scholarly journals Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2247
Author(s):  
Weam Alharbi ◽  
Sergei Petrovskii
Keyword(s):  
Approximate Solution ◽  
Numerical Analysis ◽  
Homotopy Perturbation Method ◽  
Series Solutions ◽  
Natural Phenomena ◽  
Homotopy Perturbation ◽  
Power Series Solutions ◽  
Approximate Power ◽  
Fractional Delay ◽  
With Memory

The fractional calculus is useful in describing the natural phenomena with memory effect. This paper addresses the fractional form of Ambartsumian equation with a delay parameter. It may be a challenge to obtain accurate approximate solution of such kinds of fractional delay equations. In the literature, several attempts have been conducted to analyze the fractional Ambartsumian equation. However, the previous approaches in the literature led to approximate power series solutions which converge in subdomains. Such difficulties are solved in this paper via the Homotopy Perturbation Method (HPM). The present approximations are expressed in terms of the Mittag-Leffler functions which converge in the whole domain of the studied model. The convergence issue is also addressed. Several comparisons with the previous published results are discussed. In particular, while the computed solution in the literature is physical in short domains, with our approach it is physical in the whole domain. The results reveal that the HPM is an effective tool to analyzing the fractional Ambartsumian equation.

scholarly journals Modified nonlinearities distribution Homotopy Perturbation method as a tool to find power series solutions to ordinary differential equations

Nova Scientia ◽  
2014 ◽  
Vol 6 (12) ◽  
pp. 13 ◽  
Author(s):  
Umberto Filobello-Nino ◽  
Héctor Vázquez-Leal ◽  
Yasir Khan ◽  
D. Pereyra-Díaz ◽  
A. Pérez-Sesma ◽  
...  
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Perturbation Method ◽  
Ordinary Differential Equations ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Series Solutions ◽  
Homotopy Perturbation ◽  
Polynomial Coefficients ◽  
Power Series Solutions

In this article, modified non-linearities distribution homotopy perturbation method (MNDHPM) is used in order to find power series solutions to ordinary differential equations with initial conditions, both linear and nonlinear. We will see that the method is particularly relevant in some cases of equations with non-polynomial coefficients and inhomogeneous non-polynomial terms

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.

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.

scholarly journals An analytical technique for solving general linear integral equations of the second kind and its application in analysis of flash lamp control circuit

2014 ◽  
Vol 62 (3) ◽  
pp. 413-421 ◽  
Author(s):  
E. Hetmaniok ◽  
D. Słota ◽  
T. Trawiński ◽  
R. Wituła
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Integral Equations ◽  
Homotopy Perturbation Method ◽  
General Linear ◽  
Practical Application ◽  
Analytical Technique ◽  
Homotopy Perturbation ◽  
Linear Integral ◽  
Linear Integral Equations

Abstract In this paper an application of the homotopy perturbation method for solving the general linear integral equations of the second kind is discussed. It is shown that under proper assumptions the considered equation possesses a unique solution and the series obtained in the homotopy perturbation method is convergent. The error of approximate solution, received by taking only the partial sum of the series, is also estimated. Moreover, there is presented an example of applying the method for approximate solution of an equation which has a practical application for charge calculation in supply circuit of the flash lamps used in cameras.

Solution for Celebrated Blasius Problem Using Homotopy Perturbation Method

2020 ◽  
Vol 12 (2) ◽  
pp. 284-287
Author(s):  
Monika Rani ◽  
Vikramjeet Singh ◽  
Rakesh Goyal
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Perturbation Method ◽  
High Speed ◽  
Homotopy Perturbation Method ◽  
Initial Slope ◽  
Moving Wall ◽  
Homotopy Perturbation ◽  
Blasius Problem ◽  
Blasius Equation

In this manuscript, we have analyzed Celebrated Blasius boundary problem with moving wall or high speed 2D laminar viscous flow over gasifying flat plate. To find the way out of this nonlinear differential equation, a version of semi-analytical homotopy perturbation method has been applied. It has been observed that the precision of the solution would be achieved with increasing approximations. On comparison with literature, our solution has been proven highly accurate and valid with faster rate of convergence. It has been revealed that the second order approximate solution of Blasius equation in terms of initial slope is obtained as 0.33315 reducing the error by 0.32%.

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 Solving Fractional-Order Logistic Equation Using a New Iterative Method

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Sachin Bhalekar ◽  
Varsha Daftardar-Gejji
Keyword(s):  
Iterative Method ◽  
Perturbation Method ◽  
Decomposition Method ◽  
Homotopy Perturbation Method ◽  
Adomian Decomposition Method ◽  
Logistic Equation ◽  
Series Solutions ◽  
Homotopy Perturbation ◽  
Adomian Decomposition ◽  
New Iterative Method

A fractional version of logistic equation is solved using new iterative method proposed by Daftardar-Gejji and Jafari (2006). Convergence of the series solutions obtained is discussed. The solutions obtained are compared with Adomian decomposition method and homotopy perturbation method.

scholarly journals Convergence and Error Estimation of a New Formulation of Homotopy Perturbation Method for Classes of Nonlinear Integral/Integro-Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2244
Author(s):  
Mohamed M. Mousa ◽  
Fahad Alsharari
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Perturbation Method ◽  
Error Estimation ◽  
Homotopy Perturbation Method ◽  
Main Concept ◽  
Convergence Theorems ◽  
Homotopy Perturbation ◽  
New Formulation

In this work, the main concept of the homotopy perturbation method (HPM) was outlined and convergence theorems of the HPM for solving some classes of nonlinear integral, integro-differential and differential equations were proved. A theorem for estimating the error in the approximate solution was proved as well. The proposed HPM convergence theorems were confirmed and the efficiency of the technique was explored by applying the HPM for solving several classes of nonlinear integral/integro-differential equations.

Sign in / Sign up

Export Citation Format

Share Document