The use of variational iteration method and Adomian decomposition method to solve the Eikonal equation and its application in the reconstruction problem

Abstract

SUMMARYIn this paper, the variational iteration method (VIM) and the Adomian decomposition method (ADM) are applied to solve the Eikonal equation. In addition, we apply the VIM using He's polynomials to find a semi-analytic solution of the studied model. The VIM is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. On the other hand, in the ADM the approximate solution is considered as an infinite series usually converging to the accurate solution. In addition, the results obtained using the two methods are presented. Comparison of the results reveals that the present methods are very efficient and convenient.

Solution of a nonlinear time-delay model in biology via semi-analytical approaches

“…In [38], a numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations is presented. The Adomian decomposition method is used for finding the solution of the ordinary differential equations which arise from problems of calculus of variations [67]. In [68] the solution of the second Painlevé equation is investigated by means of Adomian decomposition method.…”

A seminumeric approach for solution of the Eikonal partial differential equation and its applications

“…The comparison between the results demonstrates that modified form of He's homotopy perturbation method corresponds to Adomian's decomposition method for the Eikonal equation, and both methods have the same number of steps to obtain the solution. However, in the MHPM, and the ADM, we need to calculate the Adomian and He's polynomials but the variational iteration technique [31] provides the solution of the problem without calculating any polynomial.…”

“…Several examples are given to show the efficiency of these methods for solving the studied model. Also, the authors of the current article investigated the variational iteration method in another paper [31].…”

On the solutions of a class of nonlinear ordinary differential equations by the Bessel polynomials

“…Most of these equations have no analytical solution and numerical techniques may be used to obtain approximate ones. For example, some nonlinear differential equations have been solved using the Taylor matrix methods [19], the closed-form [33], the Chebyshev polynomial approximations [11], the constructive polynomial approximation [23], the iterative approximation [15], the constructing accurate polynomial approximations [10], a new method for generating general solution [31], the variational iteration method [13,18], the He's variational iteration method [1], the cubic B-spline scaling functions and Chebyshev cardinal functions [26], the homotopy perturbation method [2,22], the Adomian decomposition method [9,14,35], an efficient method [6], the variational iteration method and Adomian decomposition method [12], the 4q-Boubaker Poly- * Department of Mathematics, Faculty of Science, Mugla University, Mugla, 48000, Turkey 56 Ş. Yüzbaşı and N. Şahin nomials Expansion Scheme (BPES) and the Homotopy Perturbation Method (HPM) [25], and the He's parameter-expanding method (HPEM) and the 4q-Boubaker Polynomials Expansion Scheme (BPES) [24].…”