Two-dimensional differential transform method, Adomian's decomposition method, and variational iteration method for partial differential equations

Application of a hybrid numerical method to the nonlinear dynamic analysis of a micro gas bearing system

“…The modified Reynolds equation is solved using a hybrid numerical method combining the differential transformation method (DTM) and the finite difference method (FDM). This hybrid numerical method might be not the best candidate compared with variational iteration method or Adomian method [15,16], but the validity of the hybrid method is confirmed by comparing the results obtained for the orbits of the rotor center with those obtained using the conventional FDM scheme. The proposed method is then applied to analyze the nonlinear dynamic response of the rotor for values of the rotor mass and bearing number in the ranges 0.01-0.16 and 1.0-5.0, respectively.…”

“…Applying the DTM method to (15), the dynamic equation of the system of interest can be solved over the entire period spanned by the first sub-domain. The end point of function Z(t) in the first sub-domain is Z 1 .…”

“…The hybrid numerical method proposed in this study is commenced by using the differential transformation method (DTM) [15,16] to discretize the modified Reynolds equation given in (1) with respect to time. DTM is one of the most widely used techniques for solving both linear and nonlinear differential equations due to its rapid convergence rate and minimal calculation error.…”

Multivariate padé approximation for solving partial differential equations (PDE)

“…In recent years, much research has been focused on the numerical solution of PDEs. Some numerical methods have been developed, using the Taylor series, two-dimensional differential transformation, three-dimensional differential transformation, Adomian decomposition and variational iteration method [1][2][3][4][5][6]. The purpose of this paper is to consider the numerical solution of PDEs by using multivariate padé approximations.…”

Implementation to the Different Differential Equations of Homotopy Analysis, Differential Transformed and Adomian Decomposition Method

“…When the non-linearity is strong, these semi-analytical approaches, often gives unsatisfactory results.The solution of this type of the problems which gives the opportunity to control the region of the convergence and the speed of the solution series.Homotopy analysis method (HAM) is given by Shijun Liao [1]- [5] in 1992. At the same time, this method is the generalization of the non-perturbative method given earlier such as Adomian decomposition method [6], [7]- [11], Lyapunov artificial small parameter method [12], d-expansion method [13], differential transform method [1], [8], [14], [15] and homotopy perturbation method [16]. In other words, this can be thought as a theory of generalized or combined of the previous methods.…”