scholarly journals A study of the convergence of and error estimation for the homotopy perturbation method for the Volterra–Fredholm integral equations

2013 ◽  
Vol 26 (1) ◽  
pp. 165-169 ◽  
Author(s):  
Edyta Hetmaniok ◽  
Iwona Nowak ◽  
Damian Słota ◽  
Roman Wituła
Keyword(s):  
Perturbation Method ◽  
Integral Equations ◽  
Error Estimation ◽  
Homotopy Perturbation Method ◽  
Fredholm Integral Equations ◽  
Homotopy Perturbation

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 Error Estimation of the Homotopy Perturbation Method to Solve Second Kind Volterra Integral Equations with Piecewise Smooth Kernels: Application of the CADNA Library

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1730 ◽  
Author(s):  
Samad Noeiaghdam ◽  
Aliona Dreglea ◽  
Jihuan He ◽  
Zakieh Avazzadeh ◽  
Muhammad Suleman ◽  
...  
Keyword(s):  
Perturbation Method ◽  
Integral Equations ◽  
Error Estimation ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Energy System ◽  
Volterra Integral Equations ◽  
Stochastic Arithmetic ◽  
Homotopy Perturbation ◽  
Cadna Library

This paper studies the second kind linear Volterra integral equations (IEs) with a discontinuous kernel obtained from the load leveling and energy system problems. For solving this problem, we propose the homotopy perturbation method (HPM). We then discuss the convergence theorem and the error analysis of the formulation to validate the accuracy of the obtained solutions. In this study, the Controle et Estimation Stochastique des Arrondis de Calculs method (CESTAC) and the Control of Accuracy and Debugging for Numerical Applications (CADNA) library are used to control the rounding error estimation. We also take advantage of the discrete stochastic arithmetic (DSA) to find the optimal iteration, optimal error and optimal approximation of the HPM. The comparative graphs between exact and approximate solutions show the accuracy and efficiency of the method.

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