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

a comparison of adomian s decomposition method and wavelet galerkin method for solving integro differential equations this paper aims to introduce a comparison of adomian decomposition method and wavelet galerkin method for the solution of integro differential equations from the computational viewpoint the comparison shows that the adomian decomposition method is efficient and easy to use

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

a numerical application of the decomposition method for the combined kdv mkdv equation we consider solitary wave solutions of the combined korteweg de vries modified korteweg de vries kdv mkdv equation we prove the convergence of adomian decomposition method adm applied to the combined kdv mkdv equation then we obtain the exact solitary wave solutions and numerical solutions of the combined kdv mkdv equation for the initial conditions the numerical solutions are compared with the known analytical solutions their remarkable accuracy are finally demonstrated for the combined kdv mkdv equation

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

a modification of the adomian decomposition method for a class of nonlinear singular boundary value problems in this paper the adomian decomposition method is modified to solve a class of nonlinear singular boundary value problems which arise as nonlinear normal modal equations in nonlinear conservative vibratory systems the effectiveness of the modified method is verified by three examples

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

the adomian decomposition method for computing eigenelements of sturm liouville two point boundary value problems we will consider the adomian decomposition method for computing the eigenelements of sturm liouville two point boundary value problem the method proved to be very successful and powerful in computing such elements numerical examples showed the competitive nature of the method comparison with the results of others will also be presented

a study on a boundary layer equation arising in an incompressible fluid in this paper a differential equation of mixing layer that arises in viscous incompressible fluid will be investigated the series solution method will be effectively implemented to a related integro differential equation the condition at infinity will be applied to a related pade approximants of the obtained series solution

approximate solutions for boundary value problems of time fractional wave equation in this article we implement relatively a new analytical technique the adomian decomposition method for solving the boundary value problems of time fractional wave equation the fractional derivative is described in the caputo sense the decomposition method is used to construct analytical approximate solutions of time fractional wave equation subject to specified boundary conditions the solutions are calculated in the form of a convergent series with easily computable components some examples are given the results reveal that the adomian method is very effective and convenient

discontinuous fractional sturm liouville problems with transmission conditions abstract the main purpose of this study is the investigation of discontinuous sturm liouville problem with fractional derivatives we give an operator theoretic framework of the problem under consideration namely we define an operator a in the hilbert space l 2 1 1 the eigenvalues and corresponding eigenfunctions of which coincide with the eigenvalues and corresponding eigenfunctions of the boundary value problem respectively then we establish the characteristic function and prove that the eigenvalues of the considered problem coincide with the roots of this characteristic function

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

a novel fast modification of the adomian decomposition method to solve integral equations of the first kind with hypersingular kernels abstract in this paper using the modified adomian decomposition method a novel fast scheme has been presented being interactive to be used for solving a class of hypersingular integral equations of the first kind compared to the standard adomian decomposition method this method expedites the quick accumulation of a series of solutions different problems for exemplification have been solved in order to display the accuracy and validity of the aforementioned method according to the results it gets known that the novel fast modified adomian decomposition method is so efficient and simple which gives the exact solution and also shuns the complicated function of long theoretical calculations of collocation methods based on polynomial producing the exact solution

identifying a control function in parabolic partial differential equations from overspecified boundary data determination of an unknown time dependent function in parabolic partial differential equations plays a very important role in many branches of science and engineering in the current investigation the adomian decomposition method is used for finding a control parameter p t in the quasilinear parabolic equation u t u x x p t email protected in 0 1 x 0 t with known initial and boundary conditions and subject to an additional condition in the form of 0 1k x u x t dx e t email protected email protected t which is called the boundary integral overspecification the main approach is to change this inverse problem to a direct problem and then solve the resulting equation using the well known adomian decomposition method the decomposition procedure of adomian provides the solution in a rapidly convergent series where the series may lead to the solution in a closed form furthermore due to the rapid convergence of adomian s method a truncation of the series solution with sufficiently large number of implemented components can be considered as an accurate approximation of the exact solution this method provides a reliable algorithm that requires less work if compared with the traditional techniques some illustrative examples are presented to show the efficiency of the presented method

remark on modifications of adomian decomposition method a vital exception to the superiority of the chebyshev polynomial modification of the adomian decomposition method to other recent modifications is shown in this article also a new modification of the decomposition method is proposed and shown to be reliable

numerical comparison of methods for solving fractional differential algebraic equations fdaes in this paper the variational iteration method vim and the adomian decomposition method adm are implemented to give approximate solutions for fractional differential algebraic equations fdaes both methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations this paper presents a numerical comparison between these two methods and the homotopy analysis method ham for solving fdaes numerical results reveal that the vim and the adm are quite accurate and applicable

analytic solution of nonlinear boundary value problems in several dimensions by decomposition abstract nonlinear partial differential equations are systematically solved by the decomposition method of g adomian for general boundary conditions described by boundary operator equations linear partial differential equations and linear or nonlinear ordinary differential equations become special cases all dealt with by a single procedure

a geometric approach to complete reducibility let g be a connected reductive linear algebraic group we use geometric methods to investigate g completely reducible subgroups of g giving new criteria for g complete reducibility we show that a subgroup of g is g completely reducible if and only if it is strongly reductive in g this allows us to use ideas of r w richardson and hilbert mumford kempf from geometric invariant theory we deduce that a normal subgroup of a g completely reducible subgroup of g is again g completely reducible thereby providing an affirmative answer to a question posed by j p serre and conversely we prove that the normalizer of a g completely reducible subgroup of g is again g completely reducible some rationality questions and applications to the spherical building of g are considered many of our results extend to the case of non connected g

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

numerical solution of singularly perturbed fifth order two point boundary value problem we will consider adomain decomposition method and the homotopy method to solve a fifth order singularly perturbed bvp arising in viscoelastic flows the success and pitfalls of the methods will be investigated numerical testing will be provided to show the efficiency of the methods proposed comparison with the work of others will also be done

thin film flow of magnetohydrodynamic mhd johnson segalman fluid on vertical surfaces using the adomian decomposition method abstract in this paper the magnetohydrodynamic mhd thin film flow of johnson segalman fluid for lifting and drainage problems on a vertical surfaces are investigated the nonlinear differential equations arising from the mathematical modeling of the johnson segalman fluid the laws of momentum mass and suitable boundary conditions are solved analytically using the adomian decomposition method adm a constant magnetic field is applied transversely to the direction of flow and particular attention is given to the effects of magnetic parameters on the velocity the influence on the velocity field of various non newtonian parameters such as the weissenberg number stokes number and applied magnetic field are investigated several graphs illustrate the influence of various pertinent parameters on the velocity it is observed that increasing the applied magnetic field will generally reduce the flow velocity in the lifting case while in the drainage case the magnetic field and the velocity field are shown to have a direct relation

a novel expansion iterative method for solving linear partial differential equations of fractional order in this manuscript we implement a relatively new analytic iterative technique for solving time space fractional linear partial differential equations subject to given constraints conditions based on the generalized taylor series formula the solution methodology is based on generating the multiple fractional power series expansion solution in the form of a rapidly convergent series with minimum size of calculations this method can be used as an alternative to obtain analytic solutions of different types of fractional linear partial differential equations applied in mathematics physics and engineering some numerical test applications were analyzed to illustrate the procedure and to confirm the performance of the proposed method in order to show its potentiality generality and accuracy for solving such equations with different constraints conditions numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the suggested algorithm

derivation of the adomian decomposition method using the homotopy analysis method abstract adomian decomposition method has been used intensively to solve nonlinear boundary and initial value problems it has been proved to be very efficient in generating series solutions of the problem under consideration under the assumption that such series solution exits however very little has been done to address the mathematical foundation of the method and its error analysis in this article the mathematical derivation of the method using the homotopy analysis method is presented in addition an error analysis is addressed as well as the convergence criteria

equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations abstract we consider the solution of partial differential equations for initial boundary conditions using the decomposition method the partial solutions obtained from the seperate equations for the highest ordered linear operator terms are shown to be identical when the boundary conditions are general and asymptotically equal when the boundary conditions in one independent variable are independet of other variables

a comparison between adomian decomposition method and taylor series method in the series solutions in this work we will compare the performance of the adomian decomposition method and the taylor series method applied to the solution of linear and nonlinear ordinary differential equations the comparison shows that the decomposition method is reliable efficient and easy to use from a computational viewpoint although the two methods provide the solution in an infinite series the decomposition method provides a fast convergent series of easily computable components and eliminates cumbersome computational work needed by taylor series method

numerical computation of the nonlinear feedback operators for the nonquadratic time variant optimal control problems in this paper a reliable numerical algorithm for the nonlinear feedback operators for the time variant optimal control with nonquadratic criteria is constructed our approach rests mainly on the modified adomian decomposition method the convergence of the decomposition method as applied to the synthesis equations is investigated the proposed scheme avoids the complexity provided by using other techniques

a dynamic compensator for in domain control of burgers equation this paper addresses the problem of output regulation control of a burgers equation under pointwise in domain actuation a dynamic compensator that generates the in domain control is developed for

the modified decomposition method and pade approximants for solving the thomas fermi equation in this paper we propose a new approach to develop a nonperturbative approximate solution for the thomas fermi equation this approach rests on the recently developed modification of the adomian decomposition method the initial slope of the thomas fermi potential y 0 is computed by converting the obtained series solution into several diagonal pade approximants the proposed scheme is presented in a general way so that it can be used in applied sciences

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

on the stability of rational runge kutta methods abstract this paper is concerned with the stability of rational two stage runge kutta methods for the numerical solution of stiff differential systems with a stability analysis based on linear diagonal systems of arbitrary dimension we find necessary and sufficient conditions for the coefficients of a method to be a and a stable extending previous results on this subject given by hairer 1 and wambecq 3 4 5

numerical experiments on the rational runge kutta method abstract the rational runge kutta rrk method is a non linear explicit a stable scheme for the numerical treatment of initial value problems for systems of ordinary differential equations odes the method may be expected to be suitable for the treatment of stiff systems when the number of odes is very large we have studied theoretically and implemented numerically the rrk scheme on problems of the type d x t d t a x t w t under stiff conditions we have paid particular attention to the case when the system of odes originates from the semi discretization of an evolution problem for a partial differential equation the scheme is shown to perform poorly when the eigenvalues of a are widely spread on the other hand its performance is reasonable when the eigenvalues of a are packed around an average negative value and the forcing term w t evolves slowly with respect to this value

necessary conditions for the appearance of noise terms in decomposition solutions series in this paper the phenomena of the noise terms that appear in the components of the series solution introduced by adomian and rach 1 will be examined extensively the phenomena is useful in demonstrating a fast convergence of the exact solution however the condition set by adomian and rach 1 in that these noise terms appear for every nonhomogeneous equation is not always true and the nonhomogeneity condition is not sufficient another necessary condition will be established in this study to ensure the appearance of noise terms in specific cases several differential and integral equations will be tested to confirm the newly developed condition

rational runge kutta methods are not suitable for stiff systems of odes abstract the present paper shows that rational rk methods are not very appropriate to solve stiff differential equations the ca 0 stability i e componentwise contractivity is defined and the non existence of ca 0 stable rational rk methods is demonstrated furthermore it is shown that the stepsizes which can be expected when solving a stiff differential system with a rational or with an explicit linear rk method are of the same order of magnitude

analytical approximations and pade approximants for volterra s population model in this paper an analytic approximation for volterra s model for population growth of a species in a closed system is presented the nonlinear integro differential model includes an integral term that characterizes accumulated toxicity on the species in addition to the terms of the logistic equation the series solution method and the decomposition method are implemented independently to the model and to a related ode the pade approximants that often show superior performance over series approximations are effectively used in the analysis to capture the essential behavior of the population u t of identical individuals

a comparison between adomian s decomposition methods and perturbation techniques for nonlinear random differential equations abstract comparison of the adomian decomposition method and regular perturbation techniques applied to the solution of nonlinear vector random differential equations shows that the decomposition method is superior and generally applicable even for the cases where perturbation is applicable the comparison shows that the decomposition method is easier to compute and supplies quantitatively reliable results

the decomposition method for approximate solution of the goursat problem abstract in this paper the adomian s decomposition method is effectively implemented to establish an analytic solution and a reliable numerical approximation to the goursat partial differential equation linear and nonlinear goursat examples are examined and the method supplied quantitatively reliable results for these types of problems the accuracy level of the results obtained indicates the superiority of the decomposition method over existing numerical methods that were applied to the goursat problem

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

comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations in this article the homotopy perturbation method proposed by j h he is adopted for solving linear fractional partial differential equations the fractional derivatives are described in the caputo sense comparison of the results obtained by the homotopy perturbation method with those obtained by the variational iteration method reveals that the present methods are very effective and convenient

a two step adomian decomposition method the adomian decomposition method adm is a nonnumerical method for solving a wide variety of functional equations and usually gets the solution in a series form in this paper we propose an efficient modification of adm namely two step adomian decomposition method tsadm that will facilitate the calculations we then conduct a comparative study between the tsadm and previous methods with the help of several illustrative examples the results obtained indicate that the tsadm is effective and promising

a new modification of the homotopy perturbation method for linear and nonlinear operators in this paper we present an efficient modification of the homotopy perturbation method that will facilitate the calculations we then conduct a comparative study between the new modification and the homotopy perturbation method numerical illustrations are investigated to show the features of the technique the modified method accelerates the rapid convergence of the series solution and reduces the size of work the new modification introduces a promising tool for many linear and nonlinear problems

a new approach to the nonlinear advection problem an application of the decomposition method abstract in this paper the nonlinear advection problem is solved easily and elegantly by using the adomian s decomposition method the exact solution is quickly obtained by observing the appearance of the self canceling noise terms whose sum vanishes in the limit comparison with other existing techniques shows that the present approach is highly accurate

adomian decomposition method for approximating the solution of the korteweg devries equation in this paper the adomian decomposition method for solving the nonlinear korteweg devries equation is implemented with appropriate initial conditions we discuss the case when the problem has either one or more solitons some numerical examples are presented

adomian decomposition method with chebyshev polynomials in this paper an efficient modification of the adomian decomposition method is presented by using chebyshev polynomials the proposed method can be applied to linear and non linear models the scheme is tested for some examples and the obtained results demonstrate reliability and efficiency of the proposed method