Some multi-step iterative methods for solving nonlinear equations

Abstract

a b s t r a c tIn this paper, we develop some new iterative methods for solving nonlinear equations by using the techniques introduced in Golbabai and Javidi (2007) [1] and Rafiq and Rafiullah (2008) [20]. We establish the convergence analysis of the proposed methods and then demonstrate their efficiency by taking some test problems.

The multi-step homotopy analysis method for solving the Jaulent-Miodek equations

“…The principles of the homotopy analysis method are given in [7,8,9,10,11]. The nonlinear Jaulent-Miodek equations which will be considered in this paper has the following form:…”

The multi-step homotopy analysis method for solving fractional-order model for HIV infection of CD4+T cells

“…The MHAM is used to provide approximate solutions for nonlinear problem in terms of convergent series with easily computable components, it has been shown that the approximated solution obtained are not valid for large t for some systems. Therefore we use the MHAM, which is offers accurate solution over a longer time frame compared to the HAM [1,2,16,18,19]. In this section we need to construct the MHAM of the general form of the HIV infection of CD4 + T cells model (2.1).…”

Implicit–explicit multistep methods for general two-dimensional nonlinear Schrödinger equations

“…Full implicit schemes are unconditionally stable, however, the main disadvantage of this approach is the need to solve nonlinear system at each time step, which could be solved by iterative methods [1,25], and is rather taxing and time consuming. The error order of the time discretization of numerical scheme may be decreased if we exploit linearized technique directly [4,19].…”

“…It is possible to solve the 2D NLS equation to develop singularities at some finite time [17]. Since searching for the exact solutions of NLS equations can be a hard task, one has to make use of numerical procedures and so, a considerable advance in numerical methods for this type of equations has been attained (e.g., [1], [11], [13], [15], [25], [35]- [38] and [41]). For instance, Xu [37] presented the local discontinuous Galerkin methods for NLS equations, [13,35] studied the NLS equations using split-step method which separated the equation into linear and nonlinear parts.…”