Homotopy Perturbation Method for a Class of Nonlinear Singular System of Initial Value Problems

2020 ◽  
Vol 12 (1) ◽  
pp. 60-64
Author(s):  
Priti Pathak ◽  
Amit K. Barnwal
Keyword(s):  
Perturbation Method ◽  
Convergence Analysis ◽  
Initial Value Problems ◽  
Homotopy Perturbation Method ◽  
Singular System ◽  
Initial Value ◽  
Recursive Scheme ◽  
Homotopy Perturbation ◽  
Nonlinear Singular System

In this paper, a method based on homotopy perturbation method is used to establish the recursive scheme for the solution of nonlinear singular system of initial value problems. The convergence analysis of the proposed method is also shown. The accuracy and efficiency of the proposed method are demonstrated through various examples.

scholarly journals A local fractional homotopy perturbation method for solving the local fractional Korteweg-de Vries equations with non-homogeneous term

2019 ◽  
Vol 23 (3 Part A) ◽  
pp. 1495-1501 ◽  
Author(s):  
Yong-Ju Yang ◽  
Shun-Qin Wang
Keyword(s):  
Perturbation Method ◽  
Initial Value Problems ◽  
Homotopy Perturbation Method ◽  
Validity And Reliability ◽  
Initial Value ◽  
Homotopy Perturbation ◽  
De Vries

In this paper, a local fractional homotopy perturbation method is presented to solve the boundary and initial value problems of the local fractional Korteweg-de Vries equations with non-homogeneous term. In order to demonstrate the validity and reliability of the method, two types of the Korteweg-de Vries equations with non-homogeneous term are proposed.

scholarly journals Homotopy Perturbation Method and Convergence Analysis for the Linear Mixed Volterra-Fredholm Integral Equations

2020 ◽  
pp. 409-415
Author(s):  
Pakhshan Mohammed Ameen Hasan ◽  
Nejmaddin Abdulla Sulaiman
Keyword(s):  
Integral Equation ◽  
Perturbation Method ◽  
Convergence Analysis ◽  
Integral Equations ◽  
Error Estimation ◽  
Homotopy Perturbation Method ◽  
Fredholm Integral Equations ◽  
Homotopy Perturbation ◽  
The Absolute ◽  
Aitken Method

In this paper, the homotopy perturbation method is presented for solving the second kind linear mixed Volterra-Fredholm integral equations. Then, Aitken method is used to accelerate the convergence. In this method, a series will be constructed whose sum is the solution of the considered integral equation. Convergence of the constructed series is discussed, and its proof is given; the error estimation is also obtained. For more illustration, the method is applied on several examples and programs, which are written in MATLAB (R2015a) to compute the results. The absolute errors are computed to clarify the efficiency of the method.

scholarly journals Convergence Analysis for the Homotopy Perturbation Method for a Linear System of Mixed Volterra-Fredholm Integral Equations

2020 ◽  
Vol 17 (3(Suppl.)) ◽  
pp. 1010
Author(s):  
Pakhshan M. Hasan ◽  
Nejmaddin Abdulla Sulaiman
Keyword(s):  
Exact Solution ◽  
Linear System ◽  
Perturbation Method ◽  
Convergence Analysis ◽  
Integral Equations ◽  
Error Estimation ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Fredholm Integral Equations ◽  
Homotopy Perturbation

           In this paper, the homotopy perturbation method (HPM) is presented for treating a linear system of second-kind mixed Volterra-Fredholm integral equations. The method is based on constructing the series whose summation is the solution of the considered system. Convergence of constructed series is discussed and its proof is given; also, the error estimation is obtained. Algorithm is suggested and applied on several examples and the results are computed by using MATLAB (R2015a). To show the accuracy of the results and the effectiveness of the method, the approximate solutions of some examples are compared with the exact solution by computing the absolute errors.

scholarly journals Note on the Convergence Analysis of Homotopy Perturbation Method for Fractional Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Asma Ali Elbeleze ◽  
Adem Kılıçman ◽  
Bachok M. Taib
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Perturbation Method ◽  
Convergence Analysis ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Homotopy Perturbation

We apply the homotopy perturbation method to obtain the solution of partial differential equations of fractional order. This method is powerful tool to find exact and approximate solution of many linear and nonlinear partial differential equations of fractional order. Convergence of the method is proved and the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. The fractional derivatives are described in the Caputo sense. Some examples are presented to verify convergence hypothesis and simplicity of the method.

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