An Explicit Analytic Solution to the Thomas-Fermi Equation by the Improved Differential Transform Method

Abstract

In this paper, a newly proposed analytical scheme by the authors namely the improved dierential transform method is employed to provide an explicit series solution to the ThomasFermi equation. The solution procedure is very straightforward, requiring merely elementary operations together with dierentiation, and ends up in a recursive formula involving the Adomian polynomials to aord the unknown coecients. Unlike many other methods, our approach is free of integration and hence can be of computational interest. In addition, a very good agreement between the proposed solution and the results from several well-known works in the literature is demonstrated.

Numerical computation and nonlinear dynamic analysis of ultrasonic cutting system

“…(6)(7)(8)(9), and also can be applied other numerical methods such as Adomian decomposition method (ADM), the variational iteration method (VIM) and the homotopy perturbation method (HPM) [22][23][24][25][26][27][28][29][30] to calculate the governing equations.…”

Solution of nonlinear higher-index Hessenberg DAEs by Adomian polynomials and differential transform method

“…In this section, we briefly review the Adomian polynomials and their relation with the DTM. Usually a nonlinear term N ( u ) in a differential equation is decomposed in terms of Adomian polynomials (Rach 2008 , 1984 ; Wazwaz 2000 ; Duan 2010a , b , 2011 ) as where are generated for all forms of nonlinearity from and where denote the components used in the expansion There are several algorithms to compute Adomian polynomials but recently a convenient recursion to calculate Adomian polynomials for the m -variable case is proposed in (Duan 2011 ) Also an extension of the differential transform to nonlinear terms of any type, known as the improved DTM, was given in (Fatoorehchi and Abolghasemi 2013a , 2014b ) using Adomian polynomials where …”

“…The ADM, for example, was used in computing solutions of algebraic equations (Adomian and Rach 1985 ; Fatoorehchi et al. 2014a , b , 2015 ; Fatoorehchi and Abolghasemi 2014a , b ; Fatoorehchi et al. 2015b , d , c ).…”

The homogeneous balance of undetermined coefficients method and its application

“…Seeking the exact solutions of nonlinear PDEs has long been an interesting topic in the nonlinear mathematical physics. With the development of soliton theory, various methods for obtaining the exact solutions of nonlinear PDEs have been presented, such as the inverse scattering method [1], the Bäcklund and Darboux transformation method [2], the homotopy perturbation method [3], the first integral method [4], the variational iteration method [5], the Riccati-Bernoulli sub-ODE method [6], the Jacobi elliptic function method [7], the tanhsech method [8], the .G 0 =G/-expansion method [9,10], the Hirota's method [11], the homogeneous balance method (HBM) [12,13], the differential transform method (DTM) [14][15][16][17] and so on.…”