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.     

Approximate solutions to the conformable Rosenau-Hyman equation using the two-step Adomian decomposition method with Padé approximation

This paper adopts the Adomian decomposition method and the Padé approximation techniques to derive the approximate solutions of a conformable Rosenau-Hyman equation by considering the new definition of the Adomian polynomials. The Padé approximate solutions are derived along with interesting figures showing both the analytic and approximate solutions.     

A New Efficient Transform Mechanism with Convergence Analysis of the Space-Fractional Telegraph Equations

This article"s goal is to investigate the space-fractional telegraph equation using an effective method called the Adomian natural decomposition method (ANDM), which is a combination of the Adomian decomposition method (ADM) and the natural transform method (NTM). Using the Banach fixed point theorem, we explore proofs for the existence and uniqueness theorems applying it to a nonlinear differential equation. Using our method, exact solutions of the space-fractional telegraph equation and time-fractional diffusion problems have been obtained. To demonstrate the effectiveness of the suggested scheme, four examples are provided.     

On comparison of series and numerical solutions for second Painlevé equation

This attempt presents the series solution of second Painlevé equation by homotopy analysis method (HAM). Comparison of HAM solution is provided with that of the Adomian decomposition method (ADM), homotopy perturbation method (HPM), analytic continuation method, and Legendre Tau method. It is revealed that there is very good agreement between the analytic continuation and HAM solutions when compared with ADM, HPM, and Legendre Tau solutions. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A study on the systems of the Volterra integral forms of the Lane–Emden equations by the Adomian decomposition method

In this paper, we introduce systems of Volterra integral forms of the Lane–Emden equations. We use the systematic Adomian decomposition method to handle these systems of integral forms. The Volterra integral forms overcome the singular behavior at the origin x = 0. The Adomian decomposition method gives reliable algorithm for analytic approximate solutions of these systems. Our results are supported by investigating several numerical examples. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

The homotopy analysis method for solving the time-fractional Fornberg–Whitham equation and comparison with Adomian’s decomposition method

In this paper, we applied relatively new analytical techniques, the homotopy analysis method (HAM) and the Adomian’s decomposition method (ADM) for solving time-fractional Fornberg–Whitham equation. The homotopy analysis method contains the auxiliary parameter, which provides us with a simple way to adjust and control the convergence region of solution series. The fractional derivatives are described in the Caputo sense. A comparison is made the between HAM and ADM results. The present methods performs extremely well in terms of efficiency and simplicity. Numerical results for different particular cases of the problem are presented.     

A segmented and weighted Adomian decomposition algorithm for boundary value problem of nonlinear groundwater equation

Based on Adomian decomposition method, a new algorithm for solving boundary value problem (BVP) of nonlinear partial differential equations on the rectangular area is proposed. The solutions obtained by the method precisely satisfy all boundary conditions, except the small pieces near the four corners of the rectangular area. A theorem on the boundary error is given. Hence, the Adomian decomposition method is more efficiently applied to BVPs for partial differential equations. Segmented and weighted analytical solutions with a high accuracy for the BVP of nonlinear groundwater equations on a rectangular area are obtained by the present algorithm. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Numerical methods for nonlinear partial differential equations of fractional order

In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when applied to partial differential equations of fractional order.     

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.     

Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition

Adomian decomposition method has been used to obtain solutions of linear/nonlinear fractional diffusion and wave equations. Some illustrative examples have been presented.     

A MAPLE package of new ADM-Padé approximate solution for nonlinear problems

Based on the new definition of four classes of Adomian polynomials proposed by Rach in 2008, a MAPLE package of new Adomian-Padé approximate solution for solving nonlinear problems is presented. This package combines the merits of the Adomian decomposition method and the diagonal Padé technique, and may give more accurate solutions of nonlinear problems with strong nonlinearity. Besides, the package is user-friendly and efficient, one only needs to input the initial conditions, governing equation and four optional parameters, then our package will output the analytic approximate solution within a few seconds, where the equation is decomposed into three parts, they are the linear term R, nonlinear term NN and source function g, which are all in functional form. Meanwhile, several graphs generated from the above solutions are displayed and demonstrate a favorable comparison. In this paper, several different types of examples are given to illustrate the validity and promising flexibility of the package. This package provides us with a convenient and useful tool for dealing with nonlinear problems, as well as its electronic version is free to download via the journal website.     

An efficient algorithm for the multivariable Adomian polynomials

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 u₁, u₂, … , 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.     

The numerical solution of the second Painlevé equation

The Painlevé equations were discovered by Painlevé, Gambier and their colleagues during studying a nonlinear second‐order ordinary differential equation. The six equations which bear Painlevé's name are irreducible in the sense that their general solutions cannot be expressed in terms of known functions. Painlevé has derived these equations on the sole requirement that their solutions should be free from movable singularities. Many situations in mathematical physics reduce ultimately to Painlevé equations: applications including statistical mechanics, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics, and fiber optics. This fact has caused a significant interest to the study of these equations in recent years. In this study, the solution of the second Painlevé equation is investigated by means of Adomian decomposition method, homotopy perturbation method, and Legendre tau method. Then a numerical evaluation and comparison with the results obtained by the method of continuous analytic continuation are included. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Fractional green function for linear time-fractional inhomogeneous partial differential equations in fluid mechanics

This paper deals with the solutions of linear inhomogeneous time-fractional partial differential equations in applied mathematics and fluid mechanics. The fractional derivatives are described in the Caputo sense. The fractional Green function method is used to obtain solutions for time-fractional wave equation, linearized time-fractional Burgers equation, and linear time-fractional KdV equation. The new approach introduces a promising tool for solving fractional partial differential equations.     

Iterative methods for solving fourth- and sixth-order time-fractional Cahn-Hillard equation

This paper presents analytical-approximate solutions of the time-fractional Cahn-Hilliard (TFCH) equations of fourth and sixth order using the new iterative method (NIM) and q-homotopy analysis method (q-HAM). We obtained convergent series solutions using these two iterative methods. The simplicity and accuracy of these methods in solving strongly nonlinear fractional differential equations is displayed through the examples provided. In the case where exact solution exists, error estimates are also investigated.     

Mathematical studies of the solution of Burgers' equations by Adomian decomposition method

In this paper, a novel Adomian decomposition method (ADM) is developed for the solution of Burgers' equation. While high level of this method for differential equations are found in the literature, this work covers most of the necessary details required to apply ADM for partial differential equations. The present ADM has the capability to produce three different types of solutions, namely, explicit exact solution, analytic solution, and semi-analytic solution. In the best cases, when a closed-form solution exists, ADM is able to capture this exact solution, while most of the numerical methods can only provide an approximation solution. The proposed ADM is validated using different test cases dealing with inviscid and viscous Burgers' equations. Satisfactory results are obtained for all test cases, and, particularly, results reported in this paper agree well with those reported by other researchers.     

A new algorithm for solving differential equations

Different analytic methods have been proposed to solve differential equations, so far. In this paper, a novel analytic method that efficiently solves ODEs is presented. This method requires only the calculation of the first Adomian polynomial, namely A₀, and does not need to solve the functional equation in each iteration, as well as provides less computational work than other existing methods. Some important ordinary differential equations including the Lane–Emden equation of index m, the logistic nonlinear differential equation, and the Riccati equation are considered to illustrate the efficiency of the proposed algorithm. Copyright © 2012 John Wiley & Sons, Ltd.     

Comparing numerical methods for Helmholtz equation model problem

In this article, we implement a relatively new numerical technique, Adomian’s decomposition method for solving the linear Helmholtz partial differential equations. The method in applied mathematics can be an effective procedure to obtain for the analytic and approximate solutions. A new approach to a linear or nonlinear problems is particularly valuable as a tool for Scientists and Applied Mathematicians, because it provides immediate and visible symbolic terms of analytic solution as well as its numerical approximate solution to both linear and nonlinear problems without linearization [Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, 1994; J. Math. Anal. Appl. 35 (1988) 501]. It does also not require discretization and consequently massive computation. In this scheme the solution is performed in the form of a convergent power series with easily computable components. This paper will present a numerical comparison with the Adomian decomposition and a conventional finite-difference method. The numerical results demonstrate that the new method is quite accurate and readily implemented.