An Efficient Maple Program for Calculating Adomian Polynomials

The immense quest for proficient numerical schemes for the solution of mathematical models featuring nonlinear differential equations led to the realization of the Adomian decomposition method (ADM) in the 80th. Undoubtedly, the solution of nonlinear differential equations using ADM is presided over by the acquisition of Adomian polynomials, which are not always easy to find. Thus, the present study proposes easy-to-implement Maple programs for the computation of Adomian polynomials. In fact, the proposed algorithms performed remarkably on several test functions, consisting of one- and multi-variable nonlinearities. Moreover, the introduced programs are advantageous in terms of simplicity;coupled with the requirement of less computational time in comparison with what is known in the literature.

The Degenerate Form of the Adomian Polynomials in the Power Series Method for Nonlinear Ordinary Differential Equations

In this paper, we propose a new variation of the Adomian polynomials, which we call the degenerate Adomian polynomials, for the power series solutions of nonlinear ordinary differential equations with nonseparable nonlinearities. We establish efficient algorithms for the degenerate Adomian polynomials. Next we compare the results by the Adomian decomposition method using the classic Adomian polynomials with the results by the Rach-Adomian-Meyers modified decomposition method incorporating the degenerate Adomian polynomials, which itself has been shown to be a confluence of the Adomian decomposition method and the power series method. Convergence acceleration techniques including the diagonal Pade approximants are considered, and new numeric algorithms for the multistage decomposition are deduced using the degenerate Adomian polynomials. Our new technique provides a significant advantage for automated calculations when computing the power series form of the solution for nonlinear ordinary differential equations. Several expository examples are investigated to demonstrate its reliability and efficiency.

Decomposition method for solving parabolic equations in finite domains

This paper presents a comparison among Adomian decomposition method (ADM), Wavelet-Galerkin method (WGM), the fully explicit (1,7) finite difference technique (FTCS), the fully implicit (7,1) finite difference method (BTCS), (7,7) Crank-Nicholson type finite difference formula (C-N), the fully explicit method (1,13) and 9-point finite difference method, for solving parabolic differential equations with arbitrary boundary conditions and based on weak form functionals in finite domains. The problem is solved rapidly, easily and elegantly by ADM. The numerical results on a 2D transient heat conducting problem and 3D diffusion problem are used to validate the proposed ADM as an effective numerical method for solving finite domain parabolic equations. The numerical results showed that our present method is less time consuming and is easier to use than other methods. In addition, we prove the convergence of this method when it is applied to the nonlinear parabolic equation.

Double Elzaki Transform Decomposition Method for Solving Third Order Korteweg-De-Vries Equations

In this study, we used Double Elzaki Transform (DET) coupled with Adomian polynomial to produce a new method to solve Third Order Korteweg-De Vries Equations (KdV) equations. We will provide the necessary explanation for this method with addition some examples to demonstrate the effectiveness of this method.

A reliable analysis of oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics

In this paper, we investigate the diffusion of oxygen in a spherical cell including nonlin- ear uptake kinetics. The Lane-Emden boundary value problem with Michaelis-Menten kinetics is used to model the dimensionless oxygen concentration in our analysis. We first convert the Lane-Emden equation to the equivalent Volterra integral form that incorporates the boundary condition at the cell＇s center, but which still leaves one unknown constant of integration, as an intermediate step. Next we evaluate the Volterra integral form of the concentration and its flux at the cell membrane and substitute them into the remaining boundary condition to determine the unknown constant of integration by appropriate algebraic manipulations. Upon substitution we have converted the equivalent Volterra integral form to the equivalent Fredholm Volterra integral form, and use the Duan Rach modified recursion scheme to effectively decompose the unknown constant of integration by formula. The Adomian decomposition method is then applied to solve the equivalent nonlinear Fredholm-Volterra integral representation of the LaneEmden model for the concentration of oxygen within the spherical cell. Our approach shows enhancements over existing techniques.