scholarly journals HPM for Solving the Time-fractional Coupled Burgers Equations

2016 ◽  
Vol 12 (4) ◽  
pp. 6133-6138
Author(s):  
Khadijah Abu Alnaja
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Burgers Equations ◽  
Homotopy Perturbation ◽  
Special Cases ◽  
Burger's Equations

In this paper, the homotopy perturbation method is implemented to derive the explicit approximate solutions for the time-fractional coupled Burger's equations. The including fractional derivative is in the Caputo sense. Special attention is given to prove the convergence of the method. The results are compared with those obtained by the exact at special cases of the fractional derivatives. The results reveal that the proposed method is very effective and simple.

scholarly journals Study on Space-Time Fractional Nonlinear Biological Equation in Radial Symmetry

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yanqin Liu
Keyword(s):  
Nonlinear Equation ◽  
Fractional Derivative ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Radial Symmetry ◽  
Space Time ◽  
Homotopy Perturbation ◽  
Initial Stage

We consider the initial stage of space-time fractional generalized biological equation in radial symmetry. Dimensionless multiorder fractional nonlinear equation was first given, and approximate solutions were derived in the form of series using the homotopy perturbation method with a new modification. And the influence of fractional derivative is also discussed.

scholarly journals A Modification Fractional Homotopy Perturbation Method for Solving Helmholtz and Coupled Helmholtz Equations on Cantor Sets

2019 ◽  
Vol 3 (2) ◽  
pp. 30 ◽  
Author(s):  
Dumitru Baleanu ◽  
Hassan Kamil Jassim
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Cantor Sets ◽  
New Technique ◽  
Helmholtz Equations ◽  
Homotopy Perturbation ◽  
Fractional Derivative Operators ◽  
A New Technique

In this paper, we apply a new technique, namely, the local fractional Laplace homotopy perturbation method (LFLHPM), on Helmholtz and coupled Helmholtz equations to obtain analytical approximate solutions. The iteration procedure is based on local fractional derivative operators (LFDOs). This method is a combination of the local fractional Laplace transform (LFLT) and the homotopy perturbation method (HPM). 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.

Numerical solution of fractionally damped beam by homotopy perturbation method

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Diptiranjan Behera ◽  
Snehashish Chakraverty
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Numerical Solution ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Analytic Solutions ◽  
Step Function ◽  
Homotopy Perturbation ◽  
Derivatives Of ◽  
Appl Math

AbstractThis paper investigates the numerical solution of a viscoelastic continuous beam whose damping behaviours are defined in term of fractional derivatives of arbitrary order. The Homotopy Perturbation Method (HPM) is used to obtain the dynamic response. Unit step function response is considered for the analysis. The obtained results are depicted in various plots. From the results obtained it is interesting to note that by increasing the order of the fractional derivative the beam suffers less oscillation. Similar observations have also been made by keeping the order of the fractional derivative constant and varying the damping ratios. Comparisons are made with the analytic solutions obtained by Zu-feng and Xiao-yan [Appl. Math. Mech. 28, 219 (2007)] to show the effectiveness and validation of this method.

scholarly journals Homotopy Perturbation Method For Fractional Shallow Water Equations

2014 ◽  
Vol 19 (1) ◽  
pp. 74
Author(s):  
Kamel Al-Khaled ◽  
M. K. Al-Safeen
Keyword(s):  
Perturbation Method ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Homotopy Perturbation ◽  
Power Series Solutions ◽  
Homotopy Analysis

In this paper, the homotopy perturbation method is adopted to find explicit and numerical solutions for systems of non-linear fractional shallow water equations. The fractional derivatives are described in the Caputo sense. We apply both the homotopy perturbation method and the homotopy analysis method, to solve  certain shallow water equations with time-fractional derivatives, and explicitly construct convergent power series solutions. The  results obtained reveal that these  methods are  both very effective and simple for finding approximate solutions. Some numerical examples and plots are presented to illustrate the efficiency and reliability of these methods.  

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

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 The Variational Homotopy Perturbation Method for Solving((n×n)+1)Dimensional Burgers’ Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
F. A. Hendi ◽  
B. S. Kashkari ◽  
A. A. Alderremy
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Term ◽  
Burgers Equations ◽  
Homotopy Perturbation ◽  
Burgers System

The variational homotopy perturbation method VHPM is used for solvingn-dimensional Burgers’ system. Some examples are examined to validate that the method reduced the calculation size, treating the difficulty of nonlinear term and the accuracy.

Homotopy perturbation method for MHD non-Newtonian Williamson fluid over exponentially stretching sheet with viscous dissipation and convective boundary condition

2019 ◽  
Vol 30 (11) ◽  
pp. 1950088 ◽  
Author(s):  
Khadijah M. Abualnaja
Keyword(s):  
Fluid Flow ◽  
Perturbation Method ◽  
Numerical Method ◽  
Homotopy Perturbation Method ◽  
Stretching Sheet ◽  
Fluid Temperature ◽  
Williamson Fluid ◽  
Homotopy Perturbation ◽  
Exponential Stretching ◽  
Special Cases

This research is aimed at presenting the two-dimensional steady fluid flow, represented by Williamson constitutive model past a nonlinear exponential stretching sheet theoretically. The system of ODEs describing the physical problem is successfully solved numerically with the help of the homotopy perturbation method (HPM). Special attention is given to study the convergence analysis of the proposed method. The influences of the physical governing parameters acting on the fluid velocity and the fluid temperature are explained with the help of the figures and tables. Further, the presented numerical method is employed to calculate both the rate of heat transfer and the drag force for the Williamson fluid flow. In particular, it is observed that both the Eckert number and the dimensionless convective parameter have the effect of enhancing the temperature of the stretching surface, while the inverse was noted for the dimensionless mixed convection parameter. Finally, the comparison with previous numerical investigations of other authors at some special cases which is reported here proves that the results obtained via homotopy perturbation method are accurate and the numerical method is reliable.

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.

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