Reliable Numerical Algorithm for Handling Fuzzy Integral Equations of Second Kind in Hilbert Spaces

Abstract

Integral equations under uncertainty are utilized to describe various formulations of physical phenomena in nature. This paper aims to obtain analytical and approximate solutions for mixed integral equations under uncertainty. The scheme presented here is based upon the theory of reproducing kernel for fuzzy real-valued mappings. The solution methodology transforms a linear fuzzy mixed integral equation to crisp linear system of mixed integral equations. Several reproducing kernel spaces are defined to study the convergence and the error estimate in terms of uniform continuity. An iterative procedure based on generating orthonormal sets to solve the model problems that rely on Gram–Schmidt process is also given. Effectiveness of the reproducing kernel method is demonstrated using numerical experiments. The results obtained confirm the feasibility of the method.