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.

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 Approximate solution for fractional attractor one-dimensional Keller-Segel equations using homotopy perturbation sumudu transform method

2020 ◽  
Vol 9 (1) ◽  
pp. 370-381
Author(s):  
Dinkar Sharma ◽  
Gurpinder Singh Samra ◽  
Prince Singh
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Simple Technique ◽  
Nonlinear Partial Differential Equations ◽  
Transform Method ◽  
Present Scheme ◽  
One Dimensional ◽  
Homotopy Perturbation ◽  
Sumudu Transform

AbstractIn this paper, homotopy perturbation sumudu transform method (HPSTM) is proposed to solve fractional attractor one-dimensional Keller-Segel equations. The HPSTM is a combined form of homotopy perturbation method (HPM) and sumudu transform using He’s polynomials. The result shows that the HPSTM is very efficient and simple technique for solving nonlinear partial differential equations. Test examples are considered to illustrate the present scheme.

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 Analytical Solution of Space-Time Fractional Fokker-Planck Equation by Homotopy Perturbation Sumudu Transform Method

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Ravi Shanker Dubey ◽  
Badr Saad T. Alkahtani ◽  
Abdon Atangana
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Planck Equation ◽  
Space Time ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
He’S Polynomials ◽  
Sumudu Transform ◽  
Fokker Planck Equations ◽  
Fokker Planck

An efficient approach based on homotopy perturbation method by using Sumudu transform is proposed to solve some linear and nonlinear space-time fractional Fokker-Planck equations (FPEs) in closed form. The space and time fractional derivatives are considered in Caputo sense. The homotopy perturbation Sumudu transform method (HPSTM) is a combined form of Sumudu transform, homotopy perturbation method, and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. Some examples show that the HPSTM is an effective tool for solving many space time fractional partial differential equations.

Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and Mittag-Leffler kernel

2018 ◽  
Vol 13 (1) ◽  
pp. 13 ◽  
Author(s):  
H. Yépez-Martínez ◽  
J.F. Gómez-Aguilar
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Nonlinear Differential Equations ◽  
Fractional Operators ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Fractional Differential ◽  
Derivative Order

Analytical and numerical simulations of nonlinear fractional differential equations are obtained with the application of the homotopy perturbation transform method and the fractional Adams-Bashforth-Moulton method. Fractional derivatives with non singular Mittag-Leffler function in Liouville-Caputo sense and the fractional derivative of Liouville-Caputo type are considered. Some examples have been presented in order to compare the results obtained, classical behaviors are recovered when the derivative order is 1.

scholarly journals Approximate Solutions to Time-Fractional Schrödinger Equation via Homotopy Analysis Method

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Najeeb Alam Khan ◽  
Muhammad Jamil ◽  
Asmat Ara
Keyword(s):  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Higher Dimensions ◽  
Analysis Method ◽  
Transform Method ◽  
Coupled Equations ◽  
Fractional Schrödinger Equations ◽  
Differential Transform ◽  
Homotopy Analysis

We construct the approximate solutions of the time-fractional Schrödinger equations, with zero and nonzero trapping potential, by homotopy analysis method (HAM). The fractional derivatives, in the Caputo sense, are used. The method is capable of reducing the size of calculations and handles nonlinear-coupled equations in a direct manner. The results show that HAM is more promising, convenient, efficient and less computational than differential transform method (DTM), and easy to apply in spaces of higher dimensions as well.

scholarly journals Approximations for Large Deflection of a Cantilever Beam under a Terminal Follower Force and Nonlinear Pendulum

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
H. Vázquez-Leal ◽  
Y. Khan ◽  
A. L. Herrera-May ◽  
U. Filobello-Nino ◽  
A. Sarmiento-Reyes ◽  
...  
Keyword(s):  
Perturbation Method ◽  
Cantilever Beam ◽  
Homotopy Perturbation Method ◽  
Large Deflection ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Follower Force ◽  
Homotopy Perturbation ◽  
Solution Methods ◽  
Nonlinear Pendulum

In theoretical mechanics field, solution methods for nonlinear differential equations are very important because many problems are modelled using such equations. In particular, large deflection of a cantilever beam under a terminal follower force and nonlinear pendulum problem can be described by the same nonlinear differential equation. Therefore, in this work, we propose some approximate solutions for both problems using nonlinearities distribution homotopy perturbation method, homotopy perturbation method, and combinations with Laplace-Padé posttreatment. We will show the high accuracy of the proposed cantilever solutions, which are in good agreement with other reported solutions. Finally, for the pendulum case, the proposed approximation was useful to predict, accurately, the period for an angle up to179.99999999∘yielding a relative error of 0.01222747.

scholarly journals Rational Biparameter Homotopy Perturbation Method and Laplace-Padé Coupled Version

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Hector Vazquez-Leal ◽  
Arturo Sarmiento-Reyes ◽  
Yasir Khan ◽  
Uriel Filobello-Nino ◽  
Alejandro Diaz-Sanchez
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Homotopy Perturbation ◽  
Physical Phenomena

The fact that most of the physical phenomena are modelled by nonlinear differential equations underlines the importance of having reliable methods for solving them. This work presents the rational biparameter homotopy perturbation method (RBHPM) as a novel tool with the potential to find approximate solutions for nonlinear differential equations. The method generates the solutions in the form of a quotient of two power series of different homotopy parameters. Besides, in order to improve accuracy, we propose the Laplace-Padé rational biparameter homotopy perturbation method (LPRBHPM), when the solution is expressed as the quotient of two truncated power series. The usage of the method is illustrated with two case studies. On one side, a Ricatti nonlinear differential equation is solved and a comparison with the homotopy perturbation method (HPM) is presented. On the other side, a nonforced Van der Pol Oscillator is analysed and we compare results obtained with RBHPM, LPRBHPM, and HPM in order to conclude that the LPRBHPM and RBHPM methods generate the most accurate approximated solutions.

scholarly journals Numerical Solutions of Nonlinear Fractional Partial Differential Equations Arising in Spatial Diffusion of Biological Populations

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Jagdev Singh ◽  
Devendra Kumar ◽  
Adem Kılıçman
Keyword(s):  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Spatial Diffusion ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Breeding Territory ◽  
Sumudu Transform ◽  
User Friendly

The main aim of this work is to present a user friendly numerical algorithm based on homotopy perturbation Sumudu transform method for nonlinear fractional partial differential arising in spatial diffusion of biological populations in animals. The movements are made generally either by mature animals driven out by invaders or by young animals just reaching maturity moving out of their parental territory to establish breeding territory of their own. The homotopy perturbation Sumudu transform method is a combined form of the Sumudu transform method and homotopy perturbation method. The obtained results are compared with Sumudu decomposition method. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and accurate. These results reveal that the proposed method is computationally very attractive.

The Homotopy Perturbation Method for Solving the Modified Korteweg-de Vries Equation

2008 ◽  
Vol 63 (10-11) ◽  
pp. 621-626 ◽  
Author(s):  
Ahmet Yildirim
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Vries Equation ◽  
Nonlinear Differential Equations ◽  
Convergent Series ◽  
Science And Engineering ◽  
Weak Nonlinearity ◽  
Homotopy Perturbation ◽  
Engineering Problems ◽  
De Vries

The homotopy perturbation method (HPM) is employed successfully for solving the modified Korteweg-de Vries equation. In this method, the solution is calculated in the form of a convergent series with an easily computable component. This approach does not need linearization, weak nonlinearity assumptions or perturbation theory. The results show applicability, accuracy and efficiency of the HPM in solving nonlinear differential equations. It is predicted that the HPM can be widely applied in science and engineering problems.

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