Analytical Approximations to Conservative Oscillators With Odd Nonlinearity Using the Variational Iteration Method

2012 ◽  
Vol 8 (1) ◽  
Author(s):  
A. Khosrozadeh ◽  
M. A. Hajabasi ◽  
H. R. Fahham
Keyword(s):  
Fourier Series ◽  
Iteration Method ◽  
Oscillation Amplitude ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
New Technique ◽  
Variational Iteration ◽  
Analytical Approximations ◽  
A New Technique ◽  
Odd Nonlinearity

In this article, a new technique is introduced for establishing analytical approximate solutions to conservative oscillators with strong odd nonlinearity using the variational iteration method and the Fourier series. The illustrated examples show that only a few iterations can provide very accurate approximate solutions for the whole range of oscillation amplitude even for longer time ranges.

scholarly journals A modification fractional variational iteration method for solving nonlinear gas dynamic and coupled KdV equations involving local fractional operators

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 165-175 ◽  
Author(s):  
Dumitru Baleanu ◽  
Hassan Jassim ◽  
Hasib Khan
Keyword(s):  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
New Technique ◽  
Transform Method ◽  
Variational Iteration ◽  
Kdv Equations ◽  
Gas Dynamic ◽  
Coupled Kdv Equations ◽  
A New Technique

In this paper, we apply a new technique, namely local fractional variational iteration transform method on homogeneous/non-homogeneous non-linear gas dynamic and coupled KdV equations to obtain the analytical approximate solutions. The iteration procedure is based on local fractional derivative and integral operators. This method is the combination of the local fractional Laplace transform and variational iteration method. The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

scholarly journals Variational iteration method for studying perihelion precession and deflection of light in General Relativity

2016 ◽  
Vol 4 (2) ◽  
pp. 52 ◽  
Author(s):  
V.K. Shchigolev
Keyword(s):  
General Relativity ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Simple Problem ◽  
Geodesic Equations ◽  
Variational Iteration ◽  
Deflection Of Light ◽  
Perihelion Shift ◽  
Main Ideas

A new approach in studying the planetary orbits and deflection of light in General Relativity (GR) by means of the Variational iteration method (VIM) is proposed in this paper. For this purpose, a brief review of the nonlinear geodesic equations in the spherical symmetry spacetime and the main ideas of VIM are given. The appropriate correct functionals are constructed for the geodesics in the spacetime of Schwarzschild, Reissner-Nordström and Kiselev black holes. In these cases, the Lagrange multiplier is obtained from the stationary conditions for the correct functionals. Then, VIM leads to the simple problem of computation of the integrals in order to obtain the approximate solutions of the geodesic equations. On the basis of these approximate solutions, the perihelion shift and the light deflection have been obtained for the metrics mentioned above.

A modified variational iteration method for solving fractional Riccati differential equation by Adomian polynomials

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Hossein Jafari ◽  
Hale Tajadodi ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equation ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Convergence Region ◽  
Riccati Differential Equation ◽  
Adomian Polynomials ◽  
Riccati Differential Equations ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

AbstractIn this paper, we introduce a modified variational iteration method (MVIM) for solving Riccati differential equations. Also the fractional Riccati differential equation is solved by variational iteration method with considering Adomians polynomials for nonlinear terms. The main advantage of the MVIM is that it can enlarge the convergence region of iterative approximate solutions. Hence, the solutions obtained using the MVIM give good approximations for a larger interval. The numerical results show that the method is simple and effective.

scholarly journals Local Fractional Variational Iteration Method for Local Fractional Poisson Equations in Two Independent Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Li Chen ◽  
Yang Zhao ◽  
Hossein Jafari ◽  
J. A. Tenreiro Machado ◽  
Xiao-Jun Yang
Keyword(s):  
Iteration Method ◽  
Fractional Derivatives ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Mathematical Physics ◽  
Nondifferentiable Functions ◽  
Independent Variables ◽  
Poisson Equations ◽  
Variational Iteration ◽  
Two Independent Variables

The local fractional Poisson equations in two independent variables that appear in mathematical physics involving the local fractional derivatives are investigated in this paper. The approximate solutions with the nondifferentiable functions are obtained by using the local fractional variational iteration method.

scholarly journals Local Fractional Laplace Variational Iteration Method for Nonhomogeneous Heat Equations Arising in Fractal Heat Flow

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Shu Xu ◽  
Xiang Ling ◽  
Carlo Cattani ◽  
Gong-Nan Xie ◽  
Xiao-Jun Yang ◽  
...  
Keyword(s):  
Heat Flow ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Previous Method ◽  
Heat Equations ◽  
Nondifferentiable Functions ◽  
Variational Iteration ◽  
Nonhomogeneous Heat

The local fractional Laplace variational iteration method is used for solving the nonhomogeneous heat equations arising in the fractal heat flow. The approximate solutions are nondifferentiable functions and their plots are also given to show the accuracy and efficiency to implement the previous method.

scholarly journals Variational Iteration Method for Volterra Functional Integrodifferential Equations with Vanishing Linear Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Ali Konuralp ◽  
H. Hilmi Sorkun
Keyword(s):  
Exact Solutions ◽  
Iteration Method ◽  
Numerical Solutions ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Integrodifferential Equations ◽  
Test Problems ◽  
Application Process ◽  
Delay Function ◽  
Variational Iteration

Application process of variational iteration method is presented in order to solve the Volterra functional integrodifferential equations which have multi terms and vanishing delays where the delay functionθ(t)vanishes inside the integral limits such thatθ(t)=qtfor0<q<1,t≥0. Either the approximate solutions that are converging to the exact solutions or the exact solutions of three test problems are obtained by using this presented process. The numerical solutions and the absolute errors are shown in figures and tables.

scholarly journals A variational iteration method integral transform technique for handling heat transfer problems

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 55-61 ◽  
Author(s):  
Yuejin Zhou ◽  
Shun Pang ◽  
Guo Chong ◽  
Xiaojun Yang ◽  
Xiaoding Xu ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Iteration Method ◽  
Integral Transform ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Excess Temperature ◽  
Variational Iteration ◽  
Transfer Equations ◽  
Transfer Problems ◽  
Integral Transform Technique

In this paper, we consider the heat transfer equations at the low excess temperature. The variational iteration method integral transform technique is used to find the approximate solutions for the problems. The used method is accurate and efficient.

Numerical solution of the general Volterra nth-order integro-differential equations via variational iteration method

2018 ◽  
Vol 13 (02) ◽  
pp. 2050042
Author(s):  
Fernane Khaireddine
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Numerical Solution ◽  
General Formula ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Numerical Examples ◽  
Variational Iteration ◽  
Linear And Nonlinear

In this paper, we use the variational iteration method (VIM) to construct approximate solutions for the general [Formula: see text]th-order integro-differential equations. We show that his method can be effectively and easily used to solve some classes of linear and nonlinear Volterra integro-differential equations. Finally, some numerical examples with exact solutions are given.

scholarly journals Modified Fractional Variational Iteration Method for Solving the Generalized Time-Space Fractional Schrödinger Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Baojian Hong ◽  
Dianchen Lu
Keyword(s):  
Schrödinger Equation ◽  
Iteration Method ◽  
Schrodinger Equation ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Fractional Schrödinger Equation ◽  
Variational Iteration ◽  
Time Space ◽  
Generalized Time ◽  
Fractional Schrodinger Equation

Based on He’s variational iteration method idea, we modified the fractional variational iteration method and applied it to construct some approximate solutions of the generalized time-space fractional Schrödinger equation (GFNLS). The fractional derivatives are described in the sense of Caputo. With the help of symbolic computation, some approximate solutions and their iterative structure of the GFNLS are investigated. Furthermore, the approximate iterative series and numerical results show that the modified fractional variational iteration method is powerful, reliable, and effective when compared with some classic traditional methods such as homotopy analysis method, homotopy perturbation method, adomian decomposition method, and variational iteration method in searching for approximate solutions of the Schrödinger equations.

Variational Iteration Method for Solving Riccati Matrix Differential Equations

2017 ◽  
Vol 5 (3) ◽  
pp. 673
Author(s):  
Khalid Hammood Al-jiz ◽  
Noor Atinah Ahmad ◽  
Fadhel Subhi Fadhel
Keyword(s):  
Iteration Method ◽  
Necessary Conditions ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Matrix Differential Equation ◽  
Variational Iteration ◽  
Matrix Differential Equations ◽  
Matrix Differential ◽  
He’S Variational Iteration Method ◽  
Riccati Matrix

<p>Riccati matrix differential equation has long been known to be so difficult to solve analytically and/or numerically. In this connection, most of the recent studies are concerned with the derivation of the necessary conditions that ensure the existence of the solution. Therefore, in this paper, He’s Variational iteration method is used to derive the general form of the iterative approximate sequence of solutions and then proved the convergence of the obtained sequence of approximate solutions to the exact solution. This proof is based on using the mathematical induction to derive a general formula for the upper bound proved to be converge to zero under certain conditions. </p>

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