Interpolated variational iteration method for initial value problems

Abstract

In order to solve an initial value problem by the variational iteration method, a sequence of functions is produced which converges to the solution under some suitable conditions. In the nonlinear case, after a few iterations the terms of the sequence become complicated, and therefore, computing a highly accurate solution would be difficult or even impossible. In this paper, for one-dimensional initial value problems, we propose a new approach which is based on approximating each term of the sequence by a piecewise linear function. Moreover, the convergence of the method is proved. Three illustrative examples are given to show the superiority of the proposed method over the classical variational iteration method.Comment: 17 pages, 8 figures in Applied Mathematical Modelling, 201

Optimal control design for linear time-varying systems by interpolated variational iteration method

“…Using system (3a), the system can be expressed as follows:where V(t),W(t)∈double-struckR2×2. To solve this problem, we use the original IVIM proposed by Salkuyeh and Tavakoli (2016), and our proposed modified IVIM. Figure 3 shows the CPU time in terms of the number of grid points for different number of iterations.…”

“…The main idea of this paper comes from an improvement of VIM, called interpolated VIM (IVIM), made by Salkuyeh and Tavakoli (2016) for solving one-dimensional initial value problems. For the first time, we implement IVIM to solve LTV optimal control problems, by interpolating the optimal states and costates at each grid point, with B-spline basis functions.…”

Boundary-domain integral method and homotopy analysis method for systems of nonlinear boundary value problems in environmental engineering

“…Other examples include: the thermal behaviour of a spherical cloud of gas, the theory of stellar structure and the theory of thermionic currents. The systems of Lane-Emden equations arise when modelling many other physical phenomena as well (Khojasteh Salkuyeh & Tavakoli, 2016;Sun, Liu, & Keith, 2004;Wazwaz, 2001;Wazwaz & Rach, 2011). Wazwaz et al ( Wazwaz, Rach, & Duan, 2013; used the Adomian decomposition method for solving the Volterra integral form of the Lane-Emden equations with initial values and boundary conditions.…”

Variational Iteration Method for Burger’s Equation in Different Positions

“…When this Lagrange multiplier value is substituted into the functional (13), it produces the following iteration formula,…”

“…The method of VIM, that was suggested initially by [1][2][3][4], was evinced by several authors to exist as a potent tool for mathematics when it comes for problems like linear and nonlinear ones [5][6][7][8][9][10][11][12][13][14][15]. In contrast to the old-style numerical methods, the features of discretization, linearization, transformation or perturbation is no longer in need for VIM.…”