Quadrature iterative method for numerical solution of two-dimensional linear fuzzy Fredholm integral equations

Abstract

In this paper, first, we propose an iterative method based on quadrature formula for solving two-dimensional linear fuzzy Fredholm integral equations (2DLFFIE). Then, we prove the error estimation of the method. In addition, we show the numerical stability analysis of the method with respect to the choice of the first iteration. Finally, supporting examples are also provided. Keywords Fuzzy-number-valued functions Á Numerical method Á Two-dimensional linear fuzzy Fredholm integral equations Á Quadrature iterative method 1. u is normal, i.e. 9x 0 2 R; uðx 0 Þ ¼ 1. 2. uðgx þ ð1 À gÞyÞ ! minfuðxÞ; uðyÞg8x; y 2 R; 8g 2 ½0; 1 (u is called a convex fuzzy subset). 3. u is upper semicontinuous on R, i.e., 8x 0 2 R and 8 [ 0, 9 neighborhood Vðx 0 Þ : uðxÞ uðx 0 Þ þ ; 8x 2 Vðx 0 Þ.

Numerical Solution of Two-Dimensional Linear Fuzzy Fredholm Integral Equations by the Fuzzy Lagrange Interpolation

“…Also, two-dimensional fuzzy integral equations have been noticed by a lot of researchers because of their broad applications in engineering sciences. Some of the most important papers in this area are trapezoidal quadrature rule and iterative method [10][11][12], triangular functions [13], quadrature iterative [14], Bernstein polynomials [15], collocation fuzzy wavelet like operator [16], homotopy analysis method (HAM) [17], open fuzzy cubature rule [18], kernel iterative method [19], modified homotopy pertubation [20], block-pulse functions [21], optimal fuzzy quadrature formula [22], and finally, iterative method and fuzzy bivariate block-pulse functions [23]. Also, some researchers have solved one-dimensional fuzzy Fredholm integral equations by using fuzzy interpolation via iterative method such as: iterative interpolation method [9], Lagrange interpolation based on the extension principle [5], and spline interpolation [7].…”