scholarly journals The local fractional variational iteration method a promising technology for fractional calculus

2020 ◽  
Vol 24 (4) ◽  
pp. 2605-2614 ◽  
Author(s):  
Yong-Ju Yang
Keyword(s):  
Fractional Calculus ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Initial Solution ◽  
Iteration Algorithm ◽  
Promising Technology ◽  
Variational Iteration ◽  
Sumudu Transform

In order to make the local variational iteration algorithm converge faster and more effective, the Sumudu transform is adopted and a proper initial solution is chosen. Some examples are given to show that the presented method is reliable, efficient and easy to implement from a computational viewpoint.

scholarly journals An Aproximation to Solution of Space and Time Fractional Telegraph Equations by the Variational Iteration Method

2012 ◽  
Vol 2012 ◽  
pp. 1-2 ◽  
Author(s):  
Ji-Huan He
Keyword(s):  
Iteration Method ◽  
Variational Iteration Method ◽  
Iteration Algorithm ◽  
Effective Algorithm ◽  
Variational Iteration ◽  
Space And Time ◽  
Telegraph Equations

Sevimlican suggested an effective algorithm for space and time fractional telegraph equations by the variational iteration method. This paper shows that algorithm can be updated by either variational iteration algorithm-II or the fractional variational iteration method.

An Alternative Approach to Differential-Difference Equations Using the Variational Iteration Method

2010 ◽  
Vol 65 (12) ◽  
pp. 1055-1059 ◽  
Author(s):  
Naeem Faraz ◽  
Yasir Khan ◽  
Francis Austin
Keyword(s):  
Difference Equations ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Iteration Algorithm ◽  
Mathematical Tool ◽  
Variational Iteration ◽  
Alternative Approach

Although a variational iteration algorithm was proposed by Yildirim (Math. Prob. Eng. 2008 (2008), Article ID 869614) that successfully solves differential-difference equations, the method involves some repeated and unnecessary iterations in each step. An alternative iteration algorithm (variational iteration algorithm-II) is constructed in this paper that overcomes this shortcoming and promises to provide a universal mathematical tool for many differential-difference equations.

scholarly journals Comment on “Variational Iteration Method for Fractional Calculus Using He’s Polynomials”

2012 ◽  
Vol 2012 ◽  
pp. 1-2 ◽  
Author(s):  
Ji-Huan He
Keyword(s):  
Iteration Method ◽  
Variational Iteration Method ◽  
Initial Boundary ◽  
Iteration Algorithm ◽  
Initial Boundary Value Problems ◽  
Variational Iteration ◽  
He’S Polynomials ◽  
Fractional Differential ◽  
Initial Boundary Value ◽  
Modified Variational Iteration Method

Recently Liu applied the variational homotopy perturbation method for fractional initial boundary value problems. This note concludes that the method is a modified variational iteration method using He’s polynomials. A standard variational iteration algorithm for fractional differential equations is suggested.

scholarly journals Fractional Variational Iteration Method for Fractional Cauchy Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Bao Si-yuan
Keyword(s):  
Cauchy Problem ◽  
Exact Solutions ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Solution Procedure ◽  
Iteration Algorithm ◽  
Cauchy Problems ◽  
Variational Iteration ◽  
Fractional Cauchy Problem

The fractional variational iteration method is used to solve the fractional Cauchy problem. Some examples are given to elucidate the solution procedure and reliability of the obtained results. The variational iteration algorithm leads to exact solutions in the present study.

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 An analytical approach to fractional Bousinesq-Burges equations

2020 ◽  
Vol 24 (4) ◽  
pp. 2581-2588
Author(s):  
Feng Lu
Keyword(s):  
Fractional Calculus ◽  
Iteration Method ◽  
Analytical Approach ◽  
Variational Iteration Method ◽  
Fractional Complex Transform ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

This paper proposes an analytical approach to fractional calculus by the fractional complex transform and the modified variational iteration method. The fractional Bousinesq-Burges equations are used as an example to reveal the main merits of the present technology.

scholarly journals On the Solution of Generalized Space Time Fractional Telegraph Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Badr S. Alkahtani ◽  
Vartika Gulati ◽  
Pranay Goswami
Keyword(s):  
Lagrange Multiplier ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Telegraph Equation ◽  
Space Time ◽  
New Technique ◽  
Variational Iteration ◽  
Fractional Telegraph Equation ◽  
Sumudu Transform

We present the solution of generalized space time fractional telegraph equation by using Sumudu variational iteration method which is the combination of variational iteration method and Sumudu transform. We tried to overcome the difficulties in finding the value of Lagrange multiplier by this new technique.

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