Collocation Method for Fuzzy Volterra Integral Equations of the Second Kind

Abstract

In this paper we consider fuzzy Volterra integral equation of the second kind whose kernel may change sign. We give conditions for smoothness of the upper and lower functions of the solution. For numerical solution we propose the collocation method with two different basis function sets: triangular and rectangular basis. The smoothness results allow us to obtain the convergence rates of the methods. The proposed methods are illustrated by numerical examples, which confirm the theoretical convergence estimates.

F1-transform in Fuzzy Fredholm Integral Equations

“…)) 2 and R ∞ < 1. Then, for all sufficiently large n > n 0 , the solution Y n of the system (14) exists and it approximates the solution of (2).…”

“…Many researchers from AI community are interested by Integral equations with fuzzy-valued parameters, see [1,2,5,9,10]. For that reason, a fuzzy-valued function is a suitable model.…”

Collocation Method for Fuzzy Volterra Integral Equation of the Second Kind with Weakly Singular Kernels

“…are the Lagrange fundamental polynomials on [0, 1]. For fixed r ∈ [0, 1] we look for approximate solution of equation (3.2) as a spline u N ∈ (S (−1) m−1 (△ h )) 2 . We require that the equation is exactly satisfied at collocation points t n,i .…”

“…We don't know about exact solution. By Theorem 4.9 in [3] the exact solution belongs to (C m,α,p d (0, T ]) 2 . In this case we should use graded meshes with different grading parameters on [0, 1] and on [1,2].…”

“…Hence numerical methods for these equations are needed. The simplest and at the same time robust methods which are all to deal with the nonsmooth solution are collocation methods with piecewise polynomials [2,7].…”