scholarly journals Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind

2013 ◽  
Vol 67 (1) ◽  
pp. 163-185 ◽  
Author(s):  
Edyta Hetmaniok ◽  
Damian Słota ◽  
Tomasz Trawiński ◽  
Roman Wituła
Keyword(s):  
Integral Equations ◽  
Homotopy Analysis Method ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Linear Integral ◽  
Linear Integral Equations

Convergence and error estimation of homotopy analysis method for some type of nonlinear and linear integral equations

2015 ◽  
Vol 23 (4) ◽  
Author(s):  
Edyta Hetmaniok ◽  
Iwona Nowak ◽  
Damian Słota ◽  
Roman Wituła
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Homotopy Analysis Method ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Linear Integral ◽  
Special Case ◽  
Linear Integral Equations

AbstractIn this paper an application of the homotopy analysis method for some type of nonlinear and linear integral equations of the second kind is presented. A special case of considered equation is the Volterra- Fredholm integral equation. In homotopy analysis method a series is created. It has shown that if the series is convergent, its sum is the solution of the considered equation. It has been also shown that under proper assumptions the considered equation possesses a unique solution and the series obtained in homotopy analysis method is convergent. The error of the approximate solution was estimated. This approximate solution is obtained when we limit to the partial sum of the series.Application of the method is illustrated with examples.

scholarly journals Solving two-dimensional nonlinear fuzzy Volterra integral equations by homotopy analysis method

2021 ◽  
Vol 54 (1) ◽  
pp. 11-24
Author(s):  
Atanaska Georgieva
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Homotopy Analysis Method ◽  
Volterra Integral Equations ◽  
Two Dimensional ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Fuzzy Solution

Abstract The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

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