numerical integration of non linear two point boundary value problems using iterated deferred corrections i abstract in several recent papers the idea has been proposed of using implicit runge kutta formulae for the approximate integration of non linear two point boundary value problems these formulae often take a very special form so that computational efficiencies are obtained in certain well defined circumstances in this paper we examine the relative merits of these different classes of formulae and discuss their efficient implementation in a deferred correction framework a theoretical comparison of these formulae is important firstly because they call for different computational tasks to be performed and secondly because they are not always applicable to exactly the same classes of problems the most efficient of these formulae are then compared with methods based on finite differences and with methods based on spline collocation for some simple smooth test functions on the basis of operation counts and numerical experimentation it is concluded that runge kutta methods can be generally competitive with the other methods considered

on the solution of characteristic value problems arising in linear stability analysis semi analytical approach abstract the linear stability analysis is normally governed by equations that constitute an eigenvalue characteristic value problem in this paper for the first time two semi analytical algorithms 1 differential transform method dtm and 2 adomian decomposition method adm are examined for solving a characteristic value problem occurring in linear stability analysis in this paper the characteristic value problem of couette taylor flow is selected because its simple geometry continues to be a paradigm for theoretical studies of hydrodynamic stability the results show that dtm handles the solution conveniently and accurately however this paper limits the use of adm for solving characteristic value problems the results indicate that the present algorithm based on dtm could be used as a promising method for solving characteristic value problems

an efficient algorithm for the multivariable adomian polynomials abstract in this article the sum of the series of multivariable adomian polynomials is demonstrated to be identical to a rearrangement of the multivariable taylor expansion of an analytic function of the decomposition series of solutions u 1 u 2 u m about the initial solution components u 1 0 u 2 0 u m 0 of course the multivariable adomian polynomials were developed and are eminently practical for the solution of coupled nonlinear differential equations the index matrices and their simplified forms of the multivariable adomian polynomials are introduced we obtain the recurrence relations for the simplified index matrices which provide a convenient algorithm for rapid generation of the multivariable adomian polynomials another alternative algorithm for term recurrence is established in these algorithms recurrence processes do not require complicated operations such as parametrization expanding and regrouping derivatives etc as practiced in prior art the mathematica program generating the adomian polynomials based on the algorithm in this article is designed

positive and necklace solitary waves on bounded domains abstract we present new solitary wave solutions of the two dimensional nonlinear schrodinger equation on bounded domains such as rectangles circles and annuli these multi peak necklace solitary waves consist of several identical positive profiles pearls such that adjacent pearls have opposite signs they are stable at low powers but become unstable at powers well below the critical power for collapse p cr this is in contrast with the ground state single pearl solitary waves on bounded domains which are stable at any power below p cr on annular domains the ground state solitary waves are radial at low powers but undergo a symmetry breaking at a threshold power well below p cr as in the case of convex bounded domains necklace solitary waves on the annulus are stable at low powers and become unstable at powers well below p cr unlike on convex bounded domains however necklace solitary waves on the annulus have a second stability regime at powers well above p cr for example when the ratio of the inner to outer radii is 1 2 four pearl necklaces are stable when their power is between 3 1 p cr and 3 7 p cr this finding opens the possibility to propagate localized laser beams with substantially more power than was possible until now the instability of necklace solitary waves is excited by perturbations that break the antisymmetry between adjacent pearls and is manifested by power transfer between pearls in particular necklace instability is unrelated to collapse in order to compute numerically the profile of necklace solitary waves on bounded domains we introduce a non spectral variant of petviashvili s renormalization method

an innovative eigenvalue problem solver for free vibration of euler bernoulli beam by using the adomian decomposition method this paper deals with free vibration problems of euler bernoulli beam under various supporting conditions the technique we have used is based on applying the adomian decomposition method adm to our vibration problems doing some simple mathematical operations on the method we can obtain ith natural frequencies and mode shapes one at a time the computed results agree well with those analytical and numerical results given in the literature these results indicate that the present analysis is accurate and provides a unified and systematic procedure which is simple and more straightforward than the other modal analysis

numerical simulation of equal width wave equation by means of the variational iteration method the solution of the equal width wave equation is numerically obtained and compared with that obtained by the adomian decomposition method numerical results show that the variational iteration method can readily be implemented with excellent accuracy to nonlinear equations

an innovative eigenvalue problem solver for free vibration of uniform timoshenko beams by using the adomian modified decomposition method this paper deals with free vibration problems of uniform timoshenko beams under various supporting boundary conditions the technique we have used is based on applying the adomian modified decomposition method amdm to our vibration problems doing some simple mathematical operations on the method we can obtain ith natural frequencies and mode shapes one at a time the computed results agree well with those analytical and numerical results given in the literature these results indicate that the present analysis is accurate and provides a unified and systematic procedure which is simpler and more straightforward than the other modal analysis

on travelling wavefronts of nicholson s blowflies equation with diffusion this paper is devoted to the study of nicholson s blowflies equation with diffusion a kind of time delayed reaction diffusion for any travelling wavefront with speed c c c is the minimum wave speed we prove that the wavefront is time asymptotically stable when the delay time is sufficiently small and the initial perturbation around the wavefront decays to zero exponentially in space as x but it can be large in other locations the result develops and improves the previous wave stability obtained by mei et al in 2004 the new approach developed in this paper is the comparison principle combined with the technical weighted energy method numerical simulations are also carried out to confirm our theoretical results

new application of adomian decomposition method on fokker planck equation adomian decomposition method has been used for finding the explicit and numerical solutions of the reduced fokker planck equation in a force field the obtained results are compared with those found by other theories such as lie group theory the behaviour of the approximate distribution functions are shown graphically the results of lie group method confirm the correctness of those obtained by adomian decomposition method

adomian decomposition method for solving the volterra integral form of the lane emden equations with initial values and boundary conditions in this paper we use the systematic adomian decomposition method to handle the integral form of the lane emden equations with initial values and boundary conditions the volterra integral form of the lane emden equation overcomes the singular behavior at the origin x 0 we confirm our belief that the adomian decomposition method provides efficient algorithm for analytic approximate solutions of the equation our results are supported by investigating several numerical examples that include initial value problems and boundary value problems as well finally we consider the modified decomposition method of rach adomian and meyers for the volterra integral form

approximation of a nondifferentiable nonlinear problem related to mhd equilibria we present a simple method based on a variant of the implicit function theorem which leads to the existence of a part of a nontrivial solution branch of the nonlinear eigenvalue problem u u in u 1 on where is a two dimensional domain with boundary the advantage of this method is that we can apply it for analysing the approximation of the above problem by a finite element method the error analysis of the discrete problem appears immediately we give also an iteration scheme which allows to solve the approximate problem

on the numerical solution of differential equations of lane emden type in this paper a numerical method which produces an approximate polynomial solution is presented for solving lane emden equations as singular initial value problems firstly we use an integral operator yousefi 2006 4 and convert lane emden equations into integral equations then we convert the acquired integral equation into a power series finally transforming the power series into pade series form we obtain an approximate polynomial of arbitrary order for solving lane emden equations the advantages of using the proposed method are presented then an efficient error estimation for the proposed method is also introduced and finally some experiments and their numerical solutions are given and comparing between the numerical results obtained from the other methods we show the high accuracy and efficiency of the proposed method

the variational iteration method an efficient scheme for handling fractional partial differential equations in fluid mechanics variational iteration method has been used to handle linear and nonlinear differential equations the main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently in this work a general framework of the variational iteration method is presented for analytical treatment of fractional partial differential equations in fluid mechanics the fractional derivatives are described in the caputo sense numerical illustrations that include the fractional wave equation fractional burgers equation fractional kdv equation fractional klein gordon equation and fractional boussinesq like equation are investigated to show the pertinent features of the technique comparison of the results obtained by the variational iteration method with those obtained by adomian decomposition method reveals that the first method is very effective and convenient the basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus

dirichlet problem for the diffusive nicholson s blowflies equation abstract the object of this paper is to consider the asymptotic behavior of solutions of the diffusive nicholson s blowflies equation under dirichlet boundary condition a new approach is developed to study the global attractivity of the positive steady state for a reaction diffusion equation with time delay when monotone or quasi monotone conditions fail to apply this approach should be applicable to other dirichlet problems as well even though our analysis is tailored to the diffusive nicholson s blowflies equation

legendre wavelets method for solving differential equations of lane emden type a numerical method for solving the lane emden equations as singular initial value problems is presented using integral operator and convert lane emden equations to integral equations and then applying legendre wavelet approximations the properties of legendre wavelet are first presented these properties together with the gaussian integration method are then utilized to reduce the integral equations to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

partial differential equations with discrete and distributed state dependent delays this work is an attempt to treat partial differential equations with discrete concentrated state dependent delay the main idea is to approximate the discrete delay term by a sequence of distributed delay terms all with state dependent delays we study local existence and long time asymptotic behavior of solutions and prove that the model with distributed delay has a global attractor while the one with discrete delay possesses the trajectory attractor

matrices commuting with a given normal tropical matrix abstract consider the space m n nor of square normal matrices x x i j over r i e x i j 0 and x i i 0 endow m n nor with the tropical sum and multiplication fix a real matrix a m n nor and consider the set a of matrices in m n nor which commute with a we prove that a is a finite union of alcoved polytopes in particular a is a finite union of convex sets the set a a of x such that a x x a a is also a finite union of alcoved polytopes the same is true for the set a of x such that a x x a x a topology is given to m n nor then the set a a is a neighborhood of the identity matrix i if a is strictly normal then a is a neighborhood of the zero matrix in one case a is a neighborhood of a we give an upper bound for the dimension of a we explore the relationship between the polyhedral complexes span a span x and span a x when a and x commute two matrices denoted a and a arise from a in connection with a the geometric meaning of them is given in detail for one example we produce examples of matrices which commute in any dimension

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

a pseudospectral method for two point boundary value problems pseudospectral collocation is employed for the numerical solution of nonlinear two point boundary value problems with separated end conditions second order finite difference schemes are used as preconditioners for the spectral calculation and a solution of the discretized equations is obtained using versions of the defect correction principle the method and a variant based on an adaptive grid technique are tested on a variety of sample problems and are shown to provide high accuracy with low storage requirements

variational iteration method for solving burger s and coupled burger s equations by means of variational iteration method the solutions of burger s equation and coupled burger s equations are exactly obtained comparison with the adomian decomposition method is made showing that the former is more effective than the later in this paper he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomials

a computational approach to soliton solutions of the kadomtsev petviashvili equation in this paper we present a computational approach to develop soliton solutions of the nonlinear kadomtsev petviashvili equation our approach rests mainly on the adomian decomposition method to include few components of the decomposition series the proposed framework is presented in a general way so that it can be used in nonlinear evolution equations of the same type numerical examples are tested to illustrate the proposed scheme

non linear partial differential equations with discrete state dependent delays in a metric space we investigate a class of non linear partial differential equations with discrete state dependent delays the existence and uniqueness of strong solutions for initial functions from a banach space are proved to get the well posed initial value problem we restrict our study to a smaller metric space construct the dynamical system and prove the existence of a compact global attractor

effect of bacterial memory dependent growth by using fractional derivatives reaction diffusion chemotactic model in this paper numerical solutions of a reaction diffusion chemotactic model of fractional orders for bacterial growth will be present a new solution is constructed in power series the fractional derivatives are described in the caputo sense we compare the experimental result obtained with those obtained by simulation of the chemotactic model without fractional derivatives the results show that the solution continuously depends on the time fractional derivative the resulting solutions spread faster than the classical solutions and may exhibit asymmetry depending on the fractional derivative used we present results of numerical simulations to illustrate the method and investigate properties of numerical solutions the adomian s decomposition method adm is used to find the approximate solution of fractional reaction diffusion chemotactic model numerical results show that the approach is easy to implement and accurate when applied to partial differential equations of fractional order

hamiltonian formulation towards minimization of viscous fluid fingering a variational approach has been recently employed to determine the ideal time dependent injection rate q t that minimizes fingering formation when a fluid is injected in a hele shaw cell filled with another fluid of much greater viscosity however such a calculation is approximate in nature since it has been performed by assuming a high capillary number regime in this work we go one step further and utilize a hamiltonian formulation to obtain an analytical exact solution for q t now valid for arbitrary values of the capillary number moreover this hamiltonian scheme is applied to calculate the corresponding injection rate that minimizes fingering formation in a uniform three dimensional porous media an analysis of the improvement offered by these exact injection rate expressions in comparison with previous approximate results is also provided

numerical steady state and hopf bifurcation analysis on the diffusive nicholson s blowflies equation1research partially supported by natural sciences and engineering research council of canada 1 for the dirichlet boundary value problem of the diffusive nicholson s blowflies equation it was shown in ref 17 that in a certain range of the parameter space there is a unique positive steady state solution in this paper we propose a scheme to compute this steady state numerically in addition we describe an iterative procedure to locate the critical values of the delay where a hopf bifurcation of time periodic solutions takes place near the steady state some numerical simulations of both schemes are given

a theory of the optimal policy of oil recovery by secondary displacement processes we consider the stability of the displacement of a viscous fluid in a porous medium by a less viscous fluid containing a solute for the case in which the total amount of injected solute is fixed we formulate a theory for the optimal injection policy i e the policy which minimizes the growth constant of the saffman taylor chouke instability the result is a nonlinear constrained eigenvalue problem with the viscosity ratio alpha of the two fluids and the total solute n as parameters we develop perturbation solutions for large and small n and numerical solutions for all n several features of the optimal policy which have been noted or demonstrated empirically by previous workers are proven and new features of the solution are discussed

max plus definite matrix closures and their eigenspaces in this paper we introduce the definite closure operation for matrices with finite permanent reveal inner structures of definite eigenspaces and establish some facts about hilbert distances between these inner structures and the boundary of the definite eigenspace

tropical linear maps on the plane abstract in this paper we fully describe all tropical linear maps in the tropical projective plane tp 2 that is maps from the tropical plane to itself given by tropical multiplication by a real 3 3 matrix a the map f a is continuous and piecewise linear in the classical sense in some particular cases the map f a is a parallel projection onto the set spanned by the columns of a in the general case after a change of coordinates the map collapses at most three regions of the plane onto certain segments called antennas and is a parallel projection elsewhere theorem 3 in order to study f a we may assume that a is normal i e i a 0 up to changes of coordinates a given matrix a admits infinitely many normalizations our approach is to define and compute a unique normalization for a which we call lower canonical normalization theorem 1 and then always work with it due both to its algebraic simplicity and its geometrical meaning on r n any n n some aspects of tropical linear maps have been studied in 6 we work in tp 2 adding a geometric view and doing everything explicitly we give precise pictures inspiration for this paper comes from 3 5 6 8 12 26 we have tried to make it self contained our preparatory results present noticeable relationships between the algebraic properties of a given matrix a idempotent normal matrix permutation matrix etc and classical geometric properties of the points spanned by the columns of a classical convexity and others see theorem 2 and corollary 1 as a by product we compute all the tropical square roots of normal matrices of a certain type see corollary 4 this is perhaps a curious result in tropical algebra our final aim is however to give a precise description of the map f a tp 2 tp 2 this is particularly easy when two tropical triangles arising from a denoted t a and t a fit as much as possible then the action of f a is easily described on the closure of each cell of the cell decomposition c a see theorem 3 normal matrices play a crucial role in this paper the tropical powers of normal matrices of size n n satisfy a n 1 a n a n 1 this statement can be traced back at least to 26 and appears later many times such as 1 2 6 9 10 in lemma 1 we give a direct proof of this fact for n 3 but now the equality a 2 a 3 means that the columns of a 2 are three fixed points of f a and in fact any point spanned by the columns of a 2 is fixed by f a among 3 3 normal matrices the idempotent ones i e those satisfying a a 2 are particularly nice we prove that the columns of such a matrix tropically span a set which is classically compact connected and convex lemma 2 and corollary 1 in our terminology it is a good tropical triangle

adomian decomposition method for solving fractional nonlinear differential equations in this article we have discussed a new application of adomian decomposition method on time fractional nonlinear fractional differential equations three models with fractional time derivative of order a 0 a 1 are considered and solved by means of adomian decomposition method the behaviour of adomian solutions and the effects of different values of a are investigated numerical examples are tested to illustrate the pertinent feature of the proposed algorithm

the homotopy analysis method for solving the fornberg whitham equation and comparison with adomian s decomposition method in this work an analytical technique namely the homotopy analysis method ham is applied to obtain an approximate analytical solution of the fornberg whitham equation a comparison is made between the ham results and the adomian s decomposition method adm and the homotopy perturbation method hpm the results reveal that ham is very simple and effective the ham contains the auxiliary parameter h which provides us with a simple way to adjust and control the convergence region of solution series

the decomposition method for ordinary differential equations with discontinuities in this paper the decomposition method is applied to the initial value problem ivp for ordinary differential equations with discontinuities for both linear and nonlinear cases

on the numerical integration of nonlinear two point boundary value problems using iterated deferred corrections part 2 the development and analysis of highly stable deferred correction formulae iterated deferred correction methods have been very widely used for the numerical solution of general nonlinear two point boundary value problems in ordinary differential equations however there

numerical solution of duffing equation by the laplace decomposition algorithm the laplace decomposition algorithm lda is a numerical algorithm can be adapted to solve duffing equations this paper both describes the principle of lda and discusses its advantages and drawbacks concrete example are also studied to show with numerical results how the lda works efficiently

a reliable treatment for mixed volterra fredholm integral equations the main goal of this paper is to demonstrate the use of the modified decomposition method for mixed nonlinear volterra fredholm integral equations the modified method combined with the noise terms phenomena may provide the exact solution by using two iterations only two numerical illustrations are given to show the pertinent features of the technique

solitary wave solutions for the general kdv equation by adomian decomposition method in recent publications chaos solitons fractals 12 2001 2283 int j appl math 3 4 2000 361 we have dealt with the numerical solutions of the korteweg de vries kdv and modified korteweg de vries mkdv equations we extend this study to a more general nonlinear equation which is the general korteweg de vries gkdv equation in which the previous studies is a special case of it the method applied here is adomian decomposition method which has been developed by george adomian solving frontier problems of physics the decomposition method kluwer academic publishers boston ma 1994 numerical examples are tested to illustrate the pertinent feature of the proposed algorithm

adomian decomposition method for solution of nonlinear differential algebraic equations in m m hosseini adomian decomposition method with chebyshev polynomials appl math comput in press an efficient modification of the adomian decomposition method was presented by using chebyshev polynomials also in m m hosseini adomian decomposition method for solution of differential algebraic equations j comput appl math in press solution of linear differential algebraic equations was considered by adomian decomposition method in this paper standard and modified adomian decomposition methods are applied to non linear differential algebraic equations the schemes are tested for some examples and the results demonstrate reliability and efficiency of the proposed methods

viscous fluid fingering on a negatively curved surface viscous fingering formation in flat hele shaw cells is a classical and widely studied fluid mechanical problem we examine the development of viscous fluid fingering on a two dimensional surface of constant negative gaussian curvature the hyperbolic plane h 2 a perturbative mode coupling formalism is applied to study the influence of the negative surface curvature on the two most important pattern formation mechanisms of the system fingertip splitting and finger competition we also report on a time dependent control strategy placed on the injection rate which is able to minimize viscous fingering growth on h 2

a congruence on the semiring of normal tropical matrices abstract we introduce and study a congruence on the normal tropical matrix semiring m n n which is relevant to the kleene stars of normal tropical matrices we prove that this congruence is a bisemilattice congruence and give an exact description of each class in particular we show that the class e is an interval when e is a strongly regular normal tropical matrix we also present a method using floyd warshall algorithm to compute the greatest lower bound of e in such case

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 nonlinear inverse power method with shift the classical inverse power method for linear eigenvalue problems is generalized for some nonlinear equations the algorithm does not involve the evaluation of jacobian operators

analytical study of fractional bratu type equation arising in electro spun organic nanofibers elaboration abstract in this article we present a hybrid analytical scheme namely homotopy perturbation transform method to achieve the solutions of the non linear bratu type problem with fractional order derivatives the caputo type fractional derivatives are considered in the present article he s polynomial is used to tackle the nonlinearity which arise in our considered problem using the initial conditions the numerical solutions of the problem are computed the numerical procedure reveals that only a few iterations are needed for better approximation of the solutions which illustrates the effectiveness and reliability of the method effects of fractional order derivatives on the solutions for various particular cases are depicted through graphs

on commuting matrices in max algebra and in classical nonnegative algebra abstract this paper studies commuting matrices in max algebra and nonnegative linear algebra our starting point is the existence of a common eigenvector which directly leads to max analogs and nonnegative analogs of some classical results for complex matrices we also investigate frobenius normal forms of commuting matrices particularly when the perron roots of the components are distinct for the case of max algebra we show how the intersection of eigencones of commuting matrices can be described and we consider connections with boolean algebra which enables us to prove that two commuting irreducible matrices in max algebra have a common eigennode

hopf bifurcation analysis in a delayed nicholson blowflies equation abstract the dynamics of a nicholson s blowflies equation with a finite delay are investigated we prove that a sequence of hopf bifurcations occur at the positive equilibrium as the delay increases explicit algorithm for determining the direction of the hopf bifurcations and the stability of the bifurcating periodic solutions are derived using the theory of normal form and center manifold global existence of periodic solutions are established using a global hopf bifurcation result of wu trans amer math soc 350 1998 4799 and a bendixson criterion for higher dimensional ordinary differential equations due to li and muldowney j differential equations 106 1994 27

max algebra the linear algebra of combinatorics abstract let a b max a b a b a b for a b r r by max algebra we understand the analogue of linear algebra developed for the pair of operations extended to matrices and vectors max algebra which has been studied for more than 40 years is an attractive way of describing a class of nonlinear problems appearing for instance in machine scheduling information technology and discrete event dynamic systems this paper focuses on presenting a number of links between basic max algebraic problems like systems of linear equations eigenvalue eigenvector problem linear independence regularity and characteristic polynomial on one hand and combinatorial or combinatorial optimisation problems on the other hand this indicates that max algebra may be regarded as a linear algebraic encoding of a class of combinatorial problems the paper is intended for wider readership including researchers not familiar with max algebra