Further accuracy tests on Adomian decomposition method for chaotic systems

The Adomian decomposition method (ADM) is treated as an algorithm for approximating the solutions of the Lorenz and Chen systems in a sequence of time intervals, i.e. the classical ADM is converted into a hybrid analytical–numerical method. Comparisons with the seventh- and eighth-order Runge–Kutta method (RK78) reconfirm the very high accuracy of the hybrid analytical–numerical ADM.     

New technique to avoid “noise terms” on the solutions of inhomogeneous differential equations by using Adomian decomposition method

In this paper we present a new technique to get the solutions of inhomogeneous differential equations using Adomian decomposition method (ADM) without noise terms. We construct an appropriate differential equations for the inhomogeneity function which must be contains the integral variable, and convert all of these differential equations (original differential equation and the constructed differential equations) to augmented system of first-order differential equations. The ADM is using to solve the augmented system and the initial conditions are taken as initial approximations. Generally, the closed form of the exact solution or its expansion is obtained without any noise terms. Several differential equations will be tested to confirm the newly developed technique.     

Solving delay differential systems with history functions by the Adomian decomposition method

A number of nonlinear phenomena in many branches of the applied sciences and engineering are described in terms of delay differential equations, which arise when the evolution of a system depends both on its present and past time. In this work we apply the Adomian decomposition method (ADM) to obtain solutions of several delay differential equations subject to history functions and then investigate several numerical examples via our subroutines in MAPLE that demonstrate the efficiency of our new approach. In our approach history functions are continuous across the initial value and its derivatives must be equal to the initial conditions (see Section 3) so that our results are more efficient and accurate than previous works.     

Construction of solutions for mixed hyperbolic elliptic Riemann initial value system of conservation laws

In this paper, we apply Adomian decomposition method (ADM) to develop a fast and accurate algorithm for systems of conservation laws of mixed hyperbolic elliptic type. The solutions of our model equations are calculated in the form of convergent power series with easily computable components. The results obtained are compared with our Modification of Adomian decomposition method (MADM) Az-Zo’bi and Al-Khaled (2010) [1]. The study outlines the significant features of the two methods. With application to van der Waals system, we obtain the stability of the approximate solutions graphically when the system changes type with more efficiency of the MADM.     

Solutions of Emden–Fowler equations by homotopy-perturbation method

In this paper, approximate and/or exact analytical solutions of the generalized Emden–Fowler type equations in the second-order ordinary differential equations (ODEs) are obtained by homotopy-perturbation method (HPM). The homotopy-perturbation method (HPM) is a coupling of the perturbation method and the homotopy method. The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve. In this work, HPM yields solutions in convergent series forms with easily computable terms, and in some cases, only one iteration leads to the high accuracy of the solutions. Comparisons with the exact solutions and the solutions obtained by the Adomian decomposition method (ADM) show the efficiency of HPM in solving equations with singularity.     

Application of semi‐analytic methods for the Fitzhugh–Nagumo equation,which models the transmission of nerve impulses

In this work, the homotopy perturbation method (HPM), the variational iteration method (VIM) and the Adomian decomposition method (ADM) are applied to solve the Fitzhugh–Nagumo equation. Numerical solutions obtained by these methods when compared with the exact solutions reveal that the obtained solutions produce high accurate results. The results show that the HPM, the VIM and the ADM are of high accuracy and are efficient for solving the Fitzhugh–Nagumo equation. Also the results demonstrate that the introduced methods are powerful tools for solving the nonlinear partial differential equations. Copyright © 2010 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.     

Determination of the Correct Range of Physical Parameters in the Approximate Analytical Solutions of Nonlinear Equations Using the Adomian Decomposition Method

Physical parameters in dimensionless form in the governing equations of real-life phenomena naturally occur. How to control them by determining their range of validity is in general a big issue. In this paper, a mathematical approach is presented to identify the correct range of physical parameters adopting the recently popular analytic approximate Adomian decomposition method (ADM). Having found the approximate analytical Adomian series solution up to a specified truncation order, the squared residual error formula is employed to work out the threshold and the existence domain of certain physical parameters satisfying a preassigned tolerance. If the current procedure is not closely pursued, the presented results with the ADM may not be up to the desired level of accuracy (the worst is the divergent physically meaningless solutions), or much more ADM series terms need to be computed to satisfy certain accuracy. Examples reveal the necessity of the present approach to make sure that the results embark the correct range of physical parameters in the study of a physical problem containing several dominating parameters.     

A series solution of the fin problem with a temperature-dependent thermal conductivity

In this work, we apply the homotopy analysis method to solve fin problems with temperature-dependent thermal conductivity. The results are compared with the solutions obtained by the Adomian decomposition method (ADM), homotopy perturbation method (HPM) and with the Taylor series expansion method as well. The results of the comparisons have shown that the ADM and HPM are just an approximation of the present study and the accuracy of the solution obtained by the present study is much better than the latter algorithms.     

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 convergence proof of the Adomian decomposition method for a mixed hyperbolic elliptic system of conservation laws

In this paper, we propose a new convergence proof of the Adomian’s decomposition method (ADM), applied to the generalized nonlinear system of partial differential equations (PDE’s) based on new formula for Adomian polynomials. The decomposition scheme obtained from the ADM yields an analytical solution in the form of a rapidly convergent series for a system of conservation laws. Systems of conservation laws is presented, we obtain the stability of the approximate solution when the system changes type. We show with an explicit example that the latter property is true for general Cauchy problem satisfying convergence hypothesis. The results indicate that the ADM is effective and promising.     

ADM-Padé technique for the nonlinear lattice equations

ADM-Padé technique is a combination of Adomian decomposition method (ADM) and Padé approximants. We solve two nonlinear lattice equations using the technique which gives the approximate solution with higher accuracy and faster convergence rate than using ADM alone. Bell-shaped solitary solution of Belov–Chaltikian (BC) lattice and kink-shaped solitary solution of the nonlinear self-dual network equations (SDNEs) are presented. Comparisons are made between approximate solutions and exact solutions to illustrate the validity and the great potential of the technique.     

Analytical approximate solution of the cooling problem by Adomian decomposition method

The Adomian decomposition method (ADM) can provide analytical approximation or approximated solution to a rather wide class of nonlinear (and stochastic) equations without linearization, perturbation, closure approximation, or discretization methods. In the present work, ADM is employed to solve the momentum and energy equations for laminar boundary layer flow over flat plate at zero incidences with neglecting the frictional heating. A trial and error strategy has been used to obtain the constant coefficient in the approximated solution. ADM provides an analytical solution in the form of an infinite power series. The effect of Adomian polynomial terms is considered and shows that the accuracy of results is increased with the increasing of Adomian polynomial terms. The velocity and thermal profiles on the boundary layer are calculated. Also the effect of the Prandtl number on the thermal boundary layer is obtained. Results show ADM can solve the nonlinear differential equations with negligible error compared to the exact solution.     

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).     

A new analytical method for system of ODEs

In this article, a new homotopy perturbation method (NHPM) is introduced to obtain exact solutions of system of ODEs. Theoretical considerations are discussed. Comparison of the results of applying NHPM with those of homotopy perturbation method (HPM) and Adomia's decomposition method (ADM) leads to significant consequences. Three examples, including stiff system of linear and nonlinear ODEs are given to demonstrate the efficiency of the new method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Application of homotopy-perturbation method to Klein–Gordon and sine-Gordon equations

In this paper, the homotopy-perturbation method (HPM) is employed to obtain approximate analytical solutions of the Klein–Gordon and sine-Gordon equations. An efficient way of choosing the initial approximation is presented. Comparisons with the exact solutions, the solutions obtained by the Adomian decomposition method (ADM) and the variational iteration method (VIM) show the potential of HPM in solving nonlinear partial differential equations.     

An efficient method for analyzing the solutions of the Korteweg-de Vries equation

A new iterative method is applied to study the solutions of the Korteweg-de Vries (KdV) equation. The method is a modified form of the well known Adomian decomposition method (ADM), where it avoids the difficulty of computing the Adomian polynomials. We prove the existence of a unique solution of the KdV equation. And then, we show that the new method generates an infinite series which converges uniformly to the exact solution of the problem. Soliton solutions of the KdV equation are obtained by the new method. Numerical calculations indicate the effectiveness of the new method where the obtained results are very accurate and better than the ones obtained by the ADM.     

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.     

Homotopy analysis method for singular IVPs of Emden–Fowler type

In this paper, approximate and/or exact analytical solutions of singular initial value problems (IVPs) of the Emden–Fowler type in the second-order ordinary differential equations (ODEs) are obtained by the homotopy analysis method (HAM). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions. It is shown that the solutions obtained by the Adomian decomposition method (ADM) and the homotopy-perturbation method (HPM) are only special cases of the HAM solutions.     

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.