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.     

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).     

Solving System of Conformable Fractional Differential Equations by Conformable Double Laplace Decomposition Method

Herein, an approach known as conformable double Laplace decomposition method (CDLDM) is suggested for solving system of non-linear conformable fractional differential equations. The devised scheme is the combination of the conformable double Laplace transform method (CDLTM) and, the Adomian decomposition method (ADM). Obtained results from mathematical experiments are in full agreement with the results obtained by other methods. Furthermore, according to the results obtained we can conclude that the proposed method is efficient, reliable and easy to be implemented on related many problems in real-life science and engineering.     

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.     

A new analytical technique to solve Volterra's integral equations

The purpose of the present paper is to show that the well‐known homotopy analysis method for solving ordinary and partial differential equations can be applied to solve linear and nonlinear integral equations of Volterra's type with high accuracy as well. Comparison of the present method with Adomian decomposition method (ADM), a well‐known method to solve integral equations, reveals that the ADM is only especial case of the present method. Furthermore, some illustrating examples such as linear, nonlinear and singular integral equations of Volterra's type are given to show high efficiency with reliable accuracy of the method. Copyright © 2011 John Wiley & Sons, Ltd.     

Adomian solution of a nonlinear quadratic integral equation

We are concerned here with a nonlinear quadratic integral equation (QIE). The existence of a unique solution will be proved. Convergence analysis of Adomian decomposition method (ADM) applied to these type of equations is discussed. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of Adomian’s series solution. Two methods are used to solve these type of equations; ADM and repeated trapezoidal method. The obtained results are compared.     

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.     

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.     

A new analytical technique to solve Fredholm’s integral equations

This paper shows that the homotopy analysis method, the well-known method to solve ODEs and PDEs, can be applied as well as to solve linear and nonlinear integral equations with high accuracy. Comparison of the present method with Adomian decomposition method (ADM), which is well-known in solving integral equations, reveals that the ADM is only special case of the present method. Also, some linear and nonlinear examples are presented to show high efficiency and illustrate the steps of the problem resolution.     

Series Solutions of Systems of Nonlinear Fractional Differential Equations

Differential equations of fractional order appear in many applications in physics, chemistry and engineering. An effective and easy-to-use method for solving such equations is needed. In this paper, series solutions of the FDEs are presented using the homotopy analysis method (HAM). The HAM provides a convenient way of controlling the convergence region and rate of the series solution. It is confirmed that the HAM series solutions contain the Adomian decomposition method (ADM) solution as special cases.      

An improved variational iteration method for solving coupled KdV and Boussinesq-like B(m, n) equations

In this article, we implement a new analytical technique; He’s variational iteration method for solving the coupled KdV and Boussinesq-like equations. In this method, first general Lagrange multipliers are introduced to construct correction functional for the problems. The multipliers in the functional can be identified optimally via the variational theory. Next the components of obtained iteration formulae defined by partial sum of other sequence, specially constructed according to Adomian’s decomposition method (ADM). Also according to ADM we used a partial sum of Adomian polynomials instead of nonlinear terms in iteration formulae. The initial approximations can be freely chosen with possible unknown constants, which can be determined by imposing the initial conditions. The results reveal that the proposed method is very effective and can be applied for other nonlinear problems.     

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.     

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.     

Homotopy perturbation method for homogeneous Smoluchowsk's equation

In this work, homotopy perturbation method (HPM) has been used to solve homogeneous Smoluchowsk's equation. The results will be compared with Adomian decomposition method (ADM). It is shown that the results of the HPM are the same as those obtained by ADM. To illustrate the reliability of the method, some special cases of the equation have been solved as examples. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

The Adomian decomposition method with Green’s function for solving nonlinear singular boundary value problems

In this paper, we present an efficient numerical algorithm for solving a general class of nonlinear singular boundary value problems. This present algorithm is based on the Adomian decomposition method (ADM) and Green’s function. The method depends on constructing Green’s function before establishing the recursive scheme. In contrast to the existing recursive schemes based on ADM, the proposed numerical algorithm avoids solving a sequence of transcendental equations for the undetermined coefficients. The approximate series solution is calculated in the form of series with easily computable components. Moreover, the convergence analysis and error estimation of the proposed method is given. Furthermore, the numerical examples are included to demonstrate the accuracy, applicability, and generality of the proposed scheme. The numerical results reveal that the proposed method is very effective.     

A systematic method for solving differential-difference equations

A systematic method for searching travelling-wave solutions to differential-difference equations (DDEs) is proposed in the paper. First of all, we introduce Bäcklund transformations for the standard Riccati equation which generate new exact solutions by using its simple and known solutions. Then we introduce a kind of formal polynomial solutions to DDEs and further determine the explicit forms by applying the balance principle. Finally, we work out exact solutions of the DDEs via substituting the form solutions and solving over-determined algebraic equations with the help of Maple. As illustrative examples, we obtain the travelling-wave solutions of the (2 + 1)-dimensional Toda lattice equation, the discrete modified KdV (mKdV) equation, respectively.     

A novel method for nonlinear two-point boundary value problems: Combination of ADM and RKM

In this paper, a novel method is proposed for solving nonlinear two-point boundary value problems (BVPs). This method is based on a combination of the Adomian decomposition method (ADM) and the reproducing kernel method (RKM). A major advantage of this method over standard ADM is that it can avoid unnecessary computation in determining the unknown parameters. The proposed method can be applied to singular and nonsingular BVPs. Numerical results obtained using the scheme presented here show that the numerical scheme is very effective and convenient for solving nonlinear two-point boundary value problems.     

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.     

Symmetry groups of differential-difference equations and their compatibility

It is shown that the intrinsic determining equations of a given differential-difference equation (DDE) can be derived by the compatibility between the original equation and the intrinsic invariant surface condition. The (2+1)-dimensional Toda lattice, the special Toda lattice and the DD-KP equation serving as examples are used to illustrate this approach. Then, Bäcklund transformations of the (2+1)-dimensional DDEs including the special Toda lattice, the modified Toda lattice and the DD-KZ equation are presented by using the non-intrinsic direct method. In addition, the Clarkson-Kruskal direct method is developed to find similarity reductions of the DDEs.