture of an mls program copyright 2000 john wiley sons ltd

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

differential quadrature element method for static analysis of reissner mindlin polar plates abstract in this paper a new numerical technique the differential quadrature element method dqem has been developed for static analysis of the two dimensional polar reissner mindlin plate in the polar coordinate system by integrating the domain decomposition method ddm with the differential quadrature method dqm the detailed formulations for the sectorial dqem plate bending element and the compatibility conditions between each element are presented the convergence properties and the accuracy of the dqem for bending of thick polar plates are investigated through a number of numerical computations consequently the dqem has been successfully applied to analyze several annular sector plates with discontinuous loading and boundary conditions and cutouts to illustrate the simplicity and flexibility of this method for solving reissner mindlin plates in polar coordinate system which are not solvable directly using the differential quadrature method the numerical results are verified by the existing exact solutions or the fem solutions obtained using the software package ansys version 5 3

differential quadrature analysis of free vibration of rhombic plates with free edges an accurate free vibration analysis of rhombic plates with free edges is presented by using the modified differential quadrature method dqm fourteen combinations of boundary conditions with free edges and five skew angles for each plate are analyzed it is found for the first time that the problem is extremely sensitive to grid spacing and the method to applying the multiple boundary conditions when the skew angle is very large various existing non uniform grid spacing and methods for applying the multiple boundary conditions are tested it is surprising to see that dqm with only one grid spacing can yield accurate and reliable solutions besides applying boundary conditions correctly at free corners is also important for success for verifications the dq results are compared with either existing accurate solutions or data obtained by finite element method due to bending stress singularities occurred at corners having obtuse angles the rate of convergence is low for a few cases and quite different from rhombic plates without free edges some new results are presented and waiting for comparison with future data obtained by other investigators

min max theory for constant mean curvature hypersurfaces in this paper we develop a min max theory for the construction of constant mean curvature cmc hypersurfaces of prescribed mean curvature in an arbitrary closed manifold as a corollary we prove the existence of a nontrivial smooth closed almost embedded cmc hypersurface of any given mean curvature c moreover if c is nonzero then our min max solution always has multiplicity one

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

a meshless method for solving the time fractional advection diffusion equation with variable coefficients abstract in this paper an efficient and accurate meshless method is proposed for solving the time fractional advection diffusion equation with variable coefficients which is based on the moving least square mls approximation in the proposed method firstly the time fractional derivative is approximated by a finite difference scheme of order o t 2 0 1 and then the mls approach is employed to approximate the spatial derivative where time fractional derivative is expressed in the caputo sense also the validity of the proposed method is investigated in error analysis discussion the main aim is to show that the meshless method based on the mls shape functions is highly appropriate for solving fractional partial differential equations fpdes with variable coefficients the efficiency and accuracy of the proposed method are verified by solving several examples

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

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

a meshfree approach for solving 2d variable order fractional nonlinear diffusion wave equation abstract this paper is concerned with the moving least squares mls meshless approach for the numerical solution of two dimensional 2d variable order time fractional nonlinear diffusion wave equation v otfnd we on arbitrary domains the variable order fractional derivative of this equation is expressed in the caputo type the proposed method is based on the mls approximation in conjunction with the finite difference technique for spatial derivatives the mls approximation and for temporal derivative the finite difference technique are utilized to discretize the equation the aim of the present paper is to show that the meshfree method based on the mls approximation and collocation scheme is suitable for solving the variable order fractional partial differential equations pdes the proposed method is validated using some different nonlinear numerical examples with complex geometries the results obtained demonstrate the accuracy and easy implementation of the method for nonlinear variable order time fractional pdes

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

differential quadrature for multi dimensional problems the technique of differential quadrature for the solution of partial differential equations introduced by bellman et al is extended and generalized to encompass partial differential equations involving multiple space variables approximation formulae for a variety of first and second order partial derivatives and typical weighting coefficients are presented application of these formulae is demonstrated on the solution of the convection diffusion equation for the two and three dimensional space dependent cases and for both the transient and steady state dispersion of inert neutrally buoyant pollutants from continuous sources into an unbounded atmosphere

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

on visualization scaling subeigenvectors and kleene stars in max algebra abstract the purpose of this paper is to investigate the interplay arising between max algebra convexity and scaling problems the latter which have been studied in nonnegative matrix theory are strongly related to max algebra one problem is that of strict visualization scaling defined as for a given nonnegative matrix a a diagonal matrix x such that all elements of x 1 ax are less than or equal to the maximum cycle geometric mean of a with strict inequality for the entries which do not lie on critical cycles in this paper such scalings are described by means of the max algebraic subeigenvectors and kleene stars of nonnegative matrices as well as by some concepts of convex geometry

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

vibration analysis of symmetrically laminated plates based on fsdt using the moving least squares differential quadrature method in this paper we adopt the first order shear deformation theory in the moving least squares differential quadrature mlsdq procedure for predicting the free vibration behavior of moderately thick symmetrically laminated composite plates the transverse deflection and two rotations of the laminate are independently approximated with the moving least squares mls approximation the weighting coefficients used in the mlsdq approximation are obtained through the fast computation of the mls shape functions and their partial derivatives the natural frequencies of vibration are computed for various laminated plates and compared with the available published results through numerical experiments the capability and efficiency of the mlsdq method for eigenvalue problems are demonstrated and the numerical accuracy and convergence are thoughtfully examined effects of the size of support order of completeness of the basis functions and node irregularity on the numerical accuracy are also investigated

smoothing and accelerated computations in the element free galerkin method two topics in the formulation and implementation of meshless methods are considered the smoothing of the approximating functions at concave boundaries and the speedup of the calculation of the approximating functions and their derivatives these techniques are described in the context of the element free galerkin method but they are applicable to other meshless methods results are presented for some elastostatic problems which show a moderate improvement in the accuracy of the smoothed interpolant the speedup in calculating the shape functions is about a factor of two

high accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain summary based on the differential quadrature dq rule the gauss lobatto quadrature rule and the variational principle a dq finite element method dqfem is proposed for the free vibration analysis of thin plates the dqfem is a highly accurate and rapidly converging approach and is distinct from the differential quadrature element method dqem and the quadrature element method qem by employing the function values themselves in the trial function for the title problem the dqfem without using shape functions essentially combines the high accuracy of the differential quadrature method dqm with the generality of the standard finite element formulation and has superior accuracy to the standard fem and fdm and superior efficiency to the p version fem and qem in calculating the stiffness and mass matrices by incorporating the reformulated dq rules for general curvilinear quadrilaterals domains into the dqfem a curvilinear quadrilateral dq finite plate element is also proposed the inter element compatibility conditions as well as multiple boundary conditions can be implemented simply and conveniently as in fem through modifying the nodal parameters when required at boundary grid points using the dq rules thus the dqfem is capable of constructing curvilinear quadrilateral elements with any degree of freedom and any order of inter element compatibilities a series of frequency comparisons of thin isotropic plates with irregular and regular planforms validate the performance of the dqfem copyright q 2009 john wiley sons ltd

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

vibration and stability analysis of stiffened plates by semi analytic finite difference method part i consideration of bending displacements only abstract a semi analytic finite difference method developed by the author for plates has been extended to the vibration and stability analysis of stiffened plates the method which is semi analytic in nature essentially consists of substituting the displacement function satisfying boundary conditions along two opposite edges into the free vibration and stability equations of the stiffened plate and then by using suitable transformation they are reduced to ordinary differential equations with constant coefficients the effect of the stiffeners has been suitably incorporated in this first part of a two part paper the bending displacements of the plate and the stiffener only are considered rectangular stiffened plates possessing different boundary conditions mass and stiffness propertees and varying number of stiffeners have been analyzed comparison with published results indicates excellent agreement

analysis of composite plates using moving least squares differential quadrature method abstract in this work the moving least squares differential quadrature method mlsdqm is employed to analyze bending problems of composite plates based on a transverse shear theory the governing equations of the problem are derived the transverse deflection and two rotations of the plate are independently approximated with mls approximations the weighting coefficients used in the mlsdq approximation are obtained through the fast computation of the mls shape functions and their partial derivatives the obtained results are compared with the previous analytical and numerical ones further a parametric study is introduced to investigate the effects of elastic and geometric characteristics on the values of transverse deflection of the plate

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

moving least squares differential quadrature method and its application to the analysis of shear deformable plates a moving least squares differential quadrature mlsdq method is developed and employed for the analysis of moderately thick plates based on the first order shear deformation theory fsdt to carry out the analysis the governing equations in terms of the generalized displacements transverse deflection and two rotations of the plate are formulated by employing the moving least squares approximation the weighting coefficients used in the mlsdq approximation are computed through a fast computation of shape functions and their derivatives numerical examples illustrating the accuracy stability and convergence of the mlsdq method are presented effects of support size order of completeness and node irregularity on the numerical accuracy are investigated copyright 2003 john wiley sons ltd