Open quadrature rule nonlinear fuzzy Fredholm integral equations with error estimation

Abstract
Introduction

In the field of fuzzy mathematics, fuzzy integral equations play an essential role in solving problems related to fuzzy control and fuzzy modeling in economic. Fuzzy integral equations was first explored in the reformulating the initial value problem for first-order fuzzy differential equations as the fuzzy Volterra integral equation. Investigating the existence, uniqueness and boundedness of the solution related to fuzzy integral equations are necessary. The Banach fixed point theorem is a powerful tool to investigate the existence and uniqueness of the solution of fuzzy integral equations. In many cases, it is complicated to find an exact solution to the fuzzy integral equation and hence, various numerical methods for solving such equations are constructed. Many researchers introduced analytical and approximation methods to solve fuzzy integral equations. However, in practice, numerical methods are typically used. For fuzzy integral equations, the numerical methods based on successive approximations and quadrature formulas are considered. Additionally, many researchers have constructed numerical iterative methods for solving fuzzy integral equations using successive approximations and appropriate quadrature formulas. Other numerical methods have been used for fuzzy integral equations based on quadrature rules and Nystruom methods, iterative procedures and Lagrange interpolation, iterative schemes and divided and finite differences, Bernstein polynomials, fuzzy Haar wavelets, iterative schemes and block pulse functions, iterative processes and fuzzy triangular functions. Also, analytic numeric methods such as Adomian decomposition, homotopy analysis and homotopy perturbation are used for solving fuzzy integral equations. In this paper we apply an open fuzzy cubature formula to construct an iterative numerical method for solving nonlinear fuzzy Fredholm integral equations. The convergence of the method is investigated by expressing error estimate and the method is tested on two numerical experiments. In this paper, we develop operational matrix method based on Euler polynomials to solve weakly singular Ito-Volterra integral equations. Euler polynomials have received considerable attention in dealing with various problems and equations.





Material and methods

In this article, first we propose an open fuzzy quadrature formula for Lipschitz type fuzzy-number-valued functions and its error estimate is provided in terms of the Lipschitz constants and then it is obtained an iterative numerical method for nonlinear fuzzy Fredholm integral equations which combines the successive approximations technique with the open fuzzy quadrature rule.



Results and discussion

We test the method on two examples verifying our theoretical results and demonstrating accuracy and efficiency of the proposed method, then compare the proposed method with the successive approximations iterative method based on trapezoidal quadrature formula. The numerical results exhibited good agreement between approximate solution and exact solution. Also, the numerical results expressed in the tables show that the accuracy improve by increasing the number points (n) and number of iterations (m). By comparing the results in Tables, we find that the proposed iterative method gives more accurate results than the iterative method based on trapezoidal quadrature rule.



Conclusion

The following conclusions were inferred from this study.

An open fuzzy quadrature formula is constructed for Lipschitz type fuzzy-number-valued

functions and its error estimate is obtained in terms of the Lipschitz constants.


As application it is proposed an iterative numerical method for nonlinear fuzzy Fredholm

integral equations which combines the technique of successive approximations with the open

fuzzy quadrature rule.


The convergence of the method is investigated by providing the error estimate to consider the

Lipschitz condition.


The method was tested on two numerical experiments and the numerical results confirm the

convergence of the method and also the obtained results are better than those obtained

using the iterative method based on fuzzy quadrature rule.
Keywords
Iterative method
Open fuzzy quadrature rule
Fuzzy nonlinear Fredholm integral equations
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Mathematical Research
Volume 7, Issue 4 - Serial Number 19
Spring 2021
Pages 781-796