a modified decomposition abstract the maclaurin series is quite limited in comparison to the adomian series obtained in the decomposition method by adding procedures from the decomposition method and the expansion of nonlinearities using the adomian polynomials as well as a recent result of adomian and rach on transformation of series using the above polynomials the maclaurin series can be made much more useful in its applicability however the convergence is still slower than for adomian s results using decomposition

numerical approximation for functionals of reflecting diffusion processes the aim of this paper is to approximate the expectation of a large class of functionals of the solution x xi of a stochastic differential equation with normal reflection in a piecewise smooth domain of rd this also yields a monte carlo method for solving partial differential problems of parabolic type with mixed boundary conditions the approximation is based on a modified euler scheme for the stochastic differential equation the scheme can be driven by a sequence of bounded independently and identically distributed i i d random variables or when the domain is convex by a sequence of gaussian i i d random variables the order of weak convergence for both cases is given in the former case the order of convergence is 1 2 and it is shown to be exact by an example in the last section numerical tests are presented the behavior of the error as a function of the final time t for fixed values of the discretization step and as a function of the discretization step for fixed values of the final

on a class of nonlinear dynamical systems the structure of a differential operator in the application of the decomposition method abstract a suitable expresion of a differential operator is deduced allowing the explicit calculation of the general term f h in the power series expansion of a function f x t to be utilized for solving nonlinear differential equations

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

the solution of coupled burgers equations using adomian pade technique abstract the purpose of this study is to implement adm pade madm pade technique which is a combination of adomian decomposition method modified adomian decomposition method and pade approximation for solving homogeneous inhomogeneous two dimensional parabolic equation also we apply adm pade technique to solve coupled burgers equations the results obtained by adm pade madm pade techniques are compared to those obtained by using adm madm alone the numerical results demonstrate that adm pade madm pade technique gives the approximate solution with faster convergence rate and higher accuracy than using adm madm

boundary behavior of diffusion approximations to markov jump processes we show that diffusion approximations including modified diffusion approximations can be problematic since the proper choice of local boundary conditions if any exist is not obvious for a class of markov processes in one dimension we show that to leading order it is proper to use a diffusion fokker planck approximation to compute mean exit times with a simple absorbing boundary condition however this is only true for the leading term in the asymptotic expansion of the mean exit time higher order correction terms do not in general satisfy simple absorbing boundary conditions in addition the diffusion approximation for the calculation of mean exit times is shown to break down as the initial point approaches the boundary and leads to an increasing relative error by introducing a boundary layer we show how to correct the diffusion approximation to obtain a uniform approximation of the mean exit time we illustrate these considerations with a number of examples including a jump process which leads to kramers diffusion model this example represents an extension to a multivariate process

numerical integration analytic continuation and decomposition one step formulas are developed using the decomposition method to explore relationships with numerical integration analytic continuation and numerical continuation analytic continuation in one or multiple dimensions is considered for decomposition solutions and the effects of chaos

a mathematical model of adomian polynomials the adomian decomposition method is reviewed and a mathematical model of adomian polynomials is introduced to show the convenience of this model for computer programming the mathematica package is used to generate the adomian polynomials for any order

pade approximants and adomian decomposition method for solving the flierl petviashivili equation and its variants abstract in this paper we present a reliable combination of adomian decomposition algorithm and pade approximants to investigate the flierl petviashivili fp equation and its variants the approach introduces an alternative framework designed to overcome the difficulty of the singular point at x 0 we also investigate two generalized variants of the fp equation the proposed framework reveals quite a number of remarkable features of the combination of the two algorithms

brownian simulations and unidirectional flux in diffusion the prediction of ionic currents in protein channels of biological membranes is one of the central problems of computational molecular biophysics existing continuum descriptions of ionic permeation fail to capture the rich phenomenology of the permeation process so it is therefore necessary to resort to particle simulations brownian dynamics bd simulations require the connection of a small discrete simulation volume to large baths that are maintained at fixed concentrations and voltages the continuum baths are connected to the simulation through interfaces located in the baths sufficiently far from the channel average boundary concentrations have to be maintained at their values in the baths by injecting and removing particles at the interfaces the particles injected into the simulation volume represent a unidirectional diffusion flux while the outgoing particles represent the unidirectional flux in the opposite direction the classical diffusion equation defines net diffusion flux but not unidirectional fluxes the stochastic formulation of classical diffusion in terms of the wiener process leads to a wiener path integral which can split the net flux into unidirectional fluxes these unidirectional fluxes are infinite though the net flux is finite and agrees with classical theory we find that the infinite unidirectional flux is an artifact caused by replacing the langevin dynamics with its smoluchowski approximation which is classical diffusion the smoluchowski approximation fails on time scales shorter than the relaxation time 1 ensuremath gamma of the langevin equation we find that the probability of brownian trajectories that cross an interface in one direction in unit time ensuremath delta t equals that of the probability of the corresponding langevin trajectories if ensuremath gamma ensuremath delta t 2 that is we find the unidirectional flux source strength needed to maintain average boundary concentrations in a manner consistent with the physics of brownian particles this unidirectional flux is proportional to the concentration and inversely proportional to sqrt ensuremath delta t to leading order we develop a bd simulation that maintains fixed average boundary concentrations in a manner consistent with the actual physics of the interface and without creating spurious boundary layers

decomposition method for solving fractional riccati differential equations in this article we implement a relatively new analytical technique the adomian decomposition method for solving fractional riccati differential equations the fractional derivatives are described in the caputo sense in this scheme the solution takes the form of a convergent series with easily computable components the diagonal pade approximants are effectively used in the analysis to capture the essential behavior of the solution the corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations some numerical examples are presented to illustrate the efficiency and reliability of the method

analytical representations of the spread harmonic measure density we study the spread harmonic measure that characterizes the spatial distribution of reaction events on a partially reactive surface for euclidean domains in which brownian motion can be split into independent lateral and transverse displacements we derive analytical formulas for the spread harmonic measure density and analyze its asymptotic behavior this analysis is applicable to slab domains general cylindrical domains and a half space we investigate the spreading effect due to multiple reflections on the surface and the underlying role of finite reactivity we discuss further extensions and applications of analytical results to describe laplacian transfer phenomena such as permeation through semipermeable membranes secondary current distribution on partially blocking electrodes and surface relaxation in nuclear magnetic resonance

a new algorithm for calculating adomian polynomials in this paper a new algorithm for calculating adomian polynomials for nonlinear operators will be established by parametrization the algorithm requires less formula than the previous method developed by adomian nonlinear stochastic operator equations academic press 1986 g adomian r rach on composite nonlinearities and decomposition method j math anal appl 113 1986 504 509 g adomian applications of nonlinear stochastic systems theory to physics kluwer 1988 many forms of nonlinearity will be studied to illustrate the new algorithm the new algorithm will be extended to calculate adomian polynomials for nonlinearity of several variables

the theoretical foundation of the adomian method abstract in this paper we work out the theoretical basis of the adomian method this powerful method that g adomian has developed in the beginning of the 1980 s supplies analytical approximations to the solution of many kinds of equations thanks to our decomposition theory we justify the practical method and we generalize the convergence results obtained by y cherruault afterward we explain how to solve rigorously equations with linear nonlinear or composed operator and also equations with several terms or with a second member

partially reflected diffusion the radiation reactive or robin boundary condition for the diffusion equation is widely used in chemical and biological applications to express reactive boundaries the underlying trajectories of the diffusing particles are believed to be partially absorbed and partially reflected at the reactive boundary however the relation between the reaction constant in the robin boundary condition and the reflection probability is not well defined in this paper we define the partially reflected process as a limit of the markovian jump process generated by the euler scheme for the underlying ito dynamics with partial boundary reflection trajectories that cross the boundary are terminated with probability p sqrt delta t and otherwise are reflected in a normal or oblique direction we use boundary layer analysis of the corresponding master equation to resolve the nonuniform convergence of the probability density function of the numerical scheme to the solution of the fokker planck equation in a half space wi

numerical algorithms and decomposition abstract relationships are explored between conventional one step methods a new one step version resulting from a discretized adomian decomposition runge kutta and analytic continuation

an asymptotic theory of large deviations for markov jump processes we present new asymptotic methods for the analysis of markov jump processes the methods based on the wkb and other singular perturbation techniques are applied directly to the kolmogorov equations and not to approximate equations that come e g from diffusion approximations for time homogeneous processes we construct approximations to the stationary density function and the mean first passage time from a given domain examples involving a random walk and a problem in queueing theory are presented to illustrate our methods for a class of time inhomogeneous processes we construct long time approximations to the transition probability density function and the probability of large deviations from a stable state the law of large numbers is obtained as a special case

on the adomian decomposition method and comparisons with picard s method abstract this paper shows that the adomian decomposition method stochastic systems academic press new york 1983 is not to be identified with any of the earlier methods since it solves a wide class of nonlinear and stochastic equations without the often unphysical assumptions which have become customary similarities between the picard method and the decomposition method are purely superficial

solving riccati differential equation using adomian s decomposition method in this paper we suggest a method to solve the matrix riccati differential equation the suggested method which we called multistage adomian s decomposition method madm can be considered as an extension of the adomian s decomposition method adm which is very efficient in solving a variety of differential and algebraic equations the solution is introduced in a recursive form which can be used to obtain the solution for the whole time horizon comparisons are made between madm and the exact solution and further between madm and different numerical methods

a maple package of new adm pade approximate solution for nonlinear problems abstract based on the new definition of four classes of adomian polynomials proposed by rach in 2008 a maple package of new adomian pade approximate solution for solving nonlinear problems is presented this package combines the merits of the adomian decomposition method and the diagonal pade technique and may give more accurate solutions of nonlinear problems with strong nonlinearity besides the package is user friendly and efficient one only needs to input the initial conditions governing equation and four optional parameters then our package will output the analytic approximate solution within a few seconds where the equation is decomposed into three parts they are the linear term r nonlinear term nn and source function g which are all in functional form meanwhile several graphs generated from the above solutions are displayed and demonstrate a favorable comparison in this paper several different types of examples are given to illustrate the validity and promising flexibility of the package this package provides us with a convenient and useful tool for dealing with nonlinear problems as well as its electronic version is free to download via the journal website

an aftertreatment technique for improving the accuracy of adomian s decomposition method abstract adomian s decomposition method adm is a nonnumerical method which can be adapted for solving nonlinear ordinary differential equations in this paper the principle of the decomposition method is described and its advantages as well as drawbacks are discussed then an aftertreatment technique at is proposed which yields the analytic approximate solution with fast convergence rate and high accuracy through the application of pade approximation to the series solution derived from adm some concrete examples are also studied to show with numerical results how the at works efficiently

solutions of the initial value problem for nonlinear fractional ordinary differential equations by the rach adomian meyers modified decomposition method abstract in this paper we present the generalized adomian rach theorem and the generalized rach adomian meyers modified decomposition method for solving multi order nonlinear fractional ordinary differential equations we consider different classes of initial value problems for nonlinear fractional ordinary differential equations including the case of real valued orders and another case of rational valued orders which are solved by the present method this method can treat any analytic nonlinearity the coefficients of the solution in the form of a generalized power series are determined by a convenient recurrence scheme which does not involve integration operations compared with the classic adomian decomposition method

bilevel optimal control problems with pure state constraints and finite dimensional lower level this paper focuses on the development of optimality conditions for a bilevel optimal control problem with pure state constraints in the upper level and a finite dimensional parametric optimization problem in the lower level after transforming the problem into an equivalent single level problem we concentrate on the derivation of a necessary optimality condition of pontryagin type we point out some major difficulties arising from the bilevel structure of the original problem and its pure state constraints in the upper level leading to a degenerated maximum principle in the absence of constraint qualifications hence we use a partial penalization approach and a well known regularity condition for optimal control problems with pure state constraints to ensure the nondegeneracy of the derived maximum principle finally we illustrate the applicability of the derived theory by means of a small example

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

a survey of the maximum principles for optimal control problems with state constraints this paper gives a survey of the various forms of pontryagin s maximum principle for optimal control problems with state variable inequality constraints the relations between the different sets of optimality conditions arising in these forms are shown furthermore the application of these maximum principle conditions is demonstrated by solving some illustrative examples

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

boundary conditions for stochastic solutions of the convection diffusion equation stochastic methods offer an attractively simple solution to complex transport controlled problems and have a wide range of physical chemical and biological applications stochastic methods do not suffer from the numerical diffusion that plagues grid based methods but they typically lose accuracy in the vicinity of interfacial boundaries in this work we introduce some ideas and algorithms that can be used to implement boundary conditions in stochastic simulations of the convection diffusion equation with accuracies comparable to the bulk phase the algorithms have been tested in two dimensional channel flows over a range of peclet numbers and compared with independent finite difference calculations

mathematical programs with complementarity constraints we study mathematical programs with complementarity constraints several stationarity concepts based on a piecewise smooth formulation are presented and compared the concepts are related to stat

is bilevel programming a special case of a mathematical program with complementarity constraints bilevel programming problems are often reformulated using the karush kuhn tucker conditions for the lower level problem resulting in a mathematical program with complementarity constraints mpcc clearly both problems are closely related but the answer to the question posed is no even in the case when the lower level programming problem is a parametric convex optimization problem this is not obvious and concerns local optimal solutions we show that global optimal solutions of the mpcc correspond to global optimal solutions of the bilevel problem provided the lower level problem satisfies the slater s constraint qualification we also show by examples that this correspondence can fail if the slater s constraint qualification fails to hold at lower level when we consider the local solutions the relationship between the bilevel problem and its corresponding mpcc is more complicated we also demonstrate the issues relating to a local minimum through examples

lipschitz continuity of optimal controls for state constrained problems this paper provides new conditions under which optimal controls are lipschitz continuous for dynamic optimization problems with functional inequality constraints a control constraint expressed in terms of a general closed convex set and a coercive cost function it is shown that the linear independence condition on active state constraints present in the earlier literature can be replaced by a less restrictive positive linear independence condition that requires linear independence merely with respect to nonnegative weighting parameters smoothness conditions on the data imposed in earlier work are also relaxed the new conditions for lipschitz continuity of optimal controls are obtained by a detailed analysis of the implications of first order optimality conditions in the form of a nonsmooth maximum principle

optimal strategies for bilevel dynamic problems in this paper we study the bilevel dynamic problem which is a hierarchy of two dynamic optimization problems where the constraint region of the upper level problem is determined implicitly by the solutions to the lower level optimal control problem to obtain optimality conditions we reformulate the bilevel dynamic problem as a single level optimal control problem that involves the value function of the lower level problem sensitivity analysis of the lower level problem with respect to the perturbation in the upper level decision variable is given and first order necessary optimality conditions are derived by using nonsmooth analysis a constraint qualification of calmness type and a sufficient condition for the calmness are also given

a new modification of the adomian decomposition method for linear and nonlinear operators in this paper we present an efficient modification of the adomian decomposition method that will facilitate the calculations we then conduct a comparative study between the new modification and the modified decomposition method the study is conducted through illustrative examples the new modification introduces a promising tool for many linear and nonlinear models

recurrence triangle for adomian polynomials in this paper a recurrence technique for calculating adomian polynomials is proposed the convergence of the series for the adomian polynomials is discussed and the dependence of the convergent domain of the solution s decomposition series n 0 u n on the initial component function u 0 is illustrated by introducing the index vectors of the adomian polynomials the recurrence relations of the index vectors are discovered and the recurrence triangle is given the method simplifies the computation of the adomian polynomials in order to obtain a solution s decomposition series with larger domain of convergence we illustrate by examples that the domain of convergence can be changed by choosing a different u 0 and a modified iteration

new higher order numerical one step methods based on the adomian and the modified decomposition methods abstract we develop new higher order numerical one step methods and apply them to several examples to investigate approximate discrete solutions of nonlinear differential equations these new algorithms are derived from the adomian decomposition method adm and the rach adomian meyers modified decomposition method mdm to present an alternative to such classic schemes as the explicit runge kutta methods for engineering models which require high accuracy with low computational costs for repetitive simulations in contrast to a one size fits all philosophy this new approach incorporates the notion of analytic continuation which extends the region of convergence without resort to other techniques that are also used to accelerate the rate of convergence such as the diagonal pade approximants or the iterated shanks transforms hence global solutions instead of only local solutions are directly realized albeit in a discretized representation we observe that one of the difficulties in applying explicit runge kutta one step methods is that there is no general procedure to generate higher order numeric methods it becomes a time consuming tedious endeavor to generate higher order explicit runge kutta formulas because it is constrained by the traditional picard formalism as used to represent nonlinear differential equations the adm and the mdm rely instead upon adomian s representation and the adomian polynomials to permit a straightforward universal procedure to generate higher order numeric methods at will such as a 12th order or 24th order one step method if need be another key advantage is that we can easily estimate the maximum step size prior to computing data sets representing the discretized solution because we can approximate the radius of convergence from the solution approximants unlike the runge kutta approach with its intrinsic linearization between computed data points we propose new variable step size variable order and variable step size variable order algorithms for automatic step size control to increase the computational efficiency and reduce the computational costs even further for critical engineering models

an improvement to an alternate algorithm for computing adomian polynomials in special cases in a published article entitled an alternate algorithm for computing adomian polynomials in special cases by the first author and some of his collaborators the algorithm were applied in special cases which restricts the application of the algorithm in this article new ideas are suggested for applying the algorithm in any cases without any more assumptions some examples are presented to illustrate the applications of the algorithm in more comprehensive situations and also to show the ability and simplicity of the algorithm

convergence of adomian s method applied to differential equations this paper deals with a new proof of convergence of adomian s method applied to differential equations we also give new formulae and properties and we suggest a simple computational form for adomian s polynomials

on adomian s decomposition method and some comparisons with picard s iterative scheme abstract we consider the general mathematical framework of adomian s decomposition method g adomian stochastic systems academic press new york 1983 for a large class of nonlinear operator equations picard s iterative scheme is considered for the same class for equations and compared with the decomposition method the paper identifies carefully all substantial differences between the two methods and shows that various advantages exist for the decomposition method

multidimensional exponential timestepping with boundary test exponential timestepping algorithms are efficient for exit time problems because a boundary test can be performed at the end of each timestep giving high order convergence in numerical evaluation of mean exit times successive time increments are independent random variables with an exponential distribution we show how to perform exact timestepping for brownian motion in more than one dimension and consider hitting times of curved surfaces taking as examples the exit times from a circle from a ball from an ellipse and from the region bounded by the two lines of a hyperbola we report the results of numerical experiments performed with the algorithms developed in this work

exponential timestepping with boundary test for stochastic differential equations we present new numerical methods for scalar stochastic differential equations successive time increments are independent random variables with an exponential distribution we perform numerical experiments using a double well potential exponential timestepping algorithms are efficient for escape time problems because a simple boundary test can be performed at the end of each step

an integral type constraint qualification to guarantee nondegeneracy of the maximum principle for optimal control problems with state constraints abstract for optimal control problems involving ordinary differential equations and functional inequality state constraints the maximum principle may degenerate producing no useful information about minimizers this is known as the degeneracy phenomenon several non degenerate forms of the maximum principle valid under different constraint qualifications have been proposed in the literature in this paper we propose a new constraint qualification under which a nondegenerate maximum principle is validated in contrast with existing results our constraint qualification is of an integral type an advantage of the proposed constraint qualification is that it is verified on a larger class of problems with nonsmooth data and convex velocity sets

normal forms of necessary conditions for dynamic optimization problems with pathwise inequality constraints abstract there has been a longstanding interest in deriving conditions under which dynamic optimization problems are normal that is the necessary conditions of optimality nco can be written with a nonzero multiplier associated with the objective function this paper builds upon previous results on nondegenerate nco for trajectory constrained optimal control problems to provide even stronger normal forms of the conditions the nco developed may address problems with nonsmooth less regular data the particular case of calculus of variations problems is here explored to show a favorable comparison with existent results

discrete bilevel programming application to a natural gas cash out problem in this paper we present a mathematical framework for the problem of minimizing the cash out penalties of a natural gas shipper the problem is modeled as a mixed integer bilevel programming problem having one boolean variable in the lower level problem such problems are difficult to solve to obtain a more tractable problem we move the boolean variable from the lower to the upper level problem the implications of this change of the problem are investigated thoroughly the resulting lower level problem is a generalized transportation problem the formulation of conditions guaranteeing the existence of an optimal solution for this problem is also in the scope of this paper the corresponding results are then used to find a bound on the optimal function value of our initial problem

an algorithm for adomian decomposition method in this paper a reliable technique for calculating adomian polynomials for nonlinear operators is developed the algorithm is fulfilled with symbolic computation so that the adomian decomposition methods for solving differential equations can be mechanized in maple ten examples are presented to illustrate the implementation of the program

grid free simulation of diffusion using random walk methods abstract the simulation of diffusion of a continuum field by the random walk displacement of a set of particles is investigated in detail computational particles are used to transport elements of the gradients of the diffusive concentration one dimensional and quasi one dimensional cases are treated for a generalized diffusion variable different types of boundary conditions of the diffusion equation are considered as well as the extensions to a system of coupled diffusion equations the reaction diffusion equations and the convection diffusion set of equations it is shown that by using concentration gradients in the random walk process statistical errors are reduced and each realization of the numerical solution is a representation of the exact solution moreover transport of higher order derivatives can be utilized to improve the smoothness the algorithm is grid free and the computational elements move to follow the gradients thus it is self adaptive and uniform resolution is attained for all times the method is particularly suitable for the simulation of diffusion in systems which involve more than one transport mechanism when the diffusivity is small and when lagrangian elements are used to model the other mechanisms

a convenient computational form for the adomian polynomials abstract recent important generalizations by g adomian stochastic systems academic press 1983 have extended the scope of his decomposition method for nonlinear stochastic operator equations see also iterative method inverse operator method symmetrized method or stochastic green s function method very considerably so that they are now applicable to differential partial differential delay and coupled equations which may be strongly nonlinear and or strongly stochastic or linear or deterministic as subcases thus for equations modeling physical problems solutions are obtained rapidly easily and accurately the methodology involves an analytic parametrization in which certain polynomials an dependent on the nonlinearity are derived this paper establishes simple symmetry rules which yield adomian s polynomials quickly to high orders

the narrow escape problem the narrow escape problem in diffusion theory is to calculate the mean first passage time of a diffusion process to a small target on the reflecting boundary of a bounded domain the problem is equivalent to solving the mixed dirichlet neumann boundary value problem for the poisson equation with small dirichlet and large neumann parts the mixed boundary value problem which goes back to lord rayleigh originates in the theory of sound and is closely connected to the eigenvalue problem for the mixed problem and for the neumann problem in domains with bottlenecks we review here recent developments in the non standard asymptotics of the problem which are based on several ingredients a better resolution of the singularity of neumann s function resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks and the breakup of composite domains into simpler components the new methodology applies to two and higher dimensional problems selected applications are r

a singular perturbation approach to first passage times for markov jump processes we introduce singular perturbation methods for constructing asymptotic approximations to the mean first passage time for markov jump processes our methods are applied directly to the integrai equation for the mean first passage time and do not involve the use of diffusion approximations an absorbing interval condition is used to properly account for the possible jumps of the process over the boundary which leads to a wiener hopf problem in the neighborhood of the boundary a model of unimolecular dissociation is considered to illustrate our methods

a new modification of the adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations abstract in this paper we propose a new modified recursion scheme for the resolution of multi order and multi point boundary value problems for nonlinear ordinary and partial differential equations by the adomian decomposition method adm our new approach including duan s convergence parameter provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series we utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components whereas several prior recursion schemes have done so this modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots which is required to complete calculation of the solution by several prior modified recursion schemes using the adm

an iterative method for solving nonlinear functional equations an iterative method for solving nonlinear functional equations viz nonlinear volterra integral equations algebraic equations and systems of ordinary differential equation nonlinear algebraic equations and fractional differential equations has been discussed

a new algorithm for the decomposition solution of nonlinear differential equations in this paper we introduce a new algorithm for applying the adomian decomposition method to nonlinear differential and partial differential equations the proposed algorithm does not require the explicit evaluation of adomian polynomials to approximate the nonlinearity term the numerical experiments show that the proposed algorithm is more accurate at larger values for time though at the cost of more computations

a new algorithm for calculating adomian polynomials for nonlinear operators in this paper a reliable technique for calculating adomian polynomials for nonlinear operators will be developed the new algorithm offers a promising approach for calculating adomian polynomials for all forms of nonlinearity the algorithm will be illustrated by studying suitable forms of nonlinearity a nonlinear evolution model will be investigated

a reliable modification of adomian decomposition method in this paper we propose a powerful modification of adomian decomposition method that will accelerate the rapid convergence of the series solution the validity of the modified technique is verified through illustrative examples in all cases of differential and integral equations we obtained excellent performance from the modified algorithm the modification could lead to a promising approach for many applications in applied sciences

the modified decomposition method for analytic treatment of differential equations the modified decomposition method is applied for analytic treatment of nonlinear differential equations the modified method accelerates the rapid convergence of the series solution dramatically reduces the size of work and provides the exact solution by using few iterations only without any need to the so called adomian polynomials numerical illustrations that include nonlinear pdes nonlinear klein gordon equations and nonlinear lane emden equation are investigated to show the pertinent features of the technique

inversion of nonlinear stochastic operators abstract the operator theoretic method adomian and malakian j math anal appl 76 1 1980 183 201 recently extended adomian s solutions of nonlinear stochastic differential equations g adomian stochastic systems analysis in applied stochastic processes nonlinear stochastic differential equations j math anal appl 55 1 1976 441 452 on the modeling and analysis of nonlinear stochastic systems in proceeding international conf on mathematical modeling vol 1 pp 29 40 to provide an efficient computational procedure for differential equations containing polynomial exponential and trigonometric nonlinear terms n y the procedure depends on the calculation of certain quantities an and bn this paper generalizes the calculation of the an and bn to much wider classes of nonlinearities of the form n y y essentially the method provides a systematic computational procedure for differential equations containing any nonlinear terms of physical significance this procedure depends on a recurrence rule from which explicit general formulae are obtained for the quantities an and bn for any order n in a convenient form this paper also demonstrates the significance of the iterative series decomposition proposed by adomian for linear stochastic operators in 1964 and developed since 1976 for nonlinear stochastic operators since both the nonlinear and stochastic behavior is quite general the results are extremely significant for applications processes need not for example be limited to either gaussian processes white noise or small fluctuations

near field and far field approximations by the adomian and asymptotic decomposition methods the adomian decomposition method and the asymptotic decomposition method give the near field approximate solution and far field approximate solution respectively for linear and nonlinear differential equations the pade approximants give solution continuation of series solutions but the continuation is usually effective only on some finite domain and it can not always give the asymptotic behavior as the independent variables approach infinity we investigate the global approximate solution by matching the near field approximation derived from the adomian decomposition method with the far field approximation derived from the asymptotic decomposition method for linear and nonlinear differential equations for several examples we find that there exists an overlap between the near field approximation and the far field approximation so we can match them to obtain a global approximate solution for other nonlinear examples where the series solution from the adomian decomposition method has a finite convergent domain we can match the pade approximant of the near field approximation with the far field approximation to obtain a global approximate solution representing the true entire solution over an infinite domain

new ideas for decomposing nonlinearities in differential equations abstract in this paper we consider the decomposition for the nonlinearity in a differential equation for the solution by decomposition by analyzing and transforming the taylor expansion of the nonlinearity about the initial solution component the decomposition of the nonlinearity is converted to the partitions of the solution sets for a class of diophantine equations this conversion simplifies the discussion and presents a new idea for decompositions we enumerate five types of partitions and their corresponding decomposition polynomials each of the last four types contains infinitely many kinds of decomposition polynomials in the form of finite sums in types 2 3 and 4 there is a parameter q and each value of q corresponds to a class of decomposition polynomials in type 5 each positive integer sequence c j satisfying 1 c 1 c 2 and j c j for j 2 3 corresponds to a class of decomposition polynomials four classes of the adomian polynomials r rach a new definition of the adomian polynomials kybernetes 37 2008 910 955 are derived as particular cases

subdiffusion in a bounded domain with a partially absorbing reflecting boundary the exit time of a subdiffusive process from a bounded domain with a partially absorbing reflecting boundary is considered the short time and long time behaviors of the exit time probability density are investigated by using a spectral decomposition on the basis of the laplace operator eigenfunctions rotation invariant domains are analyzed in depth in order to illustrate the use of theoretical formulas and to compare them to numerical simulations the asymptotic results obtained are relevant for describing subdiffusion inside a living cell with a semipermeable membrane in a chemical reactor filled with catalytic grains of finite reactivity or in mineral or biological samples which are probed by nuclear magnetic resonance measurements subject to surface relaxation

convenient analytic recurrence algorithms for the adomian polynomials abstract in this article we present four analytic recurrence algorithms for the multivariable adomian polynomials as special cases we deduce the four simplified results for the one variable adomian polynomials these algorithms are comprised of simple orderly and analytic recurrence formulas which do not require time intensive operations such as expanding regrouping parametrization and so on they are straightforward to implement in any symbolic software and are shown to be very efficient by our verification using mathematica 7 0 we emphasize that from the summation expressions a n k 1 n u n k for the multivariable adomian polynomials and an k 1 n f k u 0 c n k for the one variable adomian polynomials we obtain the recurrence formulas for the u n k and the c n k these provide a theoretical basis for developing new algorithmic approaches such as for parallel computing in particular the recurrence process of one particular algorithm for the one variable adomian polynomials does not involve the differentiation operation but significantly only the arithmetic operations of multiplication and addition are involved precisely c n 1 u n n 1 and c n k 1 n j 0 n k j 1 u j 1 c n 1 j k 1 2 k n we also discuss several other algorithms previously reported in the literature including the adomian rach recurrence algorithm 1 and this author s index recurrence algorithm 23 36

nondegenerate necessary conditions for nonconvex optimal control problems with state constraints standard versions of the maximum principle for optimal control problems with pathwise state inequality constraints are satisfied by a trivial set of multipliers in the case when the left endpoint is fixed and lies in the boundary of the state constraint set and so give no useful information about optimal controls recent papers have addressed the problem of overcoming this degenerate feature of the necessary conditions in these papers it is typically shown that if a constraint qualification is imposed requiring existence of inward pointing velocities then sets of multipliers exist in addition to the trivial ones a simple new approach for deriving nondegenerate necessary conditions is presented which permits relaxation of hypotheses previously imposed concerning data regularity and convexity of the velocity set

a reliable algorithm for obtaining positive solutions for nonlinear boundary value problems abstract in this paper we propose an algorithm for solving the nonlinear two point boundary value problem that has at least one positive solution 1 6 for in a compatible interval our method stems mainly from combining the decomposition series solution obtained by adomian decomposition method with pade approximates the validity of the approach is verified through illustrative numerical examples