scholarly journals APPROXIMATE SOLUTION OF FORNBERG–WHITHAM EQUATION BY MODIFIED HOMOTOPY PERTURBATION METHOD UNDER NON-SINGULAR FRACTIONAL DERIVATIVE

Fractals ◽  
2021 ◽  
Author(s):  
HUSSAM ALRABAIAH
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Arbitrary Order ◽  
Numerical Illustration ◽  
Homotopy Perturbation ◽  
Non Local ◽  
Laplace Decomposition Method ◽  
Stated Problem ◽  
Whitham Equation

The basic idea of this paper is to investigate the approximate solution to a well-known Fornberg–Whitham equation of arbitrary order. We consider the stated problem under ABC fractional order derivative. The proposed derivative is non-local and contains non-singular kernel of Mittag-Leffler type. With the help of Modified Homotopy Perturbation Method (MHPM), we find approximate solution to the aforesaid equations. The required solution is computed in the form of infinite series. The method needs no discretization or collocation and easy to implement to compute the approximate solution that we intend. We also compare our results with that of the exact solution for the initial four terms approximate solution as well as with that computed by the Laplace decomposition method. We also plot the approximate solution of considered model through surface plots. For numerical illustration, we use Matlab throughout this work.

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%.

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.

scholarly journals Approximate Solution of an Infectious Disease Model Applying Homotopy Perturbation Method

2020 ◽  
Vol 12 (5) ◽  
pp. 64
Author(s):  
Terhemen Simon Atindiga ◽  
Ezike Godwin Mbah ◽  
Ndidiamaka Edith Didigwu ◽  
Adebisi Raphael Adewoye ◽  
Torkuma Bartholomew Kper
Keyword(s):  
Infectious Disease ◽  
Approximate Solution ◽  
Perturbation Method ◽  
Wave Equations ◽  
Homotopy Perturbation Method ◽  
Disease Model ◽  
Nonlinear Wave Equations ◽  
Simple Problem ◽  
Infectious Disease Model ◽  
Homotopy Perturbation

Scientists and engineers have developed the use of Homotopy Perturbation Method (HPM) in non-linear problems since this approach constantly distort the intricate problem being considered into a simple problem, thus making it much less complex to solve. The homotopy perturbation method was initially put forward by He (1999) with further development and improvement (He 2000a, He, 2006). Homotopy, which is as an essential aspect of differential topology involves a coupling of the conventional perturbation method and the homotopy method in topology (He, 2000b). The approach gives an approximate analytical result in series form and has been effectively applied by various academia for various physical systems namely; bifurcation, asymptotology, nonlinear wave equations and Approximate Solution of SIR Infectious Disease Model (Abubakar et al., 2013).

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