On the exact and numerical solution of the time-delayed Burgers equation

The time-delayed Burgers equation is introduced and the improved tanh-function method is used to construct exact multiple soliton and triangular periodic solutions. For an understanding of the nature of the exact solutions that contained the time-delay parameter, we calculated the numerical solutions of this equation by using the Adomian decomposition method to the boundary value problem.     

Adomian拆分法求解胀塑性流体边界层方程近似解

利用Adomian拆分方法及Mathematica软件的符号计算功能求解了一类胀塑性非牛顿流体边界层方程的近似解,对不同的参数给出了壁摩擦力的近似值及解的分布曲线,数值结果表明所得近似解有相当高的精度。     

求解任意波数的三维Helmholtz方程

提出通过Adomian分解法求解任意波数的三维Helmholtz方程。通过Adomian分解法可以把三维Helmholtz微分方程转换成递归代数公式,并进一步把其边界条件转换成适用符号计算的简单代数公式。利用边界条件可以很容易得到方程的解析解表达式。Adomian分解法的主要特点在于计算简单快速,并且不需要进行线性化或离散化。最后通过数值计算以验证Adomian分解法求解任意波数下三维Helmholtz方程的有效性。数值计算结果表明：Adomian分解法的计算结果非常接近精确解,并且该方法在大波数情况下还具有良好的收敛性。     

Determination of a time-dependent parameter in a one-dimensional quasi-linear parabolic equation with temperature overspecification

In this work, the Adomian decomposition method is used to find the solution to a one-dimensional quasi-linear parabolic partial differential equation with a time-dependent unknown function. A wide class of physical phenomena is modelled by non-classical parabolic initial-boundary value problems. Thus the theoretical behaviour and numerical approximation of these problems have been active areas of research. The decomposition procedure first proposed by the American mathematician G. Adomian (1923–1996) is useful for obtaining both exact solutions to, and numerical approximations of, various kinds of linear and nonlinear problem. The Adomian decomposition method, which accurately computes the series solution, is of great interest in science and engineering. It provides a solution as a convergent series with components that can be elegantly computed. Sufficient conditions for convergence and stability of the approximate solution are given and the results of numerical experiments are presented.     

ADM机械化求解DLFFC多孔阳极理论数模

提出低温液相直接进料燃料电池的多孔阳极普遍化理论数学模型,为定量分析多孔阳极的电极极化提供理论依据。该理论效模为一非线性二阶常微分方程边值问题,用以描述多孔电极中过电位和浓度的分布。文中采用ADM机械化的方法得出逼近解析解,并与基于有限差分程序BAND(J)的数值解作了比较。计算结果显示使用前4-6项ADM逼近,解析解即可达到满意的精度。可以看出,作者提出的ADM机械化方法．避免了复杂的推导过程,是求解该类数模的一个简便、快速的有效工具。     

The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure

In this article, we propose a new approach for solving an initial–boundary value problem with a non-classic condition for the one-dimensional wave equation. Our approach depends mainly on Adomian's technique. We will deal here with new type of nonlocal boundary value problems that are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The decomposition method of G. Adomian can be an effective scheme to obtain the analytical and approximate solutions. This new approach provides immediate and visible symbolic terms of analytic solution as well as numerical approximate solution to both linear and nonlinear problems without linearization. The Adomian's method establishes symbolic and approximate solutions by using the decomposition procedure. This technique is useful for obtaining both analytical and numerical approximations of linear and nonlinear differential equations and it is also quite straightforward to write computer code. In comparison to traditional procedures, the series-based technique of the Adomian decomposition technique is shown to evaluate solutions accurately and efficiently. The method is very reliable and effective that provides the solution in terms of rapid convergent series. Several examples are tested to support our study.     

A convergence analysis of the Adomian decomposition method for an abstract Cauchy problem of a system of first-order nonlinear differential equations

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.     

The Decomposition Method For Solving Of A Class Of Singular Two-Point Boundary Value Problems

In this paper, a class of nonlinear singular two-point boundary value problems are solved by using Adomian's decomposition methods. The approximate solution of this problem is calculated in the form of series with easily computable components. Finally, we give a nonlinear numerical example.     

Solutions of the Cahn–Hilliard equation

In this paper, we found some exact solutions of the Cahn–Hilliard equation and the system of the equations by considering a modified extended tanh function method. A numerical solution to a Cahn–Hilliard equation is obtained using a homotopy perturbation method (HPM) combined with the Adomian decomposition method (ADM). The comparisons are given in the tables.     

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.     

Approximate analytical solutions of systems of PDEs by homotopy analysis method

In this paper, the homotopy analysis method (HAM) is applied to obtain series solutions to linear and nonlinear systems of first- and second-order partial differential equations (PDEs). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. It is shown in particular that the solutions obtained by the variational iteration method (VIM) are only special cases of the HAM solutions.     

An efficient numerical technique for the solution of nonlinear singular boundary value problems

In this work, a new technique based on Green’s function and the Adomian decomposition method (ADM) for solving nonlinear singular boundary value problems (SBVPs) is proposed. The technique relies on constructing Green’s function before establishing the recursive scheme for the solution components. In contrast to the existing recursive schemes based on the ADM, the proposed technique avoids solving a sequence of transcendental equations for the undetermined coefficients. It approximates the solution in the form of a series with easily computable components. Additionally, the convergence analysis and the error estimate of the proposed method are supplemented. The reliability and efficiency of the proposed method are demonstrated by several numerical examples. The numerical results reveal that the proposed method is very efficient and accurate.     

Adomian decomposition method by Gegenbauer and Jacobi polynomials

In this paper, orthogonal polynomials on [–1,1] interval are used to modify the Adomian decomposition method (ADM). Gegenbauer and Jacobi polynomials are employed to improve the ADM and compared with the method of using Chebyshev and Legendre polynomials. To show the efficiency of the developed method, some linear and nonlinear examples are solved by the proposed method, results are compared with other modifications of the ADM and the exact solutions of the problems.     

Taylor series for the Adomian decomposition method

In this paper, we analyse the exact solutions to scalar partial differential equations obtained, thanks to the summable Taylor series provided by Adomian's decomposition method. We propose a modification of the method which makes the calculations of Taylor coefficients easier and more direct. The difference is essential, for instance, in the case of non-homogenous equations or initial conditions and is illustrated by some examples.     

一类分数阶非线性振子方程的特性研究

将分数阶微分算子引入到黏弹性介质中的阻尼振动中建立分数阶非线性振动方程。利用Adomian分解方法借助Mathematica软件的符号计算功能求解了该类分数阶阻尼振动方程的近似解,研究了振子运动与方程中分数阶导数的关系。     

Solving nonlinear Volterra–Fredholm integro-differential equations using He's variational iteration method

In this paper, a nonlinear Volterra–Fredholm integro-differential equation is solved by using He's variational iteration method. The approximate solution of this equation is calculated in the form of a sequence where its components are computed easily. The accuracy of the proposed numerical scheme is examined by comparing with the modified Adomian decomposition method. The existence and uniqueness of the solution and convergence of the proposed method are proved.     

Iterative methods improving newton's method by the decomposition method

In this paper, we present a sequence of iterative methods improving Newton's method for solving nonlinear equations. The Adomian decomposition method is applied to an equivalent coupled system to construct the sequence of the methods whose order of convergence increases as it progresses. The orders of convergence are derived analytically, and then rederived by applying symbolic computation of Maple. Some numerical illustrations are given.     

Construction of solitary wave solutions for variants of the K(n,n) equation

In this paper, variational iteration method and Adomian decomposition are implemented to construct solitary solutions for variants of K(n, n) equation. In these schemes the solution takes the form of a convergent series with easily computable components. The chosen initial solution or trial function plays a major role in changing the physical structure of the solution. Comparison between the two methods is made and many models are approached. The obtained results reveal that the two methods are very effective and convenient for constructing solitary solutions.     

On a boundedness-preserving semi-linear discretization of a two-dimensional nonlinear diffusion–reaction model

Departing from a method to approximate the solutions of a two-dimensional generalization of the well-known Fisher's equation from population dynamics, we extend this computational technique to calculate the solutions of a FitzHugh–Nagumo model and derive conditions under which its positive and bounded analytic solutions are estimated consistently by positive and bounded numerical approximations. The constraints are relatively flexible, and they are provided exclusively in terms of the model coefficients and the computational parameters. The proofs are established with the help of the theory of M-matrices, using the facts that such matrices are non-singular, and that the entries of their inverses are positive numbers. Some numerical experiments are performed in order to show that our method is capable of preserving the positivity and the boundedness of the numerical solutions. The simulations evince a good agreement between the numerical estimations and the corresponding exact solutions derived in this work.