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This work numerically studies bright and dark optical solitons that emerge from the dispersive concatenation model, having the Kerr law of nonlinear refractive index, using the Laplace-Adomian decomposition scheme. The simulations, surface plots, and contour plots are presented. The error measure is observed to be infinitesimally small. [ABSTRACT FROM AUTHOR]

In this study, we investigate a system of coupled nonlinear Lane-Emden equations that arise from catalytic diffusion reactions. A highly effective algorithm, which heavily relies on the differential transform method, is proposed to solve this system. The algorithm produces a convergent series solution with components that can be easily computed. The Adomian polynomials corresponding to the given system are utilized for calculating the differential transforms of its nonlinearities with multiple variables. A practical numerical example is provided to validate the effectiveness and accuracy of the present scheme. The numerical results obtained by our developed approach show a significantly lower error rate compared to other existing approaches.—In this study, we investigate a system of coupled nonlinear Lane-Emden equations that arise from catalytic diffusion reactions. A highly effective algorithm, which heavily relies on the differential transform method, is proposed to solve this system. The algorithm produces a convergent series solution with components that can be easily computed. The Adomian polynomials corresponding to the given system are utilized for calculating the differential transforms of its nonlinearities with multiple variables. A practical numerical example is provided to validate the effectiveness and accuracy of the present scheme. The numerical results obtained by our developed approach show a significantly lower error rate compared to other existing approaches. [ABSTRACT FROM AUTHOR]

This paper retrieves numerically the bright and dark 1-soliton solutions for the newly constructed concatenation model using the Laplace-Adomian decomposition technique. The errors analysis is also conducted, and they are of the order of 10-9. The surface, sectional, and error plots are exhibited for bright and dark solitons. [ABSTRACT FROM AUTHOR]

In this paper, the Adomian decomposition method (ADM) is successfully applied to find the approximate solutions for the system of fuzzy Fredholm integral equations (SFFIEs) and we also study the convergence of the technique. A consistent way to reduce the size of the computation is given to reach the exact solution. One of the best methods adopted to determine the behavior of the approximate solutions. Finally, the problems that have been addressed confirm the validity of the method applied in this research using a comparison by combining numerical methods such as the Trapezoidal rule and Simpson rule with ADM. [ABSTRACT FROM AUTHOR]

We apply the Elzaki-Adomian Decomposition Method (EADM) in this study to solve nonlinear Benjamin-Bona-Mahony (BBM) and Fisher's partial differential equations (PDE). This method, being an integral transform, is a hybrid of two well-known and efficient methods: the Elzaki transform and the Adomian decomposition method. The method is demonstrated by solving two special cases of the BBM Equation and one special case of Fisher's partial differential equation. Because of its high convergence rate in approximating exact solutions, this approach is very dependable. The method can also produce numerical solutions without the usage of restrictive assumptions or the discretization typical of numerical methods; making it free of round-off errors. The ElzakiAdomian Decomposition method employs a straightforward computation that leads to effectiveness. The efficiency of EADM is demonstrated in the significant reduction of number of numerical computations. The effectiveness and efficiency of EADM account for its broad application, particularly for higher order PDEs. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a probabilistic relationship between the exponential Bell polynomials and the weighted sums of independent Poisson random variables. A recently established probabilistic connection between the Adomian polynomials and independent Poisson random variables can be derived from the obtained relationship. This result has importance because any known identity for the exponential Bell polynomials will generate a new identity for the Poisson random variables. We use the obtained relationship to derive several new identities for the joint distribution of weighted sums of independent Poisson random variables. Few examples are provided that substantiate the obtained identities. [ABSTRACT FROM AUTHOR]

The goal of this work is to solve an initial-value problem for a fractional differential equation that governs the ray tracing through a crystalline lens using an interesting variation of the Adomian decomposition method. A new recursive scheme is presented by combining the Adomian decomposition method with a formula and via the solutions of the well-known generalized Abel equation. It is shown that the technique used here offers advantages in computing the components y n , n = 1 , 2 , ... in an easily computed formula. [ABSTRACT FROM AUTHOR]

In this paper, the Adomian decomposition method and Modified Technique are successfully applied to find the approximate solutions of the fuzzy system of Volterra integro-differential equations. The approximate solutions obtained have been improved by using the iteration of the integral equation and the numerical solution with the Simpson rule and Trapezoidal rule. These proposed methods gave excellent results close to the exact solution. The results show that the present method is very straightforward and effective. [ABSTRACT FROM AUTHOR]

By using a nonlinear method, we try to solve partial fractional differential equations. In this way, we construct the Laguerre wavelets operational matrix of fractional integration. The method is proposed by utilizing Laguerre wavelets in conjunction with the Adomian decomposition method. We present the procedure of implementation and convergence analysis for the method. This method is tested on fractional Fisher's equation and the singular fractional Emden–Fowler equation. We compare the results produced by the present method with some well-known results. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to obtain the highly accurate numerical solution of Lane–Emden-type equations using modified Adomian decomposition method (MADM) for unequal step-size partitions. Design/methodology/approach: First, the authors describe the standard Adomian decomposition scheme and the Adomian polynomials for solving nonlinear differential equations. After that, for the fast calculation of the Adomian polynomials, an algorithm is presented based on Duan's corollary and Rach's rule. Then, MADM is discussed for the unequal step-size partitions of the domain, to obtain the numerical solution of Lane–Emden-type equations. Moreover, convergence analysis and an error bound for the approximate solution are discussed. Findings: The proposed method removes the singular behaviour of the problems and provides the high precision numerical solution in the large effective region of convergence in comparison to the other existing methods, as shown in the tested examples. Originality/value: Unlike the other methods, the proposed method does not require linearization or perturbation to obtain an analytical and numerical solution of singular differential equations, and the obtained results are more physically realistic. [ABSTRACT FROM AUTHOR]

The main objective of our analysis is to modify the reduced differential transform method (RDTM) to be applicable for a wide range of nonlinear partial differential equations. The Adomian polynomials are employed to overcome the huge size and complexity of computing reduced differential transforms, and new generalizations of transformed formulas are established. Our new approach provides a series solution with effortless components to compute. A new sufficiency convergence proof of series solution is introduced. The (1 + m)-dimensional telegraph and Burgers equations are tested to illustrate robustness and reliability of our modification. [ABSTRACT FROM AUTHOR]

The Adomian decomposition method (ADM) is a powerful tool to solve several nonlinear functional equations and a large class of initial/boundary value problems. In this paper, we discuss a probabilistic approach to compute the Adomian polynomials (AP's), which is the main part of the ADM. We provide a probabilistic interpretation for the AP's, both for the one-variable and the multivariable case. We derive some new recurrence relations for the computation AP's. Some suitable examples are discussed to show that the probabilistic approach to compute the AP's is much simpler than the analytical or combinatorial approach. [ABSTRACT FROM AUTHOR]

In this article, an iterative method suitable for inverting semilinear problems is presented. The method employs a two-way parametrization that is able to reduce semilinear boundary-value problems into simpler and linear systems governed by the Laplacian. In contrast to Newton–Picard iterations, the method prescribes a well-defined procedure for constructing initial iterates, and thus, does not suffer from lack of global convergence. With rectangular domains and periodic boundary conditions, the technique also allows fast Poisson solvers to be engaged. Theoretical justifications are provided and supported by several experiments. Numerical results show that the method yields approximations which are comparable to reference solutions. [ABSTRACT FROM AUTHOR]

In this work, a new emerging analytical techniques variational iteration method combine with Aboodh transform has been applied to find out the significant important analytical and convergent solution of some mathematical models of fractional order. These mathematical models are of great interest in engineering and physics. The derivative is in Caputo's sense. These analytical solutions are continuous that can be used to understand the physical phenomena without taking interpolation concept. The obtained solutions indicate the validity and great potential of Aboodh transform with the variational iteration method and show that the proposed method is a good scheme. Graphically, the movements of some solutions are presented at different values of fractional order. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce Sumudu decomposition method for solving Fractional Bratu-type differential equation. This method is a combination of the Sumudu transform and Adomian decomposition method. The fractional derivative is described in the Caputo sense. The Sumudu decomposition method is applied to obtain approximate analytical solution of non-linear Fractional Bratu-type differential equation. A novel combination of Sumudu transform and Adomian decomposition provides approximate solution in the form of infinite convergent series solution. The behavior of approximate analytical solutions and exact solutions for different values of are plotted graphically. The results acquired from Sumudu decomposition method indicates that the proposed method is very well founded, suitable and effective. Finally, some numerical examples are given to illustrate the efficiency and applicability of our method. [ABSTRACT FROM AUTHOR]

The second-order nonlinear oscillators have rich dynamics. We proposed a novel analytical method based on both variational iteration method and Adomian method. The variational iteration method is used to establish an equivalent integral system. So then Adomian polynomials are adopted to linearize the strong nonlinear terms in nonlinear oscillators and analytical solutions are obtained successively. [ABSTRACT FROM AUTHOR]

In financial mathematics, the procedure for finding the solutions of the Black-Scholes model defined in terms of Arithmetic Asian Option (AAO) is one of the most tasking issues when considering its corresponding analytical solutions. In this paper, such analytical solution of a continuous arithmetic Asian option is obtained through the application of a proposed semi-analytical approach known as Projected Differential transform Method (PDTM). The Asian option model in continuous arithmetic form witnesses the application of PDTM for the first time in literature (to the authors’ best of knowledge). The PDTM entails less computational work, even without neglecting the high level of accuracy. The obtained solution is in agreement with those in literature via other solution methods. In terms of recommendation, the proposed solution method will be of great interest for related versions or forms of Asian option pricing models (geometric) likewise other financial nonlinear differential models. [ABSTRACT FROM AUTHOR]

This paper explores the approximate analytical solution of non-linear Klein-Gordon equations (NKGE) by using multistep modified reduced differential transform method (MMRDTM). Through this proposed strategy, the non-linear term is substituted by associating Adomian polynomials obtained by utilization of a multistep approach. The NKGE solutions can be obtained with a reduced number of computed terms. In addition, the approximate solutions converge rapidly in a wide time region. Three examples are provided to illustrate the effectiveness of the proposed method to obtain solutions for the NKGE. Graphical results are shown to represent the behavior of the solution so as to demonstrate the validity and accuracy of the MMRDTM. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inhomogeneous Benjamin-Bona-Mahony-Peregrine-Burgers (BBMPB) equation with its initial condition. The decomposition methods such as Adomian and Laplace are applied to this equation. Then, the solution satisfying the given initial condition is gained by using each method. To demonstrate these decomposition methods are how effective and convenient for solving the nonlinear partial differential equations(NLPDEs), solutions of some NLPDEs subjected to initial conditions are attained by these method. In the application of the method, it is clearly noticed that there is no need to convert the nonlinear terms into the linear ones due to the Adomian polynomials. In addition to this simplicity, it is seen that each method gives the same solution with a fast convergence by taking the advantage of the noise terms. [ABSTRACT FROM AUTHOR]

Kepler's equation is one of the fundamental equations in orbital mechanics. It is a transcendental equation in terms of the eccentric anomaly of a planet which orbits the Sun. Determining the position of a planet in its orbit around the Sun at a given time depends upon the solution of Kepler's equation, which we will solve in this paper by the Adomian decomposition method (ADM). Several properties of the periodicity of the obtained approximate solutions have been proved in lemmas. Our calculations demonstrated a rapid convergence of the obtained approximate solutions which are displayed in tables and graphs. Also, it has been shown in this paper that only a few terms of the Adomian decomposition series are sufficient to achieve highly accurate numerical results for any number of revolutions of the Earth around the Sun as a consequence of the periodicity property. Numerically, the four-term approximate solution coincides with the Bessel-Fourier series solution in the literature up to seven decimal places at some values of the time parameter and nine decimal places at other values. Moreover, the absolute error approaches zero using the nine term approximate Adomian solution. In addition, the approximate Adomian solutions for the eccentric anomaly have been used to show the convergence of the approximate radial distances of the Earth from the Sun for any number of revolutions. The minimal distance (perihelion) and maximal distance (aphelion) approach 147 million kilometers and 152.505 million kilometers, respectively, and these coincide with the well known results in astronomical physics. Therefore, the Adomian decomposition method is validated as an effective tool to solve Kepler's equation for elliptical orbits. [ABSTRACT FROM AUTHOR]

In this work, a new practicalmethod, which is named by the Fractional Natural DecompositionMethod (FNDM), is proposed to obtain the approximate analytical solutions of fractional gas dynamics equations. The FNDM is a mixture of the Natural Transform Method and the Adomian Decomposition Method. In this method, the fractional derivatives are considered as Caputo sense and the nonlinear terms are determined by virtue of Adomian polynomials. Some test examples are given to demonstrate the efficiency and accuracy of the FNDM. [ABSTRACT FROM AUTHOR]

In this paper, we address a new computational method, which is called the decomposition- Sumudu-like-integral-transform method, to handle the 1-D Burgers equation. The proposed method enables the efficient and accurate. [ABSTRACT FROM AUTHOR]

In this paper, the Adomian decomposition method (ADM) is proposed to solve the hyperbolic Kepler equation which is often used to describe the eccentric anomaly of a comet of extrasolar origin in its hyperbolic trajectory past the Sun. A convenient method is therefore needed to solve this equation to accurately determine the radial distance and/or the Cartesian coordinates of the comet. It has been shown that Adomian's series using a few terms are sufficient to achieve extremely accurate numerical results even for much higher values of eccentricity than those in the literature. Besides, an exceptionally rapid rate of convergence of the sequence of the obtained approximate solutions has been demonstrated. Such approximate solutions possess the odd property in the mean anomaly which are illustrated through several plots. Moreover, the absolute remainder error, using only three components of Adomian's solution decreases across a specified domain, approaches zero as the eccentric anomaly tends to infinity. Also, the absolute remainder error decreases by increasing the number of components of the Adomian decomposition series. In view of the obtained results, the present method may be the most effective approach to treat the hyperbolic Kepler equation. [ABSTRACT FROM AUTHOR]

Purpose: The aim of our study is comparison of method applications based on differential transformations for solving boundary value problems which are described by non-linear ordinary differential equations. Methods: This article reviews two approaches based on differential transformations for solving non-linear boundary value problems: the modified differential transform method and the system-analogue simulation method. Results: In this paper, we present results of the numerical solution of non-linear boundary value problem by methods based on differential transformations for demonstration the effectiveness and applicability of techniques. The relative error for given solutions, obtained with using first 6 discretes of differential spectra is presented. Discussion: Comparison of numerical solutions obtained by modified differential transform method and system-analogue simulation method with exact solution shows that both methods have good agreement with exact solution of non-linear boundary value problem for small intervals. However, application of system-analogue simulation method is preferential for big intervals, on which the boundary value problem is solved. [ABSTRACT FROM AUTHOR]

A technique for extending the Laplace transform method to solve nonlinear differential equations is presented. By developing several theorems, which incorporate the Adomian polynomials, the Laplace transformation of nonlinear expressions is made possible. A number of well-known nonlinear equations including the Riccati equation, Clairaut's equation, the Blasius equation and several other ones involving nonlinearities of various types such as exponential and sinusoidal are solved for illustration. The proposed approach is analytical, accurate, and free of integration. [ABSTRACT FROM AUTHOR]

In this work, an effective numerical method is developed to solve a class of singular boundary value problems arising in various physical models by using the improved differential transform method (IDTM). The IDTM applies the Adomian polynomials to handle the differential transforms of the nonlinearities arising in the given differential equation. The relation between the Adomian polynomials of those nonlinear functions and the coefficients of unknown truncated series solution is given by a simple formula, through which one can easily deduce the approximate solution which takes the form of a convergent series. An upper bound for the estimation of approximate error is presented. Several physical problems are discussed as illustrative examples to testify the validity and applicability of the proposed method. Comparisons are made between the present method and the other existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider the fourth-order linear differential equationu(4)+f(x)u=g(x) subject to the mixed boundary conditionsu(0) =u(1) =uii(0) =uii(1) = 0.We first establish sufficient conditions onf(x) that guarantee a unique solution of this problem in the Hilbert space by using an a priori estimate. Accurate analytic solutions in series forms are obtained by a new variation of the Duan-Rach modified Adomian decomposition method (DRMA), and then extend this approach to some boundary value problems of fourth-order nonlinear beam equations. Also, a comparison of the two approximate solutions by the ADM with the Green function approach is presented. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present the application of the analytical series technique to study properties of the nonlinear chaotic dynamical systems. More specifically, Laplace-Adomian decomposition method is applied to Rössler system and the so-called generalized Lorenz system. Some advantages and possible applications of this approach are discussed. Results are illustrated by numerical computations. [ABSTRACT FROM AUTHOR]

In this article, a new method called the Natural Decomposition Method (NDM) is introduced. We use this method to obtain exact solutions for three different types of nonlinear partial differential equations (NLPDEs). The NDM is based on the Natural transform method (NTM) and the Adomian decomposition method (ADM). The theoretical analysis of the NDM is investigated for some equations and is calculated with easily computable terms. The results we obtained are compared with existing solutions obtained by other methods. One can conclude that the method is easy to use and efficient. Most of the symbolic and numerical computations were performed using Mathematica software. [ABSTRACT FROM AUTHOR]

In this paper, we discuss two simple parametrization methods for calculating Adomian polynomials for several nonlinear operators, which utilize the orthogonality of functions einx, where n is an integer. Some important properties of Adomian polynomials are also discussed and illustrated with examples. These methods require minimum computation, are easy to implement, and are extended to multivariable case also. Examples of different forms of nonlinearity, which includes the one involved in the Navier Stokes equation, is considered. Explicit expression for the n-th order Adomian polynomials are obtained in most of the examples. [ABSTRACT FROM AUTHOR]

We investigate the reactant diffusion within a porous catalytic pellet including the heat of reaction. The Lane–Emden boundary value problem with an Arrhenius reaction rate is used to model the reactant concentration. We combine the Volterra integral form with the Adomian decomposition method to solve the equivalent Fredholm–Volterra integral equation. We then estimate the reactant concentration at the center of the catalytic pellet and the iterated Shanks transform is used to improve the accuracy of the approximations. The objective error analysis formulas are used to demonstrate a high accuracy and rapid rate of convergence, which does not depend on a priori knowledge of the exact solution or comparison with an alternate approximation method. Thus low-stage approximations by the Adomian decomposition method are validated for parametric simulations of the reactant concentration profiles and the effectiveness factor profiles. Our approach demonstrates enhancements over previous investigations, and is readily extensible to more general diffusion reaction models in catalytic reactor engineering. [ABSTRACT FROM AUTHOR]

We develop a robust algorithm for isothermal flash calculations of multi-component mixtures. The algorithm provides an explicit, parametric solution for a recent variation of the Rachford–Rice equation by employing a powerful analytical technique, namely the Adomian decomposition method. The computational speed of the algorithm is shown to be further enhanced by applying the iterated Shanks transformation. Unlike previous methods, our algorithm obviates the troublesome need for an initial guess, guarantees an excellent rate of convergence and is demonstrated to be more computationally economic than prior art. Several reliable examples from the literature, including the flash calculation for a mixture with nineteen components, as well as an extensive set of hypothetical, randomly-generated problems are presented to illustrate the exceptional efficacy of our new approach. [ABSTRACT FROM AUTHOR]

In this paper, we examine a system of two coupled nonlinear differential equations that relates the concentrations of carbon dioxide CO $$_2$$ and phenyl glycidyl ether in solution. This system is subject to a set of Dirichlet boundary conditions and a mixed set of Neumann and Dirichlet boundary conditions. We apply the Adomian decomposition method combined with the Duan-Rach modified recursion scheme to analytically treat this system of coupled nonlinear boundary value problems. The rapid convergence of our analytic approximate solutions is demonstrated by graphs of the objective error analysis instead of comparison to an alternate solution technique alone. The Adomian decomposition method yields a rapidly convergent, easily computable, and readily verifiable sequence of analytic approximate solutions that is suitable for numerical parametric simulations. Thus our sequence of approximate solutions are shown to identically satisfy the original set of model equations as closely as we please. [ABSTRACT FROM AUTHOR]

In this study, an efficient iterative algorithm is devised to handle a nonlinear equation arising in estimation of thermodynamic properties at supercritical conditions. The approach is based on a synergistic combination of the classic Newton-Raphshon algorithm and the Adomian decomposition method. We demonstrate that the proposed method enjoys a higher degree of accuracy while requiring fewer iterations to reach a specific solution compared to that by the Newton-Raphson algorithm. To illustrate the efficiency of the aforementioned solution technique, several numerical examples are provided. The proposed method has been easily implemented in computer codes to provide parametric, not just numeric, solutions to the model equations. Consequently, one can derive other thermodynamic properties, which have not been treated parametrically to date, based on our new combined approach. [ABSTRACT FROM AUTHOR]

Abstract: The use of the Hankinson–Thomas–Phillips correlation for prediction of the natural gas compressibility factor is a common practice in natural gas engineering calculations. However, this equation suffers a serious deficiency from a computational viewpoint; in that it is not explicit with respect to the z-factor and hence is subject to time-consuming trial and error procedures. In this paper, we propose an explicit series expansion equivalent to the Hankinson–Thomas–Phillips equation by the aid of a powerful mathematical technique known as the Adomian decomposition method. Furthermore, we have enhanced our formula by a applying nonlinear convergence accelerator algorithm, namely the Shanks transform. The proposed equation is simple, easy to use, and is shown to be extremely accurate in reproducing the experimental PVT data of natural gases. Moreover, in contrast to the previous numerical algorithms such as the Newton–Raphson algorithm, the explicit nature of our formula obviates the need for any initial guess as an input for calculation of the z-factor. Such independence permits our formula to always quickly converge to the correct z-factor. [Copyright &y& Elsevier]

Abstract: The algebraic solution of the Lane–Emden equation is obtained by means of an algebraic operational calculus and the Adomian polynomials. [Copyright &y& Elsevier]

In this paper, we investigate the diffusion of oxygen in a spherical cell including nonlinear uptake kinetics. The Lane-Emden boundary value problem with Michaelis-Menten kinetics is used to model the dimensionless oxygen concentration in our analysis. We first convert the Lane-Emden equation to the equivalent Volterra integral form that incorporates the boundary condition at the cell's center, but which still leaves one unknown constant of integration, as an intermediate step. Next we evaluate the Volterra integral form of the concentration and its flux at the cell membrane and substitute them into the remaining boundary condition to determine the unknown constant of integration by appropriate algebraic manipulations. Upon substitution we have converted the equivalent Volterra integral form to the equivalent Fredholm-Volterra integral form, and use the Duan-Rach modified recursion scheme to effectively decompose the unknown constant of integration by formula. The Adomian decomposition method is then applied to solve the equivalent nonlinear Fredholm-Volterra integral representation of the Lane-Emden model for the concentration of oxygen within the spherical cell. Our approach shows enhancements over existing techniques. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a new approach to solve nonlocal initial-boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations subject to initial and nonlocal boundary conditions of integral type. We first transform the given nonlocal initial-boundary value problems of integral type for the linear and nonlinear parabolic and hyperbolic partial differential equations into local Dirichlet initial-boundary value problems, and then use a relatively new modified Adomian decomposition method (ADM). Furthermore we investigate the Fourier–Adomian method, which also does not require any a priori assumptions on the solution, for the solution of nonlocal initial-boundary value problems combined with our new approach. Several examples are presented to demonstrate the efficiency of the ADM. [Copyright &y& Elsevier]

In the paper 'He's Homotopy Perturbation Method for Calculating Adomian Polynomials' by A. Ghorbani and J. Saberi-Nadjafi, [International Journal of Nonlinear Sciences and Numerical Simulation, 8(2) (2007) 229-232], the authors introduced the homotopy perturbation method to calculate Adomian polynomials, which they claim makes the calculation of Adomian polynomials easy. We discuss here the restrictions on their method for calculating Adomian Polynomials for nonlinear equations. We also prove analytically that the method given by Ghorbani and Saberi-Nadjafi is equivalent to the method used in [Applied Mathematics and Computation, 138(2-3) (2003) 523-529]. [ABSTRACT FROM AUTHOR]

Abstract: The Adomian decomposition method (ADM) is one of the most effective methods for constructing analytic approximate solutions. In this paper, based on the new definitions of the Adomian polynomials, and the double decomposition method, a new algorithm is proposed to construct analytic approximations for nonlinear differential equations with (coupled) boundary conditions. Furthermore, a MAPLE software package is developed to implement our new algorithm, which is user-friendly and efficient. One only needs to input the system, boundary conditions and several necessary parameters, then our package will deliver the analytic approximate solution within a few seconds. Several different types of nonlinear examples are given to illustrate the scope and demonstrate the validity of our software package, especially for nonlinear multi-order and multi-point boundary value problems. Our package provides a helpful and easy-to-use tool in science and engineering to deal with the most frequently encountered boundary value problems. This MAPLE package is free to download on the website: http://www.cs.ecnu.edu.cn/∼lizb/index.htm. [Copyright &y& Elsevier]

In this paper, we present the Adomian decomposition method and its modifications combined with convergence acceleration techniques, such as the diagonal Padé approximants and the iterated Shanks transforms, to solve nonlinear fractional ordinary differential equations. Two nonlinear numeric examples demonstrate that either the diagonal Padé approximants or the iterated Shanks transforms can efficiently extend the effective convergence region of the decomposition series solution. [ABSTRACT FROM AUTHOR]

We present a new approach to calculate analytic approximations of blow-up solutions and their critical blow-up times. Our approach applies the Adomian decomposition-Padé method to quickly and easily compute the critical blow-up times, which comprises the Adomian decomposition method combined with the Padé approximants technique. We validate our new approach with a variety of numerical examples, including nonlinear ODEs, systems of nonlinear ODEs, and nonlinear PDEs. Furthermore, our new method is shown to be more convenient than prior art that relies on compound discretized algorithms. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A new formula for Adomian polynomials is introduced and applied to obtain truncated series solutions for fractional initial value problems with nondifferentiable functions. These kinds of equations contain a fractional single term which is examined using Jumarie fractional derivatives and fractional Taylor series for nondifferentiable functions. The property of nonlocality of these equations is examined, and the existence and uniqueness of solutions are discussed. Convergence and error analysis for the Adomian series solution are also studied. Numerical examples show the accuracy and efficiency of this formula for solving initial value problems for high-order fractional differential equations. [ABSTRACT FROM AUTHOR]

Abstract: Although being powerful, the differential transform method (DTM) yet suffers from a drawback which is the lack of a systematic methodology for derivation of the differential transforms for nonlinear expressions. In the current paper, it is shown that this defect can be overcome with the help of the Adomian polynomials perfectly. The application of the proposed technique in treatment of nonlinear differential equations is well illustrated by a number of examples. In addition, the transformed analogues of some frequent nonlinearities are presented. [Copyright &y& Elsevier]

Abstract: In this article, we proposed an auxiliary parameter method using Adomian polynomials and Laplace transformation for nonlinear differential equations. This method is called the Auxiliary Laplace Parameter Method (ALPM). The nonlinear terms can be easily handled by the use of Adomian polynomials. Comparison of the present solution is made with the existing solutions and excellent agreement is noted. The fact that the proposed technique solves nonlinear problems without any discretization or restrictive assumptions can be considered as a clear advantage of this algorithm over the numerical methods. [Copyright &y& Elsevier]

In this article, we study some fundamental results concerning the convergence of the Adomian decomposition method (ADM) for an abstract Cauchy problem of a system of first-order nonlinear differential equations. Under certain conditions, we obtain upper estimates for the norm of solutions of this system. We also obtain results about the error estimates for the approximate solutions by the ADM and discuss their applications. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we develop new numeric modified Adomian decomposition algorithms by using the Wazwaz–El-Sayed modified decomposition recursion scheme, and investigate their practicality and efficiency for several nonlinear examples. We show how we can conveniently generate higher-order numeric algorithms at will by this new approach, including, by using examples, 12th-order and 20th-order numeric algorithms. Furthermore, we show how we can achieve a much larger effective region of convergence using these new discrete solutions. We also demonstrate the superior robustness of these numeric modified decomposition algorithms including a 4th-order numeric modified decomposition algorithm over the classic 4th-order Runge–Kutta algorithm by example. The efficiency of our subroutines is guaranteed by the inclusion of the fast algorithms and subroutines as published by Duan for generation of the Adomian polynomials to high orders. [Copyright &y& Elsevier]