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.

Fractional calculus for nanoscale flow and heat transfer

2014 ◽  
Vol 24 (6) ◽  
pp. 1227-1250 ◽  
Author(s):  
Hong-Yan Liu ◽  
Ji-Huan He ◽  
Zheng-Biao Li
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Differential Equations ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Content Type ◽  
Variational Iteration ◽  
Fractal Media ◽  
Fractional Differential ◽  
Analytical Approaches

Purpose – Academic and industrial researches on nanoscale flows and heat transfers are an area of increasing global interest, where fascinating phenomena are always observed, e.g. admirable water or air permeation and remarkable thermal conductivity. The purpose of this paper is to reveal the phenomena by the fractional calculus. Design/methodology/approach – This paper begins with the continuum assumption in conventional theories, and then the fractional Gauss’ divergence theorems are used to derive fractional differential equations in fractal media. Fractional derivatives are introduced heuristically by the variational iteration method, and fractal derivatives are explained geometrically. Some effective analytical approaches to fractional differential equations, e.g. the variational iteration method, the homotopy perturbation method and the fractional complex transform, are outlined and the main solution processes are given. Findings – Heat conduction in silk cocoon and ground water flow are modeled by the local fractional calculus, the solutions can explain well experimental observations. Originality/value – Particular attention is paid throughout the paper to giving an intuitive grasp for fractional calculus. Most cited references are within last five years, catching the most frontier of the research. Some ideas on this review paper are first appeared.

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

scholarly journals The variational iteration method for solving n-th order fuzzy differential equations

Open Physics ◽  
2012 ◽  
Vol 10 (1) ◽  
Author(s):  
Hossein Jafari ◽  
Mohammad Saeidy ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Nonlinear Equations ◽  
Analytical Method ◽  
Iteration Method ◽  
Initial Conditions ◽  
Variational Iteration Method ◽  
Linear Differential Equations ◽  
Variational Iteration ◽  
Linear And Nonlinear ◽  
Linear Differential

AbstractThe variational iteration method (VIM) proposed by Ji-Huan He is a new analytical method for solving linear and nonlinear equations. In this paper, the variational iteration method has been applied in solving nth-order fuzzy linear differential equations with fuzzy initial conditions. This method is illustrated by solving several examples.

Application of Variational Iteration Method for nth-Order Integro-Differential Equations

2009 ◽  
Vol 64 (7-8) ◽  
pp. 439-444 ◽  
Author(s):  
Said Abbasbandy ◽  
Elyas Shivanian
Keyword(s):  
Differential Equations ◽  
Iteration Method ◽  
Initial Conditions ◽  
Initial Approximation ◽  
Variational Iteration Method ◽  
Integrodifferential Equations ◽  
Variational Iteration

AbstractIn this paper, the variational iteration method is proposed to solve Fredholm’s nth-order integrodifferential equations. The initial approximation is selected wisely which satisfies the initial conditions. The results reveal that this method is very effective and convenient in comparison with other methods.

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