Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method

We use Adomian decomposition method for solving the fractional nonlinear two-point boundary value problem

where D is Caputo fractional derivative, c is a constant, μ > 0, and F:[0,1]×[0,∞)→[0,∞) a continuous function. The fractional Bratu problem is solved as an illustrative example.     

Approximate analytic solutions for a two-dimensional mathematicalmodel of a packed-bed electrode using the Adomian decomposition method

In this paper we solve a system of second-order partial differential equations with non-standard boundary conditions. This system of equations is a mathematical model that describes distributions of the overpotential and reactant concentration in a working packed-bed electrode for an electrochemical reactor. To ensure the existence and uniqueness of the solutions for this model we choose standard instead of non-standard boundary conditions, and obtain approximate analytic solutions in the form of a series that rapidly converges using the Adomian decomposition method. The method is easily implemented using the symbol operations of scientific computational software such as MATLAB.     

Adomian decomposition method for nonlinear differential-difference equations

We extend Adomian decomposition method (ADM) to find the approximate solutions for the nonlinear differential-difference equations (NDDEs), such as the discretized mKdV lattice equation, the discretized nonlinear Schrödinger equation and the Toda lattice equation. By comparing the approximate solutions with the exact analytical solutions, we find the extend method for NDDEs is of good accuracy.     

Approximate analytical solutions for a mathematical model of a tubular packed-bed catalytic reactor using an Adomian decomposition method

In this paper, a mathematical model of a tubular packed-bed catalytic reactor, which is modeled by a system of strongly nonlinear second-order partial differential equations with incompatible boundary conditions, will be solved. By properly using the boundary conditions and correctly choosing the solution search direction, approximate analytic solutions for the model can be obtained by the Adomian decomposition method. When the values of the dimensionless parameters in the system are assigned within a suitable range, the solutions describe objectively the distributions of the temperature and key reactant concentration in the reactor.     

Solution of a system of nonlinear equations by Adomian decomposition method

In this paper, the Adomian decomposition method is applied to solve a system of nonlinear equations. Convergence of the method is proved and some examples are presented to illustrate the method.     

Adomian decomposition method used for solving nonlinear pull-in behavior in electrostatic micro-actuators

Nonlinear pull-in behavior for different electrostatic micro-actuators were simulated in this study. The Adomian decomposition method was employed to overcome the difficulty in the nonlinear equation of motion. Because no iteration is required in solving the nonlinear deformation, the decomposition method is one of the most efficient methods for evaluating the unstable pull-in behavior of an electrostatic micro-actuator. To investigate the feasibility of applying the Adomian decomposition method in dealing with the nonlinear deflection equation in the micro-actuator problem, different types of micro-actuators, e.g., fixed-fixed beam actuator and cantilever beam actuator were studied and analyzed. The calculated results agreed well with those from the literature.     

Free vibration analysis of stepped beams by using Adomian decomposition method

The Adomian decomposition method (ADM) is employed in this paper to investigate the free vibrations of a stepped Euler-Bernoulli beam consisting of two uniform sections. Each section is considered a substructure which can be modeled using ADM. By using boundary condition and continuity condition equations, the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The computed results for different boundary conditions, step ratios and step locations are presented. Comparing the results using ADM to those given in the literature, excellent agreement is achieved.     

Accuracy of the Adomian decomposition method applied to the Lorenz system

In this paper, the Adomian decomposition method (ADM) is applied to the famous Lorenz system. The ADM yields an analytical solution in terms of a rapidly convergent infinite power series with easily computable terms. Comparisons between the decomposition solutions and the fourth-order Runge–Kutta (RK4) numerical solutions are made for various time steps. In particular we look at the accuracy of the ADM as the Lorenz system changes from a non-chaotic system to a chaotic one.     

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.     

Solving the generalized Burgers–Huxley equation using the Adomian decomposition method

In this paper, a convergence proof of the Adomian decomposition method (ADM) applied to the generalized nonlinear Burgers–Huxley equation is presented. The decomposition scheme obtained from the ADM yields an analytical solution in the form of a rapidly convergent series. The direct symbolic–numeric scheme is shown to be efficient and accurate.     

Revisit on partial solutions in the Adomian decomposition method: Solving heat and wave equations

The Adomian decomposition method is considered in application to heat and wave equations. The so-called partial solution technique is used. It is shown that the fundamental equation of the method is well defined only for certain types of boundary conditions. In cases involving inhomogeneous boundary conditions, improper results may be obtained by former method. This paper presents a further insight into partial solutions in the decomposition method, and the resolution of such cases.     

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.     

A convenient technique for solving linear and nonlinear Abel integral equations by the Adomian decomposition method

In this paper, linear and nonlinear Abel integral equations are transformed in such a manner that the Adomian decomposition method can be applied. Some examples with closed-form solutions are studied in detail to further illustrate the proposed technique, and the results obtained indicate this approach is indeed practical and efficient.     

利用Adomain分解法求时间分数阶薛定谔方程的近似解

非线性薛定谔方程是现代科学中非常普遍的非线性模型之一。通过 Adomain分解,得到了（2＋1）维和（3＋1）维非零势阱时间分数阶薛定谔方程的近似解。利用Adomain 分解不用像相关文献中那样将解函数的实部和虚部分别去求解,从而简化了求解过程。     

Adomian decomposition method for solution of differential-algebraic equations

Solutions of differential algebraic equations is considered by Adomian decomposition method. In E. Babolian, M.M. Hosseini [Reducing index and spectral methods for differential-algebraic equations, J. Appl. Math. Comput. 140 (2003) 77] and M.M. Hosseini [An index reduction method for linear Hessenberg systems, J. Appl. Math. Comput., in press], an efficient technique to reduce index of semi-explicit differential algebraic equations has been presented. In this paper, Adomian decomposition method is applied to reduced index problems. The scheme is tested for some examples and the results demonstrate reliability and efficiency of the proposed methods.     

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.     

A comparison between the variational iteration method and Adomian decomposition method

In this paper, we present a comparative study between the variational iteration method and Adomian decomposition method. The study outlines the significant features of the two methods. The analysis will be illustrated by investigating the homogeneous and the nonhomogeneous advection problems.     

Piecewise finite series solutions of seasonal diseases models using multistage Adomian method

The aim of this paper is to apply the multistage Adomian Decomposition Method MADM to solve systems of nonautonomous nonlinear differential equations that describe several epidemic models with periodic behavior. Here the concept of the MADM is introduced and then it is employed to obtain a piecewise finite series solution. The MADM is used here as a hybrid analytical–numerical technique for approximating the solutions of the epidemic models. In order to show the efficiency of the method, the obtained numerical results are compared with the fourth-order Runge–Kutta method solutions. Numerical comparisons show that the MADM is accurate, easy to apply and the calculated solutions preserve the periodic behavior of the continuous models. Moreover, the method has the advantage of giving a functional form of the solution for any time interval. Furthermore, it is shown that if the truncation order and the time step size are not properly chosen large computational work is required and inaccurate solutions may be obtained.