Approximate Solution of Time-Fractional Chemical Engineering Equations: A Comparative Study

2010 ◽  
Vol 8 (1) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Asmat Ara ◽  
Amir Mahmood
Keyword(s):  
Approximate Solution ◽  
Fractional Derivatives ◽  
Chemical Engineering ◽  
Homotopy Perturbation Method ◽  
Variational Iteration Method ◽  
Chemical Reactor ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Homotopy Perturbation ◽  
Conventional Numerical Method

In this paper, we present the approximate solutions of the time fractional chemical engineering equations by means of the variational iteration method (VIM) and homotopy perturbation method (HPM). The fractional derivatives are described in the Caputo sense. The solutions of the chemical reactor, reaction, and concentration equations are calculated in the form of convergent series with easily computable components. We compared the HPM against the VIM; an additional comparison will be made against the conventional numerical method. The results show that HPM is more promising, convenient, and efficient than VIM.

scholarly journals Numerical Solutions of a Quadratic Integral Equations by Using Variational Iteration and Homotopy Perturbation Methods

2017 ◽  
Vol 9 (2) ◽  
pp. 134
Author(s):  
Hind Al-badrani ◽  
F. A. Hendi ◽  
Wafa Shammakh
Keyword(s):  
Integral Equations ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Variational Iteration ◽  
Homotopy Perturbation ◽  
Quadratic Integral ◽  
Quadratic Integral Equations

In this paper, the approximate solutions for  quadratic integral equations (QIEs) are given by the variational iteration method(VIM) and homotopy perturbation method (HPM). These methods produce the solutions in terms of convergent series without needing to restrictive assumptions, to illustrate the ability and credibility of the methods, we deal with some examples that show simplicity and effectiveness.

scholarly journals Analytic and Approximate Solutions of the Space-Time Fractional Schrödinger Equations by Homotopy Perturbation Sumudu Transform Method

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Sara H. M. Hamed ◽  
Eltayeb A. Yousif ◽  
Arbab I. Arbab
Keyword(s):  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Fractional Schrödinger Equations ◽  
Sumudu Transform ◽  
Fractional Nonlinear Schrödinger Equation

A combination of homotopy perturbation method and Sumudu transform is applied to find exact and approximate solution of space and time fractional nonlinear Schrödinger equation. The fractional derivatives are described in the Caputo sense. The solutions are given in the form of convergent series with easily computable components. The results show that the method is effective and convenient for solving nonlinear differential equations of fractional order.

scholarly journals Second Approximate Solution of Duffing Equation with Strong Nonlinearity by Homotopy Perturbation Method

1970 ◽  
Vol 30 ◽  
pp. 59-75
Author(s):  
M Alhaz Uddin ◽  
M Abdus Sattar
Keyword(s):  
Approximate Solution ◽  
Nonlinear Systems ◽  
Perturbation Method ◽  
Differential System ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Second Order ◽  
Strong Nonlinearity ◽  
Analytical Technique ◽  
Homotopy Perturbation

 In this paper, the second order approximate solution of a general second order nonlinear ordinary differential system, modeling damped oscillatory process is considered. The new analytical technique based on the work of He’s homotopy perturbation method is developed to find the periodic solution of a second order ordinary nonlinear differential system with damping effects. Usually the second or higher order approximate solutions are able to give better results than the first order approximate solutions. The results show that the analytical approximate solutions obtained by homotopy perturbation method are uniformly valid on the whole solutions domain and they are suitable not only for strongly nonlinear systems, but also for weakly nonlinear systems. Another advantage of this new analytical technique is that it also works for strongly damped, weakly damped and undamped systems. Figures are provided to show the comparison between the analytical and the numerical solutions. Keywords: Homotopy perturbation method; damped oscillation; nonlinear equation; strong nonlinearity. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 59-75  DOI: http://dx.doi.org/10.3329/ganit.v30i0.8504

scholarly journals The Approximate Solution of Fractional Fredholm Integrodifferential Equations by Variational Iteration and Homotopy Perturbation Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Abdelouahab Kadem ◽  
Adem Kilicman
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Iteration Method ◽  
Homotopy Perturbation Method ◽  
Perturbation Methods ◽  
Variational Iteration Method ◽  
Integrodifferential Equations ◽  
Variational Iteration ◽  
Homotopy Perturbation ◽  
Constant Coefficients

Variational iteration method and homotopy perturbation method are used to solve the fractional Fredholm integrodifferential equations with constant coefficients. The obtained results indicate that the method is efficient and also accurate.

Application of He’s Homotopy Perturbation Method for Solving Fractional Fokker-Planck Equations

2009 ◽  
Vol 64 (12) ◽  
pp. 788-794 ◽  
Author(s):  
Mohamed M. Mousa ◽  
Aidarkhan Kaltayev
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Planck Equation ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Homotopy Perturbation ◽  
External Force Field ◽  
Promising Tool ◽  
Transport Problems ◽  
Fokker Planck

Abstract The fractional Fokker-Planck equation (FFPE) has been used in many physical transport problems which take place under the influence of an external force field and other important applications in various areas of engineering and physics. In this paper, by means of the homotopy perturbation method (HPM), exact and approximate solutions are obtained for two classes of the FFPE initial value problems. The method gives an analytic solution in the form of a convergent series with easily computed components. The obtained results show that the HPM is easy to implement, accurate and reliable for solving FFPEs. The method introduces a promising tool for solving other types of differential equation with fractional order derivatives

scholarly journals Numerical Methods for Fractional Order Singular Partial Differential Equations with Variable Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Asma Ali Elbeleze ◽  
Adem Kılıçman ◽  
Bachok M. Taib
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Variational Iteration Method ◽  
Convergent Series ◽  
Variable Coefficients ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

We implement relatively analytical methods, the homotopy perturbation method and the variational iteration method, for solving singular fractional partial differential equations of fractional order. The process of the methods which produce solutions in terms of convergent series is explained. The fractional derivatives are described in Caputo sense. Some examples are given to show the accurate and easily implemented of these methods even with the presence of singularities.

An easy and computable approximation for Troesch’s problem by using the Laplace Transform-Homotopy Perturbation Method

2019 ◽  
Vol 29 ◽  
pp. 1-14
Author(s):  
U. Filobello-Nino ◽  
H. Vazquez-Leal ◽  
A. L. Herrera-May ◽  
V. M. Jimenez-Fernandez ◽  
J. Cervantes-Perez ◽  
...  
Keyword(s):  
Approximate Solution ◽  
Laplace Transform ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Absolute Relative Error ◽  
Homotopy Perturbation ◽  
Average Absolute Relative Error ◽  
The Laplace Transform ◽  
Troesch’S Problem

This work introduces the Laplace Transform-Homotopy Perturbation Method (LT-HPM) in order to provide an approximate solution for Troesch’s problem. After comparing figures between exact and approximate solutions, as well as the average absolute relative error (AARE) of the approximate solutions of this research, with others reported in the literature, it can be said that the proposed solutions are accurate and handy. In conclusion, LT-HPM is a potentially useful tool.

scholarly journals An Efficient Method for Systems of Variable Coefficient Coupled Burgers’ Equation with Time-Fractional Derivative

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Hossein Aminikhah ◽  
Nasrin Malekzadeh
Keyword(s):  
Fractional Derivative ◽  
Variable Coefficient ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Burgers Equation ◽  
Power Series Expansion ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Time Fractional Derivative ◽  
Coupled Burgers Equation

A new homotopy perturbation method (NHPM) is applied to system of variable coefficient coupled Burgers' equation with time-fractional derivative. The fractional derivatives are described in the Caputo fractional derivative sense. The concept of new algorithm is introduced briefly, and NHPM is examined for two systems of nonlinear Burgers' equation. In this approach, the solution is considered as a power series expansion that converges rapidly to the nonlinear problem. The new approximate analytical procedure depends on two iteratives. The modified algorithm provides approximate solutions in the form of convergent series with easily computable components. Results indicate that the introduced method is promising for solving other types of systems of nonlinear fractional-order partial differential equations.

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