On the Regularization-Homotopy Analysis Method for Linear and Nonlinear Fredholm Integral Equations of the First Kind

The main goal of the paper is to present an approximate method for solving of a two-dimensional nonlinear Volterra-Fredholm fuzzy integral equation (2D-NVFFIE). It is applied the homotopy analysis method (HAM). The studied equation is converted to a nonlinear system of Volterra-Fredholm integral equations in a crisp case. Approximate solutions of this system are obtained by the help with HAM and hence an approximation for the fuzzy solution of the nonlinear Volterra-Fredholm fuzzy integral equation is presented. The convergence of the proposed method is proved and the error estimate between the exact and the approximate solution is obtained. The validity and applicability of the proposed method is illustrated on a numerical example.

Download Full-text

APPLICATION OF THE HOMOTOPY ANALYSIS METHOD FOR SOLVING THE SYSTEMS OF LINEAR AND NONLINEAR INTEGRAL EQUATIONS

Mathematical Modelling and Analysis ◽

10.3846/13926292.2016.1167787 ◽

2016 ◽

Vol 21 (3) ◽

pp. 350-370 ◽

Cited By ~ 3

Author(s):

Rafal Brociek ◽

Edyta Hetmaniok ◽

Jaros law Matlak ◽

Damian Slota

Keyword(s):

Approximate Solution ◽

Integral Equations ◽

Homotopy Analysis Method ◽

System Of Equations ◽

Analysis Method ◽

Nonlinear Integral Equations ◽

Function Series ◽

Partial Sum ◽

Homotopy Analysis ◽

Linear And Nonlinear

In this paper we indicate some applications of homotopy analysis method for solving the systems of linear and nonlinear integral equations. The method is based on the concept of creating function series. If the series converges, its sum is the solution of this system of equations. The paper presents conditions to ensure the convergence of this series and estimation of the error of approximate solution obtained when the partial sum of the series is used. Application of the method will be illustrated by examples.

Download Full-text

Convergence Analysis of Parabolic Basis Functions for Solving Systems of Linear and Nonlinear Fredholm Integral Equations

TURKISH JOURNAL OF MATHEMATICS ◽

10.3906/mat-1507-8 ◽

2017 ◽

Vol 41 ◽

pp. 787-796 ◽

Cited By ~ 3

Author(s):

Yousef JAFARZADEH ◽

Bagher KERAMATI

Keyword(s):

Convergence Analysis ◽

Integral Equations ◽

Basis Functions ◽

Fredholm Integral Equations ◽

Nonlinear Fredholm Integral Equations ◽

Linear And Nonlinear ◽

Parabolic Basis

Download Full-text

Application of the CESTAC Method to Find the Optimal Iteration of the Homotopy Analysis Method for Solving Fuzzy Integral Equations

Advances in Intelligent Systems and Computing - Progress in Intelligent Decision Science ◽

10.1007/978-3-030-66501-2_49 ◽

2021 ◽

pp. 623-637

Author(s):

Samad Noeiaghdam ◽

Mohammad Ali Fariborzi Araghi

Keyword(s):

Integral Equations ◽

Homotopy Analysis Method ◽

Fuzzy Integral ◽

Analysis Method ◽

Cestac Method ◽

Homotopy Analysis

Download Full-text

Numerical Methods for Solving Fredholm Integral Equations of Second Kind

Abstract and Applied Analysis ◽

10.1155/2013/426916 ◽

2013 ◽

Vol 2013 ◽

pp. 1-17 ◽

Cited By ~ 7

Author(s):

S. Saha Ray ◽

P. K. Sahu

Keyword(s):

Integral Equation ◽

Numerical Methods ◽

Integral Equations ◽

Applied Mathematics ◽

Fredholm Integral Equations ◽

Nonlinear Fredholm Integral Equations ◽

Linear And Nonlinear ◽

Interdisciplinary Effort

Integral equation has been one of the essential tools for various areas of applied mathematics. In this paper, we review different numerical methods for solving both linear and nonlinear Fredholm integral equations of second kind. The goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on numerical methods for solving integral equations.

Download Full-text

Numerical Solution of Higher Order Boundary Value Problems

Abstract and Applied Analysis ◽

10.1155/2013/427521 ◽

2013 ◽

Vol 2013 ◽

pp. 1-12 ◽

Cited By ~ 10

Author(s):

Shahid S. Siddiqi ◽

Muzammal Iftikhar

Keyword(s):

Boundary Value Problems ◽

Homotopy Analysis Method ◽

Present Method ◽

Boundary Value ◽

Approximate Solutions ◽

Convergent Series ◽

Higher Order ◽

Analysis Method ◽

Homotopy Analysis ◽

Linear And Nonlinear

The aim of this paper is to use the homotopy analysis method (HAM), an approximating technique for solving linear and nonlinear higher order boundary value problems. Using HAM, approximate solutions of seventh-, eighth-, and tenth-order boundary value problems are developed. This approach provides the solution in terms of a convergent series. Approximate results are given for several examples to illustrate the implementation and accuracy of the method. The results obtained from this method are compared with the exact solutions and other methods (Akram and Rehman (2013), Farajeyan and Maleki (2012), Geng and Li (2009), Golbabai and Javidi (2007), He (2007), Inc and Evans (2004), Lamnii et al. (2008), Siddiqi and Akram (2007), Siddiqi et al. (2012), Siddiqi et al. (2009), Siddiqi and Iftikhar (2013), Siddiqi and Twizell (1996), Siddiqi and Twizell (1998), Torvattanabun and Koonprasert (2010), and Kasi Viswanadham and Raju (2012)) revealing that the present method is more accurate.

Download Full-text