Beyond Adomian polynomials: He polynomials

The Adomian decomposition method is widely used in approximate calculation. The main difficulty of the method is to calculate Adomian polynomials, the procedure is very complex. In order to overcome the demerit, this paper suggests an alternative approach to Adomian method, instead of Adomian polynomials, He polynomials are introduced based on homotopy perturbation method. The solution procedure becomes easier, simpler, and more straightforward.     

A note on the use of Adomian decomposition method for high-order and system of nonlinear differential equations

This paper extends an earlier work [Hosseini MM, Nasabzadeh H. Modified Adomian decomposition method for specific second order ordinary differential equations. Appl Math Comput 2007;186:117–23] to high order and system of differential equations. Solution of these problems is considered by proposed modification of Adomian decomposition method. Furthermore, with providing some examples, the aforementioned cases are dealt with numerically.     

Adomian decomposition method with orthogonal polynomials: Legendre polynomials

This paper illustrates the using of orthogonal polynomials to modify the Adomian decomposition method. The method of employing Legendre polynomials to improve the Adomian decomposition method is presented here and compared to the method of using Chebyshev polynomials. The presented modified Adomian decomposition method is validated through an example and advantage as well as efficiency of this method is verified through investigating and comparing the results. In this paper, it is concluded that both orthogonal polynomials: Chebyshev and Legendre polynomials can be successfully used for the Adomian decomposition method and comparatively the Chebyshev expansion provides the better estimation.     

A note on the use of modified Adomian decomposition method for solving singular boundary value problems of higher-order ordinary differential equations

This paper extend the work [Yahya Qaid Hasan, Liu Ming Zhu. Solving singular boundary value problems of higher-order ordinary differential equations by modified Adomian decomposition method. Commun Nonlinear Sci Numer Simul. doi :10.1016/j.cnsns.2008.09.027] to high order of singular boundary value problems. Solution of these problems is considered by proposed modification of Adomian decomposition method. The proposed method can be applied to linear and nonlinear problems. Some examples are presented to show the ability of the method for linear and non-linear ordinary differential equation.     

The solution of linear and nonlinear systems of Volterra functional equations using Adomian–Pade technique

The purpose of this study is to implement Adomian–Pade (Modified Adomian–Pade) technique, which is a combination of Adomian decomposition method (Modified Adomian decomposition method) and Pade approximation, for solving linear and nonlinear systems of Volterra functional equations. The results obtained by using Adomian–Pade (Modified Adomian–Pade) technique, are compared to those obtained by using Adomian decomposition method (Modified Adomian decomposition method) alone. The numerical results, demonstrate that ADM–PADE (MADM–PADE) technique, gives the approximate solution with faster convergence rate and higher accuracy than using the standard ADM (MADM).     

Numerical solutions of the Laplace’s equation

This paper uses the sinc methods to construct a solution of the Laplace’s equation using two solutions of the heat equation. A numerical approximation is obtained with an exponential accuracy. We also present a reliable algorithm of Adomian decomposition method to construct a numerical solution of the Laplace’s equation in the form a rapidly convergence series and not at grid points. Numerical examples are given and comparisons are made to the sinc solution with the Adomian decomposition method. The comparison shows that the Adomian decomposition method is efficient and easy to use.     

A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method

In this paper, Adomian’s decomposition method is proposed to solve the well-known Blasius equation. Comparison with homotopy perturbation method and Howarth’s numerical solution reveals that the Adomian’s decomposition method is of high accuracy.     

解分数阶微分代数系统的Adomian分解方法

研究了利用Adomian分解求解分数阶微分代数系统的方法.分析了代数约束对Adomian方法求解的影响,指出直接解出代数约束变量,将原系统转化为微分系统进行Adomian分解的困难.提出确定代数变量级数解各分量的新方法,据此进行Adomian分解,得到整个系统的级数解.特别研究了代数约束为线性的分数阶微分代数系统的Adomian解法,证明了各变量间的线性代数约束关系可以转化为相应级数解中各分量的线性关系,从而方便求解,并结合具体例子证明了该方法简便有效.     

Use of Adomian decomposition method in the study of parallel plate flow of a third grade fluid

In this paper, Adomian’s decomposition method is used to solve non-linear differential equations which arise in fluid dynamics. We study basic flow problems of a third grade non-Newtonian fluid between two parallel plates separated by a finite distance. The technique of Adomian decomposition is successfully applied to study the problem of a non-Newtonian plane Couette flow, fully developed plane Poiseuille flow and plane Couette–Poiseuille flow. The results obtained show the reliability and efficiency of this analytical method. Numerical solutions are also obtained by solving non-linear ordinary differential equations using Chebyshev spectral method. We present a comparative study between the analytical solutions and numerical solutions. The analytical results are found to be in good agreement with numerical solutions which reveals the effectiveness and convenience of the Adomian decomposition method.     

Numerical approximations for population growth models

This paper aims to introduce a comparison of Adomian decomposition method and Sinc–Galerkin method for the solution of some mathematical population growth models. From the computational viewpoint, the comparison shows that the Adomian decomposition method is efficient and easy to use.     

Application of the Adomian decomposition method for the Fokker–Planck equation

In this work we will discuss the solution of an initial value problem of parabolic type. The main objective is to propose an alternative method of solution, one not based on finite difference or finite element or spectral methods. The aim of the present paper is to investigate the application of the Adomian decomposition method for solving the Fokker–Planck equation and some similar equations. This method can successfully be applied to a large class of problems. The Adomian decomposition method needs less work in comparison with the traditional methods. This method decreases considerable volume of calculations. The decomposition procedure of Adomian will be obtained easily without linearizing the problem by implementing the decomposition method rather than the standard methods for the exact solutions. In this approach the solution is found in the form of a convergent series with easily computed components. In this work we are concerned with the application of the decomposition method for the linear and nonlinear Fokker–Planck equation. To give overview of methodology, we have presented several examples in one and two dimensional cases.     

A new improved Adomian decomposition method and its application to fractional differential equations

In this paper, a new improved Adomian decomposition method is proposed, which introduces a convergence-control parameter into the standard Adomian decomposition method and establishes a new iterative formula. The examples prove that the presented method is reliable, efficient, easy to implement from a computational viewpoint and can be employed to derive successfully analytical approximate solutions of fractional differential equations.     

A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell–Whitehead–Segel equation

In this paper, we will carry out a comparative study between the reduced differential transform method and the Adomian decomposition method. This is been achieved by handling the Newell–Whitehead–Segel equation. Two numerical examples have also been carried out to validate and demonstrate efficiency of the two methods. Furthermost, it is shown that the reduced differential transform method has an advantage over the Adomian decomposition method that it takes less time to solve the nonlinear problems without using the Adomian polynomials.     

Adomian decomposition method is a special case of Lyapunov’s artificial small parameter method

The Adomian decomposition method and Lyapunov’s artificial small parameter method are two popular analytic methods for solving nonlinear differential equations. In this short paper, we show that the Adomian decomposition method can be derived from the more general Lyapunov’s artificial small parameter method.     

Application of Laplace decomposition method on semi-infinite domain

In this article, Laplace decomposition method (LDM) is applied to obtain series solutions of classical Blasius equation. The technique is based on the application of Laplace transform to nonlinear Blasius flow equation. The nonlinear term can easily be handled with the help of Adomian polynomials. The results of the present technique have closed agreement with series solutions obtained with the help of Adomian decomposition method (ADM), variational iterative method (VIM) and homotopy perturbation method (HPM).     

Iterative methods of order four and five for systems of nonlinear equations

The Adomian decomposition is used in order to obtain a family of methods to solve systems of nonlinear equations. The order of convergence of these methods is proved to be p≥2, under the same conditions as the classical Newton method. Also, numerical examples will confirm the theoretical results.     

The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method

In this paper, a scheme is developed to study numerical solution of the space- and time-fractional Burgers equations with initial conditions by the variational iteration method (VIM). The exact and numerical solutions obtained by the variational iteration method are compared with that obtained by Adomian decomposition method (ADM). The results show that the variational iteration method is much easier, more convenient, and more stable and efficient than Adomian decomposition method. Numerical solutions are calculated for the fractional Burgers equation to show the nature of solution as the fractional derivative parameter is changed.     

Adomian decomposition and Padé approximate for solving differential-difference equation

In this paper, we apply the Adomian decomposition method and Padé-approximate to solving the differential-difference equations (DDEs) for the first time. A simple but typical example is used to illustrate the validity and the great potential of the Adomian decomposition method (ADM) in solving DDEs. Comparisons are made between the results of the proposed method and exact solutions. The results show that ADM is an attractive method in solving the differential-difference equations.     

Comparison of the Adomian decomposition method with homotopy perturbation method for the solutions of seventh order boundary value problems

The aim of this paper is to compare the Adomian decomposition method and the homotopy perturbation method for solving the linear and nonlinear seventh order boundary value problems. The approximate solutions of the problems obtained with a small amount of computation in both methods. Two numerical examples have been considered to illustrate the accuracy and implementation of the methods.     

Solution of non-linear Klein–Gordon equation with a quadratic non-linear term by Adomian decomposition method

Non-linear PDEs are systematically solved by the decomposition method of Adomian for general boundary conditions described by boundary operator equations. In the present case the solution of the non-linear Klein–Gordon equation has been considered as an illustration of the decomposition method of Adomian.