Numerical Simulation of Emden-Fowler Type Equations Using Variational Iteration Algorithm

2009 ◽  
Vol 64 (9-10) ◽  
pp. 583-587
Author(s):  
Elçin Yusufoğlu
Keyword(s):  
Numerical Simulation ◽  
Singular Point ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Iteration Algorithm ◽  
Regular Singular Point ◽  
Variational Theory ◽  
Variational Iteration ◽  
He’S Variational Iteration Method

The main objective of this article is to present a reliable algorithm to determine exact and approximate solutions of the generalized Emden-Fowler type equations. The algorithm mainly is based on He’s variational iteration method (VIM) with an alternative framework designed to overcome the difficulty of the regular singular point at x = 0. In this method, general Lagrange multipliers are introduced to construct a correction for the problem. The multipliers in the functional can be identified optimally via the variational theory. The results reveal that the proposed method is very effective and can be applied for other nonlinear problems.

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>

A general numerical algorithm for nonlinear differential equations by the variational iteration method

2020 ◽  
Vol 30 (11) ◽  
pp. 4797-4810 ◽  
Author(s):  
Ji-Huan He ◽  
Habibolla Latifizadeh
Keyword(s):  
Differential Equations ◽  
Numerical Algorithm ◽  
Iteration Method ◽  
Solution Process ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Iteration Algorithm ◽  
Content Type ◽  
Variational Iteration ◽  
Laplace Transform Technique

Purpose The purpose of this paper is to suggest a general numerical algorithm for nonlinear problems by the variational iteration method (VIM). Design/methodology/approach Firstly, the Laplace transform technique is used to reconstruct the variational iteration algorithm-II. Secondly, its convergence is strictly proved. Thirdly, the numerical steps for the algorithm is given. Finally, some examples are given to show the solution process and the effectiveness of the method. Findings No variational theory is needed to construct the numerical algorithm, and the incorporation of the Laplace method into the VIM makes the solution process much simpler. Originality/value A universal iteration formulation is suggested for nonlinear problems. The VIM cleans up the numerical road to differential equations.

scholarly journals Applying He's Variational Iteration Method for Solving Differential-Difference Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-7 ◽  
Author(s):  
Ahmet Yildirim
Keyword(s):  
Difference Equation ◽  
Iteration Method ◽  
Good Accuracy ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Variational Iteration ◽  
Differential Difference Equation ◽  
He’S Variational Iteration Method

We extend He's variational iteration method (VIM) to find the approximate solutions for nonlinear differential-difference equation. Simple but typical examples are applied to illustrate the validity and great potential of the generalized variational iteration method in solving nonlinear differential-difference equation. The results reveal that the method is very effective and simple. We find the extended method for nonlinear differential-difference equation is of good accuracy.

scholarly journals Comment on “An Approximation to Solution of Space and Time Fractional Telegraph Equations by He's Variational Iteration Method”

2013 ◽  
Vol 2013 ◽  
pp. 1-2
Author(s):  
Yi-Hong Wang ◽  
Lan-Lan Huang
Keyword(s):  
Laplace Transform ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Telegraph Equation ◽  
Variational Iteration ◽  
Space And Time ◽  
Fractional Telegraph Equation ◽  
Telegraph Equations ◽  
He’S Variational Iteration Method

The variational iteration method was applied to the time fractional telegraph equation and some variational iteration formulae were suggested in (Sevimlican, 2010). Those formulae are improved by Laplace transform from which the approximate solutions of higher accuracies can be obtained.

scholarly journals Application of He’s Variational Iteration Method to Nonlinear Integro-Differential Equations

2010 ◽  
Vol 65 (5) ◽  
pp. 418-430 ◽  
Author(s):  
Ahmet Yildirim
Keyword(s):  
Differential Equations ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Mathematical Tool ◽  
Variational Iteration ◽  
He’S Variational Iteration Method ◽  
Powerful Mathematical Tool

In this paper, an application of He’s variational iteration method is applied to solve nonlinear integro-differential equations. Some examples are given to illustrate the effectiveness of the method. The results show that the method provides a straightforward and powerful mathematical tool for solving various nonlinear integro-differential equations

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