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).     

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.     

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 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.     

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.     

A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations

In this paper we propose a new modified recursion scheme for the resolution of multi-order and multi-point boundary value problems for nonlinear ordinary and partial differential equations by the Adomian decomposition method (ADM). Our new approach, including Duan’s convergence parameter, provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems, in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series. We utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components. Thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components, whereas several prior recursion schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots, which is required to complete calculation of the solution by several prior modified recursion schemes using the ADM.     

Two-level primal–dual proximal decomposition technique to solve large scale optimization problems

We describe a primal–dual application of the proximal point algorithm to nonconvex minimization problems. Motivated by the work of Spingarn and more recently by the work of Hamdi et al. about the primal resource-directive decomposition scheme to solve nonlinear separable problems. This paper discusses some local results of a primal–dual regularization approach that leads to a decomposition algorithm.     

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.     

A new modified Adomian decomposition method and its multistage form for solving nonlinear boundary value problems with Robin boundary conditions

In this paper we propose a new modified recursion scheme for the resolution of boundary value problems (BVPs) for second-order nonlinear ordinary differential equations with Robin boundary conditions by the Adomian decomposition method (ADM). Our modified recursion scheme does not incorporate any undetermined coefficients. We also develop the multistage ADM for BVPs encompassing more general boundary conditions, including Neumann boundary conditions.     

Numerical solution of singular boundary value problems using Green’s function and improved decomposition method

In this work, an effective technique for solving a class of singular two point boundary value problems is proposed. This technique is based on the Adomian decomposition method (ADM) and Green’s function. 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 ADM, the proposed recursive scheme avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients. The approximate solution is obtained in the form of series with easily calculable components. For the completeness, the convergence and error analysis of the proposed scheme is supplemented. Moreover, the numerical examples are included to demonstrate the accuracy, applicability, and generality of the proposed scheme. The results reveal that the method is very effective, straightforward, and simple.